Self-Supervised Keyframe Discovery for Horizon-Invariant Behavior Cloning

# Self-Supervised Keyframe Discovery for Horizon-Invariant Behavior Cloning
Source: [https://arxiv.org/html/2610.10857](https://arxiv.org/html/2610.10857)
Omkar PatilAffiliation:Arizona State UniversityEmail:[opatil3@asu\.edu](mailto:)Nakul GopalanAffiliation:Arizona State UniversityEmail:[ng@asu\.edu](mailto:)

###### Abstract

Behavior cloning \(BC\) in non\-Markovian environments is a challenging problem because policies have to reason over contextual information over long horizons\. Existing policy architectures rely on recurrent or attention\-based mechanisms to capture long\-term dependencies\. However, recurrent models suffer from hidden\-state collapse and gradient instability under backpropagation through time, while attention\-based models are fundamentally limited by context length\. To address these issues, we propose Keyframe Mnemonics, a novel self\-supervised method thatdiscoversa set of information\-critical observations \(mnemonics\) by learning an objective from randomly sampled past observations and using it as a reward for keyframe selection\. We then train a BC policy that conditions on the discovered keyframes to model the action distribution\. Under certain task\-structure assumptions, our formulation provides context retention guarantees over an infinite horizon, while maintaining a small set of decision\-relevant keyframes in the policy’s working memory\. We evaluate our method on synthetic memory domains, where mnemonic\-conditioned BC policies achieve100%100\\%success rates \(SR\) and generalize to horizons orders of magnitude beyond training without performance degradation\. Additionally, we evaluate on memory\-intensive robot manipulation benchmark, achieving a 13\.9% average absolute SR improvement over the strongest baseline across 23 tasks and retaining 80% SR at 20×\\timeslonger horizons on a real robot\. Code and videos are available at[https://keyframe\-mnemonics\.github\.io](https://keyframe-mnemonics.github.io/)\.

## 1Introduction

The Markov assumption for decision processes is that the agent’s current state contains sufficient information to determine the optimal action with respect to its goal or objective\[[1](https://arxiv.org/html/2610.10857#bib.bib1)\]\. In practice, however, real\-world environments are non\-Markovian and partially observable, where decision\-making often requires retaining \(“remembering”\) information over time to provide context for future actions\[[2](https://arxiv.org/html/2610.10857#bib.bib2),[3](https://arxiv.org/html/2610.10857#bib.bib3)\]\. Recent advancements in policy learning have increasingly relied on behavior cloning \(BC\)\[[4](https://arxiv.org/html/2610.10857#bib.bib4),[5](https://arxiv.org/html/2610.10857#bib.bib5),[6](https://arxiv.org/html/2610.10857#bib.bib6)\], in which policies are highly dependent on short observation contexts with a working memory spanning only a few seconds into the past\. This makes such BC policies brittle on tasks that require remembering information cues over long horizons\[[7](https://arxiv.org/html/2610.10857#bib.bib7)\]\. Recurrent models have been the architecture of choice for behavior cloning in such memory intensive partially observable environments and are typically trained using Back\-Propagation Through Time \(BPTT\)\[[8](https://arxiv.org/html/2610.10857#bib.bib8),[9](https://arxiv.org/html/2610.10857#bib.bib9)\]\. However, BPTT is difficult to scale, as it requires sequential gradient accumulation over episode timesteps\[[10](https://arxiv.org/html/2610.10857#bib.bib10)\]\. Truncated\-BPTT tries to mitigate this issue but trades off information from distant past observations\[[11](https://arxiv.org/html/2610.10857#bib.bib11)\]\. Moreover, recurrent models suffer from hidden\-state collapse due to repeated representation updates over time\[[12](https://arxiv.org/html/2610.10857#bib.bib12),[10](https://arxiv.org/html/2610.10857#bib.bib10)\]\. Attention\-based architectures such as transformers\[[13](https://arxiv.org/html/2610.10857#bib.bib13)\]address many representation learning challenges, but their finite context length limits retention of the full history in partially observable domains\. Furthermore, increasing context through frame stacking incurs quadratic compute complexity\[[13](https://arxiv.org/html/2610.10857#bib.bib13)\], degrades at extreme horizons\[[14](https://arxiv.org/html/2610.10857#bib.bib14)\], and is prone to overfitting spurious correlations with visual observations\[[15](https://arxiv.org/html/2610.10857#bib.bib15)\]\. Many hybrid architectures have been proposed that improve information retention through dedicated memory mechanisms\[[14](https://arxiv.org/html/2610.10857#bib.bib14),[16](https://arxiv.org/html/2610.10857#bib.bib16),[17](https://arxiv.org/html/2610.10857#bib.bib17),[18](https://arxiv.org/html/2610.10857#bib.bib18),[19](https://arxiv.org/html/2610.10857#bib.bib19),[20](https://arxiv.org/html/2610.10857#bib.bib20),[21](https://arxiv.org/html/2610.10857#bib.bib21)\]or by learning recurrent summaries\[[22](https://arxiv.org/html/2610.10857#bib.bib22),[23](https://arxiv.org/html/2610.10857#bib.bib23),[24](https://arxiv.org/html/2610.10857#bib.bib24)\]\. However, these methods remain susceptible to state collapse beyond the horizons encountered during training, often leading to severe performance degradation on tasks that require generalization to longer horizons\.

![Refer to caption](https://arxiv.org/html/2610.10857v1/method3.png)Figure 1:Keyframe Mnemonics three\-stage training for horizon\-invariant behavior cloning\.\(1\) Proxy policy training\.A random binary maskΨk\\Psi\_\{k\}is applied to the observation historyhth\_\{t\}to assemble a memory bufferℬt\\mathcal\{B\}\_\{t\}of size at mostkk, on which the proxyfθ:ℬ→𝒜f\_\{\\theta\}:\\mathcal\{B\}\\\!\\to\\\!\\mathcal\{A\}is trained to predict the expert actionata\_\{t\}under a negative log\-likelihood lossℒ\\mathcal\{L\}\. Combinatorial support of training data \(Lemma 1\) biasesfθf\_\{\\theta\}toward keyframe\-based hypotheses \(Theorem 1, Proposition 1\), so the frozen objectiveJθ∗​\(ℬt\)=ℒ⁡\(fθ∗​\(ℬt\),at\)J\_\{\\theta^\{\\ast\}\}\(\\mathcal\{B\}\_\{t\}\)=\\mathcal\{L\}\(f\_\{\\theta^\{\*\}\}\(\\mathcal\{B\}\_\{t\}\),a\_\{t\}\)scores keyframe\-rich buffers favourably \(Corollary 1; Section[2\.1](https://arxiv.org/html/2610.10857#S2.SS1)\)\.\(2\) Selector policy training\.The selectorπs\\pi\_\{\\mathrm\{s\}\}is trained using PPO to populateℬt\\mathcal\{B\}\_\{t\}with keyframes usingJθ∗​\(ℬt\)J\_\{\\theta^\{\\ast\}\}\(\\mathcal\{B\}\_\{t\}\)as reward function \(Section[2\.2](https://arxiv.org/html/2610.10857#S2.SS2)\)\.\(3\) Behavior cloning\.The trained selector is used to construct a keyframe augmented dataset𝒟†\\mathcal\{D\}^\{\\dagger\}, which is then used to train the BC policyπBC​\(at∣ot,ℬt\)\\pi\_\{\\mathrm\{BC\}\}\(a\_\{t\}\\mid o\_\{t\},\\mathcal\{B\}\_\{t\}\)\(Section[2\.3](https://arxiv.org/html/2610.10857#S2.SS3)\)\. Throughout the figure, blue tiles denote the current observationoto\_\{t\}, orange tiles denote past frames inhth\_\{t\}, hatched tiles are frames not selected intoℬt\\mathcal\{B\}\_\{t\}\.To address these challenges in behavior cloning, we proposeKeyframe Mnemonics \(KM\), a policy learning method that decomposes the original non\-Markovian decision\-making problem into two subproblems: \(i\) discovering observations \(keyframes\) that contain information relevant to the action distribution, and \(ii\) training a BC policy on demonstrations augmented with the identified keyframes\. Our keyframe discovery process is self\-supervised and leverages the intrinsic simplicity bias of neural networks\[[25](https://arxiv.org/html/2610.10857#bib.bib25),[26](https://arxiv.org/html/2610.10857#bib.bib26),[27](https://arxiv.org/html/2610.10857#bib.bib27)\]to train a reward function that assigns high rewards to information\-rich keyframes in the policy’s context buffer\. The reward learning process is driven by the mutual information advantage of keyframes with the expert action distribution, which is amplified through a novel training method that uses randomly sampled past observations as training samples for the reward function\. We then train a selector policy using reinforcement learning to optimize the learned reward function and identify a set of decision\-relevant keyframes, which we callMnemonics\(“something that assists memory”\)\. Finally, we train BC policies conditioned on these mnemonics to match the expert action distribution\. The trained policy is deployed together with the selector, which identifies keyframes at runtime and provides context to disambiguate actions under partial observability\. Under certain assumptions, this architectural decomposition enableshorizon\-invariancei\.e\., memory retention beyond the training horizon, allowing BC policies to generalize on longer time scales where existing hybrid memory methods fail\. We extensively evaluate Keyframe Mnemonics on synthetic memory benchmarks \(Add\[[28](https://arxiv.org/html/2610.10857#bib.bib28)\], Scattered Copy\[[22](https://arxiv.org/html/2610.10857#bib.bib22)\], T\-Maze\[[29](https://arxiv.org/html/2610.10857#bib.bib29)\]\), grid worlds \(MiniGrid Memory\[[30](https://arxiv.org/html/2610.10857#bib.bib30)\], LTMB\-Hallway\[[31](https://arxiv.org/html/2610.10857#bib.bib31)\]\), memory\-intensive robot manipulation benchmark MIKASA\-Robo\[[7](https://arxiv.org/html/2610.10857#bib.bib7)\], and additionally on a real robot, demonstrating stable performance without degradation far beyond the training horizon\. Our contributions are:

- •We propose a novel self\-supervised algorithm to discover decision\-relevant keyframes in non\-Markovian expert demonstrations without explicit keyframe annotations \(Figure[1](https://arxiv.org/html/2610.10857#S1.F1), Algorithm[1](https://arxiv.org/html/2610.10857#alg1)\)\.
- •We provide a theoretical intuition of our method under task\-structure assumptions, derive information retention guarantees, and support our hypotheses with empirical evidence\.
- •We demonstrate horizon\-invariant performance of mnemonic\-conditioned BC policies, with MLP policies achieving100%100\\%SR in synthetic and grid memory domains, Transformer policies improving absolute SR by13\.9%13\.9\\%on the MIKASA\-Robo benchmark, and a flow\-matching policy retaining80%80\\%SR at20×20\\timeslonger test horizons on a real robot\.

## 2Methodology

We assume a partially observable Markov decision process \(POMDP\)\[[2](https://arxiv.org/html/2610.10857#bib.bib2)\]ℳ=\(𝒮,𝒜,𝒪,𝒯,Ω,ρ0\)\\mathcal\{M\}=\(\\mathcal\{S\},\\mathcal\{A\},\\mathcal\{O\},\\mathcal\{T\},\\Omega,\\rho\_\{0\}\), whereρ0∈Δ⁡\(𝒮\)\\rho\_\{0\}\\in\\Delta\(\\mathcal\{S\}\)denotes the initial state distribution\. At timestept∈\{1,…,T\}t\\in\\\{1,\\ldots,T\\\}with horizonT∈ℕ\+T\\in\\mathbb\{N\}^\{\+\}, the unobserved state is denoted byst∈𝒮s\_\{t\}\\in\\mathcal\{S\}, the agent receives an observationot∈𝒪o\_\{t\}\\in\\mathcal\{O\}, and takes an actionat∈𝒜a\_\{t\}\\in\\mathcal\{A\}based on a belief over the state,bt=P⁡\(st∣ht\)b\_\{t\}=P\(s\_\{t\}\\mid h\_\{t\}\), whereht=\{o1,…,ot\}h\_\{t\}=\\\{o\_\{1\},\\ldots,o\_\{t\}\\\}denotes the history of observations up to timett\. Here,𝒯⁡\(st\+1∣st,at\)\\mathcal\{T\}\(s\_\{t\+1\}\\mid s\_\{t\},a\_\{t\}\)andΩ⁡\(ot\+1∣st\+1,at\)\\Omega\(o\_\{t\+1\}\\mid s\_\{t\+1\},a\_\{t\}\)denote the transition dynamics and observation function respectively\. We omit the reward functionℛ\\mathcal\{R\}and discount factorγ\\gammafrom the notation, as they are not available in standard behavior cloning\. Additionally, we assume access to an expert demonstration dataset𝒟=\{τi\}i=1N\\mathcal\{D\}=\\\{\\tau\_\{i\}\\\}\_\{i=1\}^\{N\}, where each trajectory is given byτi=\{\(ot,at\)\}t=1Ti\\tau\_\{i\}=\\\{\(o\_\{t\},a\_\{t\}\)\\\}\_\{t=1\}^\{T\_\{i\}\}\. We make the following structural assumptions over this dataset, whereI⁡\(⋅,⋅\)I\(\\cdot;\\cdot\)andH⁡\(⋅\)H\(\\cdot\)denote mutual information and entropy, respectively\.

Assumption 1 \(Approximate belief sufficiency\)\.The trajectory observations admit a partition into two disjoint subsets: a sparse subset𝒳key\\mathcal\{X\}\_\{\\mathrm\{key\}\}of sizeC≪TC\\ll T\(keyframes\) and the remainingT−CT\-Cobservations𝒳bg\\mathcal\{X\}\_\{\\mathrm\{bg\}\}\(background frames\)\. The keyframes𝒳key,t⊂ht\\mathcal\{X\}\_\{\\mathrm\{key\},t\}\\subset h\_\{t\}up to timettcontain observations with invariant task information and causal cues that remain consistent across the demonstration dataset, thereby forming an approximate sufficient statistic for the belief statebtb\_\{t\}\. In contrast, the background subset𝒳bg,t⊂ht\\mathcal\{X\}\_\{\\mathrm\{bg\},t\}\\subset h\_\{t\}contributes minimal information toward the belief state\. This assumption imposes a sparse keyframe structure on observations in the demonstration dataset, which we empirically observe in several memory intensive partially observable domains evaluated in Section[3](https://arxiv.org/html/2610.10857#S3)\.

Assumption 2 \(Action disambiguation by keyframes\)\.Any ambiguity in the action distribution is primarily captured by the keyframes𝒳key\\mathcal\{X\}\_\{\\mathrm\{key\}\}, with the following relationship in the mutual informationI⁡\(at,𝒳key,t\)\>I⁡\(at,𝒳bg,t\)I\(a\_\{t\};\\mathcal\{X\}\_\{\\mathrm\{key\},t\}\)\>I\(a\_\{t\};\\mathcal\{X\}\_\{\\mathrm\{bg\},t\}\)\. That is, keyframes strictly provide more information for disambiguating the action than the background frames\. Section[3](https://arxiv.org/html/2610.10857#S3)provides empirical evidence consistent with this assumption in the evaluated memory\-intensive POMDPs\.

We formally defineMnemonicsas the set of keyframes up to timett, denoted by𝒳key,t⊂ht\\mathcal\{X\}\_\{\\mathrm\{key\},t\}\\subset h\_\{t\}, which provide sufficient information for a behavior cloning policy to generate actions unambiguously\. Our goal is to learn a selector policyπs\\pi\_\{\\mathrm\{s\}\}that identifies keyframes from a continuous stream of observations and adds them to a memory bufferℬt\\mathcal\{B\}\_\{t\}, maximizing its overlap with𝒳key,t\\mathcal\{X\}\_\{\\mathrm\{key\},t\}\. These mnemonics are then used to condition a behavior cloning policyπBC\\pi\_\{\\mathrm\{BC\}\}that models the expert action distribution\.

### 2\.1Leveraging Simplicity\-bias for Keyframe Discovery

We formulate keyframediscoveryas a self\-supervised buffer optimization problem\. Specifically, we define an objectiveJ⁡\(⋅,f\)J\(\\cdot,f\)over a proxy policyffmapping observation frames in the buffer, which collectively represent the belief state, to expert actions,f:ℬ→𝒜f:\\mathcal\{B\}\\rightarrow\\mathcal\{A\}, and optimize the bufferℬ\\mathcal\{B\}subject to a memory budgetkkmuch smaller than the observation history \(k≪Tk\\ll T\)\. ForJ⁡\(⋅,f\)J\(\\cdot,f\)to serve as an effective objective, it must provide a stronger learning signal \(i\.e\., lower objective value\) whenℬ\\mathcal\{B\}contains a higher proportion of keyframes than background frames\. In this section, we introduce a training strategy for the proxy policyffthat increases the complexity of the input–output mapping, thereby exploiting the simplicity bias of deep neural networks to encourage learning a hypothesis that favors keyframes over background frames due to their greater informational content\. The resulting biased model can then be used to learn a selectorπs\\pi\_\{\\mathrm\{s\}\}for keyframe discovery\.

Proxy Policy Training:To train the proxy policyfθf\_\{\\theta\}, a neural network parameterized byθ\\theta, we construct input buffers using a random binary maskΨk∈\{0,1\}\|ht\|\\Psi\_\{k\}\\in\\\{0,1\\\}^\{\|h\_\{t\}\|\}that independently selects past observations from the historyhth\_\{t\}, with cardinality bounded by the memory budgetkk\. We call thismasked\-samplingpast observations that constructs the training\-time memory bufferℬt=\{oi∈ht∣Ψk​\(i\)=1\}\\mathcal\{B\}\_\{t\}=\\\{o\_\{i\}\\in h\_\{t\}\\mid\\Psi\_\{k\}\(i\)=1\\\}, which serves as input to the proxy policyfθ:ℬ→𝒜f\_\{\\theta\}:\\mathcal\{B\}\\rightarrow\\mathcal\{A\}with\|ℬt\|≤k\|\\mathcal\{B\}\_\{t\}\|\\leq k, whereℬ\\mathcal\{B\}denotes the space of all possible memory buffers\. Algorithm[1](https://arxiv.org/html/2610.10857#alg1)shows the masked\-sampling method for training the proxy policyfθf\_\{\\theta\}\. Treating\(ℬt,at\)\(\\mathcal\{B\}\_\{t\},a\_\{t\}\)as i\.i\.d\. samples, we now analyze the hypothesis spaceℋ\\mathcal\{H\}explored byfθf\_\{\\theta\}during training through the lens of Kolmogorov complexity\[[32](https://arxiv.org/html/2610.10857#bib.bib32)\], denotedK⁡\(f\)K\(f\)forf∈ℋf\\in\\mathcal\{H\}111For notational simplicity, we use the same variableffto denote both the proxy policy and the hypothesis learned by the proxy policy, and use Kolmogorov complexity as a measure of hypothesis description length\[[33](https://arxiv.org/html/2610.10857#bib.bib33)\]\.\. Letℋ∗⊂ℋ\\mathcal\{H\}^\{\*\}\\subset\\mathcal\{H\}denote the set of interpolating hypotheses achieving zero empirical risk on the training dataset induced by masked\-sampling\. Assumeℋ∗\\mathcal\{H\}^\{\*\}partitions into an invariant classℋin\\mathcal\{H\}\_\{\\mathrm\{in\}\}, corresponding to hypotheses conditioned only on keyframes, and a non\-invariant classℋ¬in\\mathcal\{H\}\_\{\\neg\\mathrm\{in\}\}, which depend on background frames\. Letfkey∈ℋinf\_\{\\mathrm\{key\}\}\\in\\mathcal\{H\}\_\{\\mathrm\{in\}\}denote keyframe\-based andfbg∈ℋ¬inf\_\{\\mathrm\{bg\}\}\\in\\mathcal\{H\}\_\{\\neg\\mathrm\{in\}\}denote background\-based hypotheses defined by mappingsfkey:𝒳key→𝒜f\_\{\\mathrm\{key\}\}:\\mathcal\{X\}\_\{\\mathrm\{key\}\}\\rightarrow\\mathcal\{A\}andfbg:𝒳bg→𝒜f\_\{\\mathrm\{bg\}\}:\\mathcal\{X\}\_\{\\mathrm\{bg\}\}\\rightarrow\\mathcal\{A\}, where the inputs are constrained to at mostkkframes, consistent withfθf\_\{\\theta\}\.

Algorithm 1Masked\-sampling training of the proxy policyfθf\_\{\\theta\}1:dataset

𝒟\\mathcal\{D\}, training memory budget

kk, proxy policy

fθf\_\{\\theta\}
2:foreach training stepdo

3:Sample

τ∼𝒟\\tau\\sim\\mathcal\{D\}and timestep

t∈\{1,…,\|τ\|\}t\\in\\\{1,\\ldots,\|\\tau\|\\\}⊳\\trianglerightuniform sampling of timestep

4:

n←min⁡\(Binomial⁡\(t,0\.5\),k\)n\\leftarrow\\min\\\!\\big\(\\mathrm\{Binomial\}\(t,\\,0\.5\),\\;k\\big\)⊳\\trianglerightbuffer occupancy under budgetkk

5:Draw

ℐ⊂\{1,…,t\}\\mathcal\{I\}\\subset\\\{1,\\ldots,t\\\}with

\|ℐ\|=n\|\\mathcal\{I\}\|=n⊳\\trianglerightuniform sampling without replacement

6:

Ψk\(i\)←𝟏\[i∈ℐ\]\\Psi\_\{k\}\(i\)\\leftarrow\\mathbf\{1\}\[\\,i\\in\\mathcal\{I\}\\,\]for

i=1,…,ti=1,\\ldots,t⊳\\trianglerightinduced binary mask onhth\_\{t\}

7:

ℬt←\{oi∈ht∣Ψk​\(i\)=1\}\\mathcal\{B\}\_\{t\}\\leftarrow\\\{\\,o\_\{i\}\\in h\_\{t\}\\mid\\Psi\_\{k\}\(i\)=1\\,\\\}ordered by

ii⊳\\triangleright\|ℬt\|≤k\|\\mathcal\{B\}\_\{t\}\|\\leq k

8:

a^t←fθ​\(ℬt\)\\hat\{a\}\_\{t\}\\leftarrow f\_\{\\theta\}\(\\mathcal\{B\}\_\{t\}\)⊳\\trianglerightpredict expert action

9:

θ←θ−η​∇θℒ​\(a^t,at\)\\theta\\leftarrow\\theta\-\\eta\\,\\nabla\_\{\\theta\}\\,\\mathcal\{L\}\(\\hat\{a\}\_\{t\},\\,a\_\{t\}\)⊳\\trianglerightupdate proxy using NLL loss

10:endfor

Overview:First, we show that masked\-sampling of past observations during training offθf\_\{\\theta\}induces combinatorially large support of training pairs \(Lemma 1\), which artificially increases the complexity of background\-based hypothesesfbgf\_\{\\mathrm\{bg\}\}and effectively restricts the feasible hypothesis space \(Theorem 1\)\. Assuming that deep neural networks exhibit an implicit bias toward simpler hypotheses, we show that keyframe\-based hypothesesfkeyf\_\{\\mathrm\{key\}\}are learned with high probability \(Proposition 1\)\. Finally, we demonstrate that the prediction error of the trained proxy policy provides an optimizable signal to identify keyframe observations \(Corollary 1\)\.

Lemma 1 \(Combinatorial cardinality of the sample space\)\.*The total cardinalityMMof the space of distinct timestep\-indexed training pairs\(ℬt,at\)\(\\mathcal\{B\}\_\{t\},a\_\{t\}\)is given by:*

M=∑i=1N\(Ti\+∑j=1k\(Ti\+1j\+1\)\)\.M=\\sum\_\{i=1\}^\{N\}\\left\(T\_\{i\}\+\\sum\_\{j=1\}^\{k\}\\binom\{T\_\{i\}\+1\}\{j\+1\}\\right\)\.\(1\)
Masked\-sampling induces a combinatorially large finite support of training pairs, whose cardinalityMMadmits a closed\-form expression via the Hockey\-stick identity\[[34](https://arxiv.org/html/2610.10857#bib.bib34)\]\. \(Proof in Appendix[A](https://arxiv.org/html/2610.10857#A1)\)\.□\\square

Assumption 3 \(Realizability of conditional programs by interpolating hypotheses\)\.ForZ∈\{key,bg\}Z\\in\\\{\\mathrm\{key\},\\mathrm\{bg\}\\\}, letfZ∗=argminf∈ℋZ:f⁡\(𝒳Z,ℓ\)=Aℓ​∀ℓK\(f\)f\_\{Z\}^\{\*\}=\\arg\\min\_\{f\\in\\mathcal\{H\}\_\{Z\}:\\,f\(\\mathcal\{X\}\_\{Z,\\ell\}\)=A\_\{\\ell\}\\ \\forall\\ell\}K\(f\)denote the minimum\-complexity interpolating hypothesis using input sourceZZ\. We assume that the shortest conditional program mapping the training input sequence𝒳Z,1:M\\mathcal\{X\}\_\{Z,1:M\}to the action sequenceA1:MA\_\{1:M\}is realizable by the hypothesis classℋ\\mathcal\{H\}up to an additive constant:\|K\(fZ∗\)−K\(A1:M∣𝒳Z,1:M\)\|≤cZ,\\left\|K\(f\_\{Z\}^\{\*\}\)\-K\(A\_\{1:M\}\\mid\\mathcal\{X\}\_\{Z,1:M\}\)\\right\|\\leq c\_\{Z\},wherecZc\_\{Z\}is an overhead constant independent of the sample sizeMM\. This assumption combines the standard interpretation of conditional Kolmogorov complexity as the shortest program mapping inputs to outputs\[[35](https://arxiv.org/html/2610.10857#bib.bib35),[32](https://arxiv.org/html/2610.10857#bib.bib32)\]with the finite\-sample expressivity of overparameterized neural networks, which can interpolate or memorize arbitrary training labels\[[36](https://arxiv.org/html/2610.10857#bib.bib36),[37](https://arxiv.org/html/2610.10857#bib.bib37)\]\.

Theorem 1 \(Expected gap in hypothesis complexity\)\.*Under the assumption that training pairs are sampled i\.i\.d\. from a sufficiently large support of sizeMM\(Lemma 1\), the expected Kolmogorov complexity of the minimum\-complexity background\-based hypothesisfbg∗f^\{\*\}\_\{\\mathrm\{bg\}\}is strictly greater than that of the keyframe\-based hypothesisfkey∗f^\{\*\}\_\{\\mathrm\{key\}\}\.*

𝔼⁡\[K⁡\(fbg∗\)\]\>𝔼⁡\[K⁡\(fkey∗\)\]\.\\mathbb\{E\}\[K\(f\_\{\\mathrm\{bg\}\}^\{\*\}\)\]\>\\mathbb\{E\}\[K\(f\_\{\\mathrm\{key\}\}^\{\*\}\)\]\.\(2\)Using results from\[[35](https://arxiv.org/html/2610.10857#bib.bib35),[32](https://arxiv.org/html/2610.10857#bib.bib32)\]\(Assumption 3\), we show that, under the stated structural assumptions, the complexity gap between background\-based and keyframe\-based hypotheses grows linearly with the masking\-induced support sizeMMaccording toM⁡\(I⁡\(A,𝒳key\)−I⁡\(A,𝒳bg\)\)M\\big\(I\(A;\\mathcal\{X\}\_\{\\mathrm\{key\}\}\)\-I\(A;\\mathcal\{X\}\_\{\\mathrm\{bg\}\}\)\\big\)\. Thus, any information advantage of keyframes is amplified by the large support sizeMM, rendering background\-based hypotheses intractable for the proxy policyfθf\_\{\\theta\}\. \(Proof in Appendix[A](https://arxiv.org/html/2610.10857#A1)\)\.□\\square

Proposition 1 \(Conditional simplicity\-bias preference\)\.*Under the simplicity\-bias assumption and the complexity gap established in Theorem 1, Stochastic Gradient Descent \(SGD\) is biased toward the invariant hypothesis classℋin\\mathcal\{H\}\_\{\\mathrm\{in\}\}\. Given a masking\-induced sample distribution with support sizeMM\(Lemma 1\), the probability of selecting an invariant hypothesis is approximated by*

PSGD​\(ℋin∣𝒟\)≈11\+2−\[M⁡\(I⁡\(A,𝒳key\)−I⁡\(A,𝒳bg\)\)−𝒪⁡\(1\)\]\.P\_\{\\mathrm\{SGD\}\}\(\\mathcal\{H\}\_\{\\mathrm\{in\}\}\\mid\\mathcal\{D\}\)\\approx\\frac\{1\}\{1\+2^\{\-\\left\[M\\big\(I\(A;\\mathcal\{X\}\_\{\\mathrm\{key\}\}\)\-I\(A;\\mathcal\{X\}\_\{\\mathrm\{bg\}\}\)\\big\)\-\\mathcal\{O\}\(1\)\\right\]\}\}\.\(3\)
Neural networks and SGD are biased toward Kolmogorov\-simple functions\[[26](https://arxiv.org/html/2610.10857#bib.bib26),[38](https://arxiv.org/html/2610.10857#bib.bib38),[27](https://arxiv.org/html/2610.10857#bib.bib27)\]\. Hence, the positive hypothesis complexity gap from Theorem 1 drivesPSGD​\(ℋin∣𝒟\)P\_\{\\mathrm\{SGD\}\}\(\\mathcal\{H\}\_\{\\mathrm\{in\}\}\\mid\\mathcal\{D\}\)close to11, biasingfθf\_\{\\theta\}toward learning an invariant keyframe\-based hypothesis\. \(Proof in Appendix[A](https://arxiv.org/html/2610.10857#A1)\)\.□\\square

Corollary 1 \(Keyframe recovery from the frozen loss objective\)\.*Let us define a subset\-selection objective function*

Jθ∗​\(ℬ\)=𝔼⁡\[ℒ⁡\(fθ∗​\(ℬ\),A\)\]J\_\{\\theta^\{\*\}\}\(\\mathcal\{B\}\)=\\mathbb\{E\}\\\!\\left\[\\mathcal\{L\}\(f\_\{\\theta^\{\*\}\}\(\\mathcal\{B\}\),A\)\\right\]\(4\)*over a frozen predictorfθ∗f\_\{\\theta^\{\*\}\}obtained under Proposition 1 after training with negative log\-likelihood lossℒ\\mathcal\{L\}, wherefθ∗f\_\{\\theta^\{\*\}\}represents a near\-optimal invariant hypothesis inℋin\\mathcal\{H\}\_\{\\mathrm\{in\}\}\. Consider candidate buffer subsets𝒞k=\{ℬt⊆ht:\|ℬt\|≤k\}\\mathcal\{C\}\_\{k\}=\\\{\\mathcal\{B\}\_\{t\}\\subseteq h\_\{t\}:\|\\mathcal\{B\}\_\{t\}\|\\leq k\\\}, and letℬopt∈arg⁡minℬ∈𝒞k​Jθ∗​\(ℬ\)\\mathcal\{B\}\_\{\\mathrm\{opt\}\}\\in\\arg\\min\_\{\\mathcal\{B\}\\in\\mathcal\{C\}\_\{k\}\}J\_\{\\theta^\{\*\}\}\(\\mathcal\{B\}\)\. If keyframe subsetsℬ⊂𝒳key\\mathcal\{B\}\\subset\\mathcal\{X\}\_\{\\mathrm\{key\}\}have a positive action\-information margin over all non\-keyframe candidate subsets, and this margin dominates the approximation error of the frozen predictorfθ∗f\_\{\\theta^\{\*\}\}, then the loss objectiveJθ∗​\(ℬ\)J\_\{\\theta^\{\*\}\}\(\\mathcal\{B\}\)recovers the keyframes up to information\-equivalent subsets\. In particular, if𝒳key\\mathcal\{X\}\_\{\\mathrm\{key\}\}is unique under the memory budgetkk, thenℬopt=𝒳key\\mathcal\{B\}\_\{\\mathrm\{opt\}\}=\\mathcal\{X\}\_\{\\mathrm\{key\}\}\.*

OptimizingJθ∗​\(ℬ\)J\_\{\\theta^\{\*\}\}\(\\mathcal\{B\}\)is equivalent to minimizing the information gapI⁡\(A,𝒳key\)−I⁡\(A,ℬ\)I\(A;\\mathcal\{X\}\_\{\\mathrm\{key\}\}\)\-I\(A;\\mathcal\{B\}\)over𝒞k\\mathcal\{C\}\_\{k\}\. Under Assumption 2, this drivesℬ\\mathcal\{B\}toward𝒳key\\mathcal\{X\}\_\{\\mathrm\{key\}\}\. \(Proof in Appendix[A](https://arxiv.org/html/2610.10857#A1)\)\.□\\square

In summary, we assume a structural partition of dataset observations based on a mutual information gap with the action distribution and propose a training procedure to learn an objective function that can be optimized todiscoverinformative keyframes\.This mechanism is analogous to data augmentation along the temporal dimension, where our masking strategy increases the difficulty of learning spurious correlations and biases the model toward invariant temporal mappings\. Theorem 1 provides theoretical intuition for this phenomenon; see Section[3\.1](https://arxiv.org/html/2610.10857#S3.SS1), where we provide empirical evidence across synthetic benchmark domains\.

### 2\.2Selector Optimization for Keyframe Identification

Given the learned objective function, we train a selector policyπs\\pi\_\{\\mathrm\{s\}\}to identify keyframes\. The selector serves two purposes: \(i\) extract keyframes from the demonstration dataset for constructing mnemonics that can be used to train a downstream policyπBC\\pi\_\{\\mathrm\{BC\}\}, and \(ii\) identify keyframes from a continuous stream of observations during inference to populate the memory buffer forπBC\\pi\_\{\\mathrm\{BC\}\}\. However, note that the objectiveJθ∗​\(ℬ\)J\_\{\\theta^\{\*\}\}\(\\mathcal\{B\}\)is defined over buffer inputs rather than individual observations\. Furthermore, the number of possible buffer states is combinatorially large \(Lemma 1\), making exhaustive evaluation non\-trivial\. To address this, we formulate keyframe identification as a deterministic Markov Decision Process \(MDP\) over the demonstration dataset, modeling the bufferℬ\\mathcal\{B\}as a priority queue and using a selector policyπs\\pi\_\{\\mathrm\{s\}\}to assign priority scores that update the buffer\. This reformulates the problem from optimizing over buffer states to learning a selector policy that prioritizes keyframe observations\.

The selection MDP is defined by the tuple⟨𝒪,𝒫,𝒦,R,γ⟩\\langle\\mathcal\{O\},\\mathcal\{P\},\\mathcal\{K\},R,\\gamma\\rangleover offline demonstration trajectoriesτi∈𝒟\\tau\_\{i\}\\in\\mathcal\{D\}, whereτi=\{\(ot,at\)\}t=1Ti\\tau\_\{i\}=\\\{\(o\_\{t\},a\_\{t\}\)\\\}\_\{t=1\}^\{T\_\{i\}\}\. At each timestept∈\{1,…,Ti\}t\\in\\\{1,\\ldots,T\_\{i\}\\\}, the selector policyπs​\(pt∣ot\)\\pi\_\{\\mathrm\{s\}\}\(p\_\{t\}\\mid o\_\{t\}\)receives an observationot∈𝒪o\_\{t\}\\in\\mathcal\{O\}and outputs a priority actionpt∈𝒫p\_\{t\}\\in\\mathcal\{P\}, which is used to update the bufferℬ\\mathcal\{B\}, modeled as a priority queue with memory budgetkk\. The MDP tuple is specified as follows\.State \(𝒪\\mathcal\{O\}\), is the current observationoto\_\{t\}from a trajectoryτi∈𝒟\\tau\_\{i\}\\in\\mathcal\{D\}\. AnAction \(𝒫\\mathcal\{P\}\)is a priority scorept∈\[0,1\]p\_\{t\}\\in\[0,1\]\.Transition \(𝒦\\mathcal\{K\}\): The transition is deterministic, with the next state given by the subsequent observationot\+1o\_\{t\+1\}inτi\\tau\_\{i\}\. The buffer is updated asℬt=ℬt−1​\(ot,pt\)\\mathcal\{B\}\_\{t\}=\\mathcal\{B\}\_\{t\-1\}\(o\_\{t\},p\_\{t\}\)according to an eviction rule that determines whetheroto\_\{t\}is inserted into or rejected from the buffer\. We use aDiscount Factor \(γ\\gamma\)=0\.99=0\.99\. TheReward \(RR\)function depends on the updated bufferℬt\\mathcal\{B\}\_\{t\}and a normalization constantλ∈\(0,1\)\\lambda\\in\(0,1\)\(typically a value close to00\)\.

R⁡\(ℬt\)=Ti⋅eJθ∗j=t/κ∑j=1TieJθ∗j/κ⏟surprise⋅log⁡\(Jθ∗​\(ℬt\)\+λ\)log⁡\(λ\)⏟reward\.R\(\\mathcal\{B\}\_\{t\}\)=\\underbrace\{T\_\{i\}\\cdot\\frac\{e^\{J^\{j=t\}\_\{\\theta^\{\*\}\}/\\kappa\}\}\{\\sum\_\{j=1\}^\{T\_\{i\}\}e^\{J^\{j\}\_\{\\theta^\{\*\}\}/\\kappa\}\}\}\_\{\\text\{surprise\}\}\\cdot\\underbrace\{\\frac\{\\log\\big\(J\_\{\\theta^\{\*\}\}\(\\mathcal\{B\}\_\{t\}\)\+\\lambda\\big\)\}\{\\log\\big\(\\lambda\\big\)\}\}\_\{\\text\{reward\}\}\.\(5\)
We use a normalized logarithmic reward which increases sharply toward11as the objective value approaches00, providing a strong optimization signal even for small changes in the memory buffer\. This formulation encouragesπs\\pi\_\{\\mathrm\{s\}\}to assign higher priorities to observations that minimize the learned objective\. The reward is further scaled by a surprise\[[22](https://arxiv.org/html/2610.10857#bib.bib22)\]coefficient weighted by episode length, which captures action uncertainty\. Intuitively, this assigns higher reward weight to timesteps with larger objective values, which corresponds to higher action entropy under the proxy policy\. Such timesteps are typically decision\-critical, where actions depend strongly on the memory buffer\. Hence, emphasizing them focuses optimization on resolving keyframes where they matter most\.

The surprise coefficients are precomputed for each trajectory using a temperature \(κ\\kappa\)\-scaled softmax over the expected objective values across timesteps\. The expected objective values are computed over a few epochs of masked\-sampled input buffers by evaluating the objectiveJθ∗​\(ℬ\)J\_\{\\theta^\{\*\}\}\(\\mathcal\{B\}\)at each timestep in the demonstration dataset𝒟\\mathcal\{D\}\. We optimizeπs\\pi\_\{\\mathrm\{s\}\}using Proximal Policy Optimization \(PPO\)\[[39](https://arxiv.org/html/2610.10857#bib.bib39)\], which enables stable training while promoting exploration through entropy regularization\. The memory buffer employs different priority strategies during training and inference\. During training, a FIFO queue penalizes the selector policy for assigning low priority to observations that have long\-term influence on actions, whereas during inference, a LIFO queue preserves high\-priority observations, ensuring their retention over long horizons\. See Appendix[B\.1](https://arxiv.org/html/2610.10857#A2.SS1)for algorithms on priority queue eviction rules used during training and inference\.

### 2\.3Training BC Policy with Keyframe Mnemonics

A trained selector policy is used to generate mnemonics, represented by the buffer stateℬt\\mathcal\{B\}\_\{t\}with\|ℬt\|≤k\|\\mathcal\{B\}\_\{t\}\|\\leq k, which approximates the keyframes up to timett, i\.e\.,𝒳key,t⊂ht\\mathcal\{X\}\_\{\\mathrm\{key\},t\}\\subset h\_\{t\}\. We now present a conditional guarantee for context retention in memory buffer assuming an optimal selector, which enables mnemonics\-conditioned BC policies to generalize to an infinite horizon \(Theorem 2\)\.

Theorem 2 \(Infinite\-horizon keyframe retention\)\.*Assume a POMDP in which the expert behavior strictly satisfies Assumption 1 and 2\. Letπs∗\\pi\_\{\\mathrm\{s\}\}^\{\*\}be an optimal selector minimizing the frozen objectiveJθ∗​\(ℬt\)J\_\{\\theta^\{\*\}\}\(\\mathcal\{B\}\_\{t\}\)\(Corollary 1\), and suppose a unique keyframe set𝒳key\\mathcal\{X\}\_\{\\mathrm\{key\}\}exists that has the highest mutual information with the action and fits within the memory budgetkk\. Then, asT→∞T\\to\\infty,πs∗\\pi\_\{\\mathrm\{s\}\}^\{\*\}assigns higher priority to keyframes than to background frames, ensuring that the priority queue retains the keyframe set for the entire horizonTTwithout eviction, i\.e\.,ℬT=𝒳key\\mathcal\{B\}\_\{T\}=\\mathcal\{X\}\_\{\\mathrm\{key\}\}\.*

An optimal selector minimizes the learned objectiveJθ∗​\(ℬt\)J\_\{\\theta^\{\*\}\}\(\\mathcal\{B\}\_\{t\}\)at every time steptt, which corresponds to recovering the keyframe set𝒳key,t\\mathcal\{X\}\_\{\\mathrm\{key\},t\}at eachtt\. \(Proof in Appendix[A](https://arxiv.org/html/2610.10857#A1)\)\.□\\square

Ideally, the memory buffer should retain the full context needed to disambiguate actions at each timesteptt\. However, in complex domains such as robot manipulation, our assumptions may not hold strongly, and actions may have some dependency on the background observation set\. To address this, we additionally include the current observationoto\_\{t\}and construct an augmented dataset𝒟†\\mathcal\{D\}^\{\\dagger\}with trajectoriesτi†=\{\(ℬt,ot,at\)\}t=1Ti\\tau^\{\\dagger\}\_\{i\}=\\\{\(\\mathcal\{B\}\_\{t\},o\_\{t\},a\_\{t\}\)\\\}\_\{t=1\}^\{T\_\{i\}\}, which is used to train the BC policyπBC​\(at∣ot,ℬt\)\\pi\_\{\\mathrm\{BC\}\}\(a\_\{t\}\\mid o\_\{t\},\\mathcal\{B\}\_\{t\}\)\. See Appendix[C](https://arxiv.org/html/2610.10857#A3)and[D](https://arxiv.org/html/2610.10857#A4)for details on policy training\.

## 3Experiments and Result Analysis

In this section, we present experiments validating the effectiveness of the Keyframe Mnemonics \(KM\) method\. We first provide empirical evidence supporting our theoretical analysis of keyframe discovery from Section[2\.1](https://arxiv.org/html/2610.10857#S2.SS1)\. We then evaluate mnemonic\-conditioned BC policies along two central questions:\(1\)How effectively does the selector identify keyframes across diverse memory benchmarks, ranging from synthetic domains to robot manipulation tasks?\(2\)Can mnemonic\-conditioned BC policies generalize beyond the training horizon and exhibit horizon\-invariant behavior cloning?

Figure 2:Empirical evidence for Keyframe Mnemonics\.\(a\) Addingoraclekeyframes to the context buffer of the proxy policy leads to decrease in objective value across three problem domains \(Corollary 1\)\. \(b\) Active memory regret remains high for small masked\-sampling support sizesMMand converges to zero as the support size increases \(Theorem 1\)\. \(c\) Passive memory regret grows withMMas the proxy policy fits the keyframe\-based hypothesis\. Note: Here,M=1\.0×M=1\.0\\timescorresponds to a calibrated value, chosen empirically \(Appendix[C\.4](https://arxiv.org/html/2610.10857#A3.SS4)\), that is sufficient according to Theorem 1 and yields mnemonics for horizon\-invariant BC policies, as shown in Figure[3](https://arxiv.org/html/2610.10857#S3.F3)\.Figure 3:Horizon\-invariance in synthetic and grid domains\.Top row: Add, Scattered\-Copy, T\-Maze\. Bottom row: MiniGrid\-Memory, LTMB\-Hallway\. Each curve shows mean±\\pmstd over 5 seeds\. Band width is one std\-dev\. Vertical dashed line marks the maximum training horizon\. Lower is better for MSE; higher is better for Success Rate\. KM maintains near\-perfect performance across orders of magnitude of horizon scaling\. Most baselines degrade outside the training range\.### 3\.1Synthetic Domain Benchmarks and Convergence Analysis

Domains\.We evaluate on three synthetic memory sequence\-to\-sequence tasks and two grid\-based environments\. These domains satisfy Assumptions 1 and 2, making them suitable for empirical analysis of our method under the stated assumptions\.Add\[[28](https://arxiv.org/html/2610.10857#bib.bib28)\]requires the policy to emit the sum of two marked floating point numbers from a sequence\.Scattered Copy\[[22](https://arxiv.org/html/2610.10857#bib.bib22)\]interleaves a small set ofnnrelevant tokens with noise tokens, and the policy must recover the relevant tokens in same order as observed\. InActive T\-Maze\[[29](https://arxiv.org/html/2610.10857#bib.bib29)\]the agent must remember an oracle observation from the start of a long corridor to make a binary decision at the final junction\.MiniGrid\-Memory\[[30](https://arxiv.org/html/2610.10857#bib.bib30)\]is a similar memory task with a more diverse grid\-based observation space\.LTMB\-Hallway\[[31](https://arxiv.org/html/2610.10857#bib.bib31)\]is a visual memory benchmark where the agent must match a key observation seen at the episode start to one of several targets uniformly placed along a long hallway\. The policy must compare each observed door with the remembered visual cue and select the matching one\. See Appendix[C\.1](https://arxiv.org/html/2610.10857#A3.SS1)for details on ground truth keyframes in each domain\.

Proxy policy objective\.Figure[2](https://arxiv.org/html/2610.10857#S3.F2)\(a\) shows the learned objective valueJθ∗​\(ℬ\)J\_\{\\theta^\{\*\}\}\(\\mathcal\{B\}\)as a function of the number of keyframes in the input buffer\. Across three synthetic tasks, we observe a decrease in the objective value as more keyframes are added to the buffer, supporting Corollary 1\.

Memory regret metrics\.Letℒtπ\\mathcal\{L\}^\{\\pi\}\_\{t\}denote the per\-timestep proxy policy loss when the buffer is populated by a selection policyπ\\pi\. We compare the trained selectorπs\\pi\_\{\\mathrm\{s\}\}against an*oracle*selectorπ⋆\\pi^\{\\star\}that assigns priority11to keyframes and00otherwise, and a*passive*baselineπrnd\\pi\_\{\\mathrm\{rnd\}\}that samples priorities from𝒰⁡\[0,1\]\\mathcal\{U\}\[0,1\]\. For memory retention analysis, we define active memory regret asractive=𝔼⁡\[1T​∑t=1T\(ℒtπs−ℒtπ⋆\)\]r\_\{\\mathrm\{active\}\}=\\mathbb\{E\}\\\!\\left\[\\frac\{1\}\{T\}\\sum\_\{t=1\}^\{T\}\(\\mathcal\{L\}^\{\\pi\_\{\\mathrm\{s\}\}\}\_\{t\}\-\\mathcal\{L\}^\{\\pi^\{\\star\}\}\_\{t\}\)\\right\]and passive memory regret asrpassive=𝔼⁡\[1T​∑t=1T\(ℒtπrnd−ℒtπ⋆\)\]r\_\{\\mathrm\{passive\}\}=\\mathbb\{E\}\\\!\\left\[\\frac\{1\}\{T\}\\sum\_\{t=1\}^\{T\}\(\\mathcal\{L\}^\{\\pi\_\{\\mathrm\{rnd\}\}\}\_\{t\}\-\\mathcal\{L\}^\{\\pi^\{\\star\}\}\_\{t\}\)\\right\]\.rpassiver\_\{\\mathrm\{passive\}\}characterizes the penalty incurred by not actively selecting what to retain, whileractiver\_\{\\mathrm\{active\}\}measures how close the trained selector is to the oracle\. The desired regime for long horizon information retention isrpassive≫ractive≈0r\_\{\\mathrm\{passive\}\}\\gg r\_\{\\mathrm\{active\}\}\\approx 0\. We vary the masked\-sampling support sizeMMacross three synthetic domains and report the memory regret incurred using the learned objective in Figure[2](https://arxiv.org/html/2610.10857#S3.F2)\(b,c\)\. With sufficiently large calibrated support, the active regret drops to zero, while passive regret gradually increases withMM, providing empirical evidence for Theorem 1\.

Figure 4:Results across the 23MIKASA\-Robotasks\.Keyframe Mnemonics improves on the strongest baseline \(ELMUR\) by an average of\+13\.9\+13\.9absolute success rate, outperformingELMURon1717of the2323tasks\. Results for tasksRememberShape3/5/9,RememberColor3/5/9andRememberShapeColorare aggregated, and the full table of results can be found in the Appendix[D](https://arxiv.org/html/2610.10857#A4)\.Baselines and horizon\-invariance experiment\.We compare KM \(mnemonic\-conditioned MLP/Transformer policies\) against seven competitive sequence models, all trained on BC datasets with favorable parameter budgets\. The set comprises LSTM \(BPTT/TBPTT\)\[[28](https://arxiv.org/html/2610.10857#bib.bib28)\], an LSTM trained with backpropagation through time, Decision\-LSTM \(DLSTM\)\[[40](https://arxiv.org/html/2610.10857#bib.bib40)\], Decision Transformer \(DT\)\[[41](https://arxiv.org/html/2610.10857#bib.bib41)\], Decision\-Mamba \(DMamba\) a DT\-like state\-space model\[[42](https://arxiv.org/html/2610.10857#bib.bib42)\], Transformer\-XL \(TrXL\) with relative position encoding and a recurrent memory cache\[[43](https://arxiv.org/html/2610.10857#bib.bib43)\], ELMUR, a long\-horizon mixture\-of\-experts memory transformer\[[14](https://arxiv.org/html/2610.10857#bib.bib14)\], and Random, a stochastic baseline that we include as a noise floor\. For each task domain, we evaluate across a range of horizons that includes both in\-distribution and out\-of\-distribution settings, spanning orders of magnitude beyond training distribution\. Results in Figure[3](https://arxiv.org/html/2610.10857#S3.F3)shows that*KM maintains near\-perfect performance across all horizons on every task, whereas all baseline policies collapse beyond the training range*\. To probe the limits of generalization we also evaluated KM at two further horizons10510^\{5\}and10610^\{6\}\. KM retained*equivalent performance*to its in\-distribution behavior on all three synthetic tasks; see Appendix[E](https://arxiv.org/html/2610.10857#A5), Table[11](https://arxiv.org/html/2610.10857#A5.T11)\.

### 3\.2Robot Manipulation Benchmarks and Real\-world Experiment

Memory benchmark\.We evaluate the performance of our method on robot\-manipulation memory tasks\.MIKASA\-Robo\[[7](https://arxiv.org/html/2610.10857#bib.bib7)\]is a recent benchmark of 32 tabletop manipulation tasks specifically designed to evaluate the object, spatial, sequential, and capacity memory capabilities of policies\. In each task the agent observes critical information at moments when the cues are visible, and must then act on that information after the cues are no longer in the observation\. For example, inRememberColor3, the agent is briefly shown a colored cube as a cue; after the cue is removed from the observation, the robot must reach to and touch the matching cube among three differently\-colored distractors, requiring the policy to retain the cue across the gap between observation and action\.MIKASA\-Robothus evaluates the policy’s ability to develop a relevant context from past observations to inform its action at the current state\. To further test the memory\-retention capabilities of policies, that is, how long the policy can hold the right context for decision\-making, we evaluate the policies onRememberColor3\-Δ​T\\Delta Twith cue\-to\-choice time delaysΔ​T\\Delta Tlarger than any seen during training\.

Baselines\.For memory benchmarking, we compare our method on all 32 tasks of theMIKASA\-Roboagainst a broad set of baselines including sequence models \(Decision Transformer\[[41](https://arxiv.org/html/2610.10857#bib.bib41)\]\), offline RL \(Conservative Q\-Learning\[[44](https://arxiv.org/html/2610.10857#bib.bib44)\]\), Behavior Cloning, Diffusion Policies\[[5](https://arxiv.org/html/2610.10857#bib.bib5)\]and the recent state of the art methodELMUR\[[14](https://arxiv.org/html/2610.10857#bib.bib14)\]\. Here,RATE\[[45](https://arxiv.org/html/2610.10857#bib.bib45)\]andELMUR\[[14](https://arxiv.org/html/2610.10857#bib.bib14)\]serve as strong hybrid memory architecture baselines for comparison on tasks that demand memory capabilities from policies\. For evaluation of memory retention capabilities, we compare our method withELMURandPTP\[[46](https://arxiv.org/html/2610.10857#bib.bib46)\]\.PTPsub\-samples frames from past observations for policy conditioning during training, in comparison to our learned approach of selecting relevant keyframes\.

Experimental setup\.We train Keyframe Mnemonics on the default dataset released byMIKASA\-Robo\. All results are reported as mean and standard deviation over33different seeds with100100evaluations per seed\. For task designed to evaluate memory retention in manipulation tasksRememberColor3\-Δ​T\\Delta T, we train all baselines with the same dataset and aΔ​T=15\\Delta T=15\. During evaluation, we evaluate each baseline fromΔ​T=15\\Delta T=15toΔ​T=150\\Delta T=150at spread of2020timesteps, and then atΔ​T=200,250\\Delta T=200,250\. We refer the reader to the Appendix[D](https://arxiv.org/html/2610.10857#A4)for more details on the training configuration and the hyperparameters\.

Results onMIKASA\-Robo\.Figure[4](https://arxiv.org/html/2610.10857#S3.F4)summarizes the absolute success\-rate improvement of Keyframe Mnemonics over the strongest published baseline,ELMUR, on the2323MIKASA\-Robotasks\. Keyframe Mnemonics outperformsELMURon 21 of the 23 memory\-intensive visual robot manipulation tasks and improves average absolute SR by 13\.9%; see full table of results in Appendix[D](https://arxiv.org/html/2610.10857#A4)\. These results show that our self\-supervised approach to keyframe identification yields policies that effectively leverage past observational cues from the environment to inform their actions\.

\(a\)Memory retention onRememberColor3\-Δ​T\\Delta Tunder cue\-to\-choice delays beyond the training horizon \(marked at 15\)\.ELMURcollapses immediately,PTPdegrades gradually, while Keyframe Mnemonics sustains its training\-time success rate up to 250 steps\.\(b\)Keyframe Mnemonics matches the diffusion\-policy \(DP\)\[[5](https://arxiv.org/html/2610.10857#bib.bib5)\]baseline on threeRobomimictasks \(Lift,Can,Square\) where memory provides no benefit, confirming it does not regress in the no\-memory regime\.
Figure 5:Horizon\-invariance in robot manipulation domain and results on Markovion tasks\.![Refer to caption](https://arxiv.org/html/2610.10857v1/results.png)Figure 6:Horizon\-invariance in real\-world and qualitative keyframes across domains\.From left\-right: Real\-world task SR over 20 trials per method\. Performance ofDP\[[5](https://arxiv.org/html/2610.10857#bib.bib5)\]andPTP\[[46](https://arxiv.org/html/2610.10857#bib.bib46)\]deteriorates while KM retains average80%80\\%SR at20×20\\timeslonger test horizons\. Mnemonics in grid, robot manipulation and real\-world domains\. Frames with orange border shows discovered keyframes tracked by the memory bufferℬt\\mathcal\{B\}\_\{t\}, while the frame with blue border shows the current observation\. Together, they provide sufficient context for BC policies to generate actions unambiguously\. See the webpage[https://keyframe\-mnemonics\.github\.io](https://keyframe-mnemonics.github.io/)for policy rollout videos\.Horizon\-invariance experiment\.Figure[5\(a\)](https://arxiv.org/html/2610.10857#S3.F5.sf1)evaluates how long each policy can hold the relevant cue once it is no longer visible\. All three methods are trained atΔ​T=15\\Delta T=15, but their out\-of\-distribution behavior diverges sharply:ELMURcollapses to near\-zero success byΔ​T=35\\Delta T=35,PTPretains roughly half its training performance throughΔ​T=75\\Delta T=75before degrading, while Keyframe Mnemonics sustains its training\-time success rate of∼70%\\sim\\\!70\\%across the full evaluated range up toΔ​T=250\\Delta T=250\. This highlights the advantages of our learned approach to keyframe identification over sub\-sampling of frames from the past, as performance ofPTPdeteriorates once the cue recedes from its observational context, while Keyframe Mnemonics keeps holding it in the policy’s context\.

No regression onRobomimictasks\.Figure[5\(b\)](https://arxiv.org/html/2610.10857#S3.F5.sf2)shows that Keyframe Mnemonics matches the performance of diffusion\-policy baseline trained without keyframes on three standardRobomimicmanipulation tasks \(Lift,Can,Square\), confirming that the keyframe\-conditioning does not deteriorate performance on Markovian benchmarks\.

Figure 7:Training time\.Real\-world robot experiment\. We evaluate KM on theRememberColor2task, where the robot observes a red or green cube for 3–5 s\. The cue is then removed, and after a delay, two cubes appear 30 cm apart\. The robot must reach for the previously observed color\. We collect 30 VR\-teleoperated demonstrations \(15 per color\) with∼5\{\\sim\}5s delays and evaluate at delays up to20×20\\timeslonger\. Results shown in Figure[6](https://arxiv.org/html/2610.10857#S3.F6)\.

Compute and training time\.Table[7](https://arxiv.org/html/2610.10857#S3.F7)reports training time forRememberColor3\-Δ​T\\Delta Ton a single workstation with an NVIDIA RTX 3090 GPU and 12th\-Gen Intel Core i9 CPU, with KM decomposed by training stages\. KM trains faster thanPTP, which conditions on subsampled frame buffers, whileELMURis the fastest to train\. We also trainedELMURfor∼\\sim10 hours to match KM’s training budget, but it remained at0%0\\%SR forΔ​T\>15\\Delta T\>15and failed to generalize beyond the training horizon even with further training\.

## 4Related Work

Explicit memory modeling\.Behavior cloning policies with dedicated memory have been developed using embedding caching\[[14](https://arxiv.org/html/2610.10857#bib.bib14)\], learned memory compression models\[[22](https://arxiv.org/html/2610.10857#bib.bib22),[23](https://arxiv.org/html/2610.10857#bib.bib23),[17](https://arxiv.org/html/2610.10857#bib.bib17),[16](https://arxiv.org/html/2610.10857#bib.bib16),[20](https://arxiv.org/html/2610.10857#bib.bib20),[21](https://arxiv.org/html/2610.10857#bib.bib21)\], temporal state aggregation\[[18](https://arxiv.org/html/2610.10857#bib.bib18),[46](https://arxiv.org/html/2610.10857#bib.bib46)\], and hybrid combinations of these approaches\[[19](https://arxiv.org/html/2610.10857#bib.bib19),[24](https://arxiv.org/html/2610.10857#bib.bib24)\]\. However, these methods generally construct compressed state representations that can blur information over long horizons\. In contrast, our method preserves information\-rich keyframes that remain unaffected by horizon length\. Other approaches use 2D point tracking\[[47](https://arxiv.org/html/2610.10857#bib.bib47)\]or 3D workspace mapping\[[48](https://arxiv.org/html/2610.10857#bib.bib48)\], but are limited by the performance of the underlying vision foundation model\.

BC with keyframe annotations\.Prior work on keyframe\-conditioned behavior cloning has relied on manual keyframe annotations\[[31](https://arxiv.org/html/2610.10857#bib.bib31),[49](https://arxiv.org/html/2610.10857#bib.bib49),[50](https://arxiv.org/html/2610.10857#bib.bib50),[51](https://arxiv.org/html/2610.10857#bib.bib51)\]or task\-specific prompts for VLM\-based selection\[[52](https://arxiv.org/html/2610.10857#bib.bib52)\]\. However, such approaches require additional supervision derived from oracle reasoning for each task\. In contrast, our keyframe identification method is self\-supervised, grounded in statistical learning principles, and requires no task\-level oracle annotations\. Therefore, we do not compare against methods that require additional data beyond expert demonstrations for policy training\.

Learning to manage policy context\.Closer to our work, RL has been used to learn keyframe selection policies for optimizing context buffers\[[53](https://arxiv.org/html/2610.10857#bib.bib53),[54](https://arxiv.org/html/2610.10857#bib.bib54)\]\. However, these methods have only been studied in simple grid domains and MuJoCo control environments, and their scalability to complex robot manipulation tasks remains unclear\. To the best of our knowledge, Keyframe Mnemonics is the first self\-supervised keyframe identification method for non\-Markovian behavior cloning\.

Please see Appendix[H](https://arxiv.org/html/2610.10857#A8)for an extended related work section\.

## 5Conclusion

We presented a novel method for enabling horizon\-invariant behavior cloning in partially observable memory intensive environments\. Our approach uses a training strategy that samples random past observations to train a reward model, which is then optimized with reinforcement learning to learn a selector policy that discovers decision\-relevant keyframes in a fully self\-supervised manner\. We provided theoretical analysis and intuition for our algorithm, along with empirical evidence supporting the proposed theory\. Furthermore, we evaluated standard behavior cloning policy architectures conditioned on learned mnemonics, achieving state\-of\-the\-art performance across diverse memory\-intensive benchmarks, ranging from synthetic tasks to complex robot manipulation environments\. We hope our work provides a step toward more reliable long\-horizon behavior cloning by enabling policies to retain compact, decision\-relevant context over extended horizons\.

Limitations\.The memory budgetkkis a hyperparameter that is difficult to predetermine, which can lead to multiple information\-equivalent buffer states and limit the number of correct keyframes retained in the buffer\. In such cases, allowing the inference\-time priority queue to retain more observations can provide richer context for the policy\. Each demonstration used in this work must contain at least one keyframe\. This limits the current algorithm in multi\-task or “play demonstration” settings, where keyframes may vary significantly across tasks or may not be present in every trajectory\. We aim to extend the method to support these settings in future work\.

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## Appendix

A[Proofs](https://arxiv.org/html/2610.10857#A1)⋅\\cdot[A](https://arxiv.org/html/2610.10857#A1)

B[Algorithms](https://arxiv.org/html/2610.10857#A2)⋅\\cdot[B](https://arxiv.org/html/2610.10857#A2)

B\.1[Buffer priority queue](https://arxiv.org/html/2610.10857#A2.SS1)⋅\\cdot[B\.1](https://arxiv.org/html/2610.10857#A2.SS1)

B\.2[Proxy policy training](https://arxiv.org/html/2610.10857#A2.SS2)⋅\\cdot[B\.2](https://arxiv.org/html/2610.10857#A2.SS2)

B\.3[Selector RL environment](https://arxiv.org/html/2610.10857#A2.SS3)⋅\\cdot[B\.3](https://arxiv.org/html/2610.10857#A2.SS3)

B\.4[End\-to\-end training and inference](https://arxiv.org/html/2610.10857#A2.SS4)⋅\\cdot[B\.4](https://arxiv.org/html/2610.10857#A2.SS4)

C[Synthetic and Grid Domain Details](https://arxiv.org/html/2610.10857#A3)⋅\\cdot[C](https://arxiv.org/html/2610.10857#A3)

C\.1[Synthetic environments](https://arxiv.org/html/2610.10857#A3.SS1)⋅\\cdot[C\.1](https://arxiv.org/html/2610.10857#A3.SS1)

C\.2[Grid environments](https://arxiv.org/html/2610.10857#A3.SS2)⋅\\cdot[C\.2](https://arxiv.org/html/2610.10857#A3.SS2)

C\.3[Training dataset](https://arxiv.org/html/2610.10857#A3.SS3)⋅\\cdot[C\.3](https://arxiv.org/html/2610.10857#A3.SS3)

C\.4[Training Keyframe Mnemonics policies](https://arxiv.org/html/2610.10857#A3.SS4)⋅\\cdot[C\.4](https://arxiv.org/html/2610.10857#A3.SS4)

C\.5[Baselines and benchmarking details](https://arxiv.org/html/2610.10857#A3.SS5)⋅\\cdot[C\.5](https://arxiv.org/html/2610.10857#A3.SS5)

C\.6[Parameter\-budget comparison](https://arxiv.org/html/2610.10857#A3.SS6)⋅\\cdot[C\.6](https://arxiv.org/html/2610.10857#A3.SS6)

C\.7[Benchmark result analysis](https://arxiv.org/html/2610.10857#A3.SS7)⋅\\cdot[C\.7](https://arxiv.org/html/2610.10857#A3.SS7)

C\.8[Hardware and wall\-clock](https://arxiv.org/html/2610.10857#A3.SS8)⋅\\cdot[C\.8](https://arxiv.org/html/2610.10857#A3.SS8)

D[Robot Manipulation Domain Details](https://arxiv.org/html/2610.10857#A4)⋅\\cdot[D](https://arxiv.org/html/2610.10857#A4)

D\.1[MIKASA\-Roboenvironments](https://arxiv.org/html/2610.10857#A4.SS1)⋅\\cdot[D\.1](https://arxiv.org/html/2610.10857#A4.SS1)

D\.2[Pipeline overview](https://arxiv.org/html/2610.10857#A4.SS2)⋅\\cdot[D\.2](https://arxiv.org/html/2610.10857#A4.SS2)

D\.3[Architectures and parameter counts](https://arxiv.org/html/2610.10857#A4.SS3)⋅\\cdot[D\.3](https://arxiv.org/html/2610.10857#A4.SS3)

D\.4[Optimizer and schedule](https://arxiv.org/html/2610.10857#A4.SS4)⋅\\cdot[D\.4](https://arxiv.org/html/2610.10857#A4.SS4)

D\.5[Stage\-1: Proxy policyfθf\_\{\\theta\}](https://arxiv.org/html/2610.10857#A4.SS5)⋅\\cdot[D\.5](https://arxiv.org/html/2610.10857#A4.SS5)

D\.6[Stage\-2: Selector policyπs\\pi\_\{\\mathrm\{s\}\}](https://arxiv.org/html/2610.10857#A4.SS6)⋅\\cdot[D\.6](https://arxiv.org/html/2610.10857#A4.SS6)

D\.7[Stage\-3: BC policyπBC\\pi\_\{\\mathrm\{BC\}\}](https://arxiv.org/html/2610.10857#A4.SS7)⋅\\cdot[D\.7](https://arxiv.org/html/2610.10857#A4.SS7)

D\.8[Camera\-image data augmentation](https://arxiv.org/html/2610.10857#A4.SS8)⋅\\cdot[D\.8](https://arxiv.org/html/2610.10857#A4.SS8)

D\.9[MIKASA\-Roboall results](https://arxiv.org/html/2610.10857#A4.SS9)⋅\\cdot[D\.9](https://arxiv.org/html/2610.10857#A4.SS9)

D\.10[Hardware and wall\-clock](https://arxiv.org/html/2610.10857#A4.SS10)⋅\\cdot[D\.10](https://arxiv.org/html/2610.10857#A4.SS10)

E[Experimental Details](https://arxiv.org/html/2610.10857#A5)⋅\\cdot[E](https://arxiv.org/html/2610.10857#A5)

E\.3[Reproducibility](https://arxiv.org/html/2610.10857#A5.SS3)⋅\\cdot[E\.3](https://arxiv.org/html/2610.10857#A5.SS3)

F[Ablation Study](https://arxiv.org/html/2610.10857#A6)⋅\\cdot[F](https://arxiv.org/html/2610.10857#A6)

F\.1[Learned selector against a random selector](https://arxiv.org/html/2610.10857#A6.SS1)⋅\\cdot[F\.1](https://arxiv.org/html/2610.10857#A6.SS1)

F\.2[Selector discount factor](https://arxiv.org/html/2610.10857#A6.SS2)⋅\\cdot[F\.2](https://arxiv.org/html/2610.10857#A6.SS2)

F\.3[Surprise coefficient](https://arxiv.org/html/2610.10857#A6.SS3)⋅\\cdot[F\.3](https://arxiv.org/html/2610.10857#A6.SS3)

F\.4[Current observation in proxy training](https://arxiv.org/html/2610.10857#A6.SS4)⋅\\cdot[F\.4](https://arxiv.org/html/2610.10857#A6.SS4)

F\.5[Queue tie\-break rule](https://arxiv.org/html/2610.10857#A6.SS5)⋅\\cdot[F\.5](https://arxiv.org/html/2610.10857#A6.SS5)

F\.6[Memory buffer size](https://arxiv.org/html/2610.10857#A6.SS6)⋅\\cdot[F\.6](https://arxiv.org/html/2610.10857#A6.SS6)

G[Additional Analysis](https://arxiv.org/html/2610.10857#A7)⋅\\cdot[G](https://arxiv.org/html/2610.10857#A7)

G\.1[Interpretability of discovered keyframes](https://arxiv.org/html/2610.10857#A7.SS1)⋅\\cdot[G\.1](https://arxiv.org/html/2610.10857#A7.SS1)

G\.2[Selector priority behavior](https://arxiv.org/html/2610.10857#A7.SS2)⋅\\cdot[G\.2](https://arxiv.org/html/2610.10857#A7.SS2)

G\.3[BC Policy activation map](https://arxiv.org/html/2610.10857#A7.SS3)⋅\\cdot[G\.3](https://arxiv.org/html/2610.10857#A7.SS3)

G\.4[Sensitivity to memory budgetkk](https://arxiv.org/html/2610.10857#A7.SS4)⋅\\cdot[G\.4](https://arxiv.org/html/2610.10857#A7.SS4)

G\.5[PPO reward plots](https://arxiv.org/html/2610.10857#A7.SS5)⋅\\cdot[G\.5](https://arxiv.org/html/2610.10857#A7.SS5)

G\.6[Limitation discussion](https://arxiv.org/html/2610.10857#A7.SS6)⋅\\cdot[G\.6](https://arxiv.org/html/2610.10857#A7.SS6)

H[Extended Related Works](https://arxiv.org/html/2610.10857#A8)⋅\\cdot[H](https://arxiv.org/html/2610.10857#A8)

## Appendix AProofs

Lemma 1 \(Combinatorial cardinality of the sample space\)\.*The total cardinalityMMof the space of distinct timestep\-indexed training pairs\(ℬt,at\)\(\\mathcal\{B\}\_\{t\},a\_\{t\}\)is given by:*

M=∑i=1N\(Ti\+∑j=1k\(Ti\+1j\+1\)\)\.M=\\sum\_\{i=1\}^\{N\}\\left\(T\_\{i\}\+\\sum\_\{j=1\}^\{k\}\\binom\{T\_\{i\}\+1\}\{j\+1\}\\right\)\.
*Proof\.*Assume a stochastic training procedure that samples a trajectoryτi\\tau\_\{i\}, a timestept∈\[1,Ti\]t\\in\[1,T\_\{i\}\], and a maskΨk\\Psi\_\{k\}selecting a subset ofj≤kj\\leq kobservations from the historyhth\_\{t\}without replacement\. The masked\-sampling process generates an observation subset representing the bufferℬ\\mathcal\{B\}by selectingjjobservation frames, wherej∼min⁡\(Binomial⁡\(t,0\.5\),k\)j\\sim\\min\(\\mathrm\{Binomial\}\(t,0\.5\),k\)\. Because this spansj∈\{0,…,min⁡\(t,k\)\}j\\in\\\{0,\\dots,\\min\(t,k\)\\\}, the sample space includes all subsets of size up tokk, including the empty set forj=0j=0\.

For any timesteptt, the number of unique combinations of sizej≥1j\\geq 1is\(tj\)\\binom\{t\}\{j\}evaluating to00ift<jt<j\. Summing these over all timestepsttyields the total combinations for sizejj:

∑t=1Ti\(tj\)=\(Ti\+1j\+1\),\\sum\_\{t=1\}^\{T\_\{i\}\}\\binom\{t\}\{j\}=\\binom\{T\_\{i\}\+1\}\{j\+1\},which simplifies via the combinatorial Hockey\-stick identity\[[34](https://arxiv.org/html/2610.10857#bib.bib34)\]\.

For the case wherej=0j=0, there is exactly\(t0\)=1\\binom\{t\}\{0\}=1unique combination at each timestep\. Summing this across allTiT\_\{i\}timesteps adds exactlyTiT\_\{i\}combinations:

∑t=1Ti\(t0\)=Ti\.\\sum\_\{t=1\}^\{T\_\{i\}\}\\binom\{t\}\{0\}=T\_\{i\}\.Summing the combinations for all possible mask sizesj∈\{0,…,k\}j\\in\\\{0,\\dots,k\\\}gives the sample space cardinality for a single trajectory:

Mi=Ti\+∑j=1k\(Ti\+1j\+1\)\.M\_\{i\}=T\_\{i\}\+\\sum\_\{j=1\}^\{k\}\\binom\{T\_\{i\}\+1\}\{j\+1\}\.Finally, summing across allNNindependent trajectories in𝒟\\mathcal\{D\}yields the total cardinalityMMof the training sample space:

M=∑i=1N\(Ti\+∑j=1k\(Ti\+1j\+1\)\)\.M=\\sum\_\{i=1\}^\{N\}\\left\(T\_\{i\}\+\\sum\_\{j=1\}^\{k\}\\binom\{T\_\{i\}\+1\}\{j\+1\}\\right\)\.□\\square

Theorem 1 \(Expected gap in hypothesis complexity\)\.*Under the assumption that training pairs are sampled i\.i\.d\. from a sufficiently large support of sizeMM\(Lemma 1\), the expected Kolmogorov complexity of the minimum\-complexity background\-based hypothesisfbg∗f^\{\*\}\_\{\\mathrm\{bg\}\}is strictly greater than that of the keyframe\-based hypothesisfkey∗f^\{\*\}\_\{\\mathrm\{key\}\}\.*

𝔼⁡\[K⁡\(fbg∗\)\]\>𝔼⁡\[K⁡\(fkey∗\)\]\.\\mathbb\{E\}\[K\(f\_\{\\mathrm\{bg\}\}^\{\*\}\)\]\>\\mathbb\{E\}\[K\(f\_\{\\mathrm\{key\}\}^\{\*\}\)\]\.
*Proof\.*We first analyze complexity using established equivalence results from algorithmic information theory\[[35](https://arxiv.org/html/2610.10857#bib.bib35)\]\. LetP⁡\(X,Y\)P\(X,Y\)denote a computable joint probability distribution generating data pairs\(X,Y\)\(X,Y\)with marginal entropyH⁡\(Y\)H\(Y\)\.

According to Theorem 10 in[Grünwald et al\. \[35\]](https://arxiv.org/html/2610.10857#bib.bib35), the expected Kolmogorov complexityK⁡\(⋅\)K\(\\cdot\)of a random variable equals its Shannon entropy up to an additive term bounded by the algorithmic complexity of the data\-generating distribution itself:

𝔼⁡\[K⁡\(Y\)\]=H⁡\(Y\)\+O⁡\(K⁡\(P\)\)\.\\mathbb\{E\}\[K\(Y\)\]=H\(Y\)\+O\(K\(P\)\)\.\(6\)
Similarly, Theorem 11 in[Grünwald et al\. \[35\]](https://arxiv.org/html/2610.10857#bib.bib35)establishes that expected algorithmic mutual informationI⁡\(Y:X\)I\(Y:X\)equals the probabilistic Shannon mutual informationI⁡\(Y,X\)I\(Y;X\)up to an additive term bounded by the same algorithmic complexity:

𝔼⁡\[I⁡\(Y:X\)\]=I⁡\(Y,X\)\+O⁡\(K⁡\(P\)\)\.\\mathbb\{E\}\[I\(Y:X\)\]=I\(Y;X\)\+O\(K\(P\)\)\.\(7\)
The algorithmic conditional complexity is defined via algorithmic mutual information \(Equation 20 in[Grünwald et al\. \[35\]](https://arxiv.org/html/2610.10857#bib.bib35)\) asK⁡\(y∣x∗\)=K⁡\(y\)−I⁡\(y:x\)K\(y\\mid x^\{\*\}\)=K\(y\)\-I\(y:x\)\. Taking the expectation over the data distribution and substituting Equations[6](https://arxiv.org/html/2610.10857#A1.E6)and[7](https://arxiv.org/html/2610.10857#A1.E7), we derive expected conditional Kolmogorov complexity in terms of conditional Shannon entropy222For notational simplicity, we suppress the distinction between conditioning onXXand on its shortest programX∗X^\{\*\}, absorbing the resulting additive terms into theO⁡\(⋅\)O\(\\cdot\)overhead\.:

𝔼⁡\[K⁡\(Y\)\]−𝔼⁡\[I⁡\(Y:X\)\]\\displaystyle\\mathbb\{E\}\[K\(Y\)\]\-\\mathbb\{E\}\[I\(Y:X\)\]=H⁡\(Y\)−I⁡\(Y,X\)\+O⁡\(K⁡\(P\)\),\\displaystyle=H\(Y\)\-I\(Y;X\)\+O\(K\(P\)\),\(8\)𝔼⁡\[K⁡\(Y∣X\)\]\\displaystyle\\mathbb\{E\}\[K\(Y\\mid X\)\]=H⁡\(Y∣X\)\+O⁡\(K⁡\(P\)\)\.\\displaystyle=H\(Y\\mid X\)\+O\(K\(P\)\)\.
We now apply this to our setting, whereP=P⁡\(𝒟\)P=P\(\\mathcal\{D\}\)denotes the distribution induced by the POMDP and expert policy that generate dataset𝒟\\mathcal\{D\}, and we takeK⁡\(𝒟\)=K⁡\(P⁡\(𝒟\)\)K\(\\mathcal\{D\}\)=K\(P\(\\mathcal\{D\}\)\)to represent the constant algorithmic overhead of this data\-generating process\. During training,\(𝒳,A\)\(\\mathcal\{X\},A\)pairs are sampled from a finite support of cardinalityMMinduced by the masked\-sampling procedure \(Lemma 1\)333Note that we use𝒳\\mathcal\{X\}to denote the input domain of interpolating hypothesesf∈ℋf\\in\\mathcal\{H\}, whileℬ\\mathcal\{B\}denotes a buffer subset used as input to the neural networkfθf\_\{\\theta\}; in both cases, the input size is bounded bykk\.\. Extending Equation[8](https://arxiv.org/html/2610.10857#A1.E8)to the entire sequence ofMMsamples, the expected conditional Kolmogorov complexity of the sequence of actions given the sequence of sampled frames is bounded by the joint Shannon entropy plus the one\-time algorithmic overhead of the data\-generating distribution:

𝔼\[K\(A1:M∣𝒳1:M\)\]=H\(A1:M∣𝒳1:M\)\+O\(K\(𝒟\)\)\.\\mathbb\{E\}\[K\(A\_\{1:M\}\\mid\\mathcal\{X\}\_\{1:M\}\)\]=H\(A\_\{1:M\}\\mid\\mathcal\{X\}\_\{1:M\}\)\+O\(K\(\\mathcal\{D\}\)\)\.\(9\)
Because theMMsampled training pairs are treated as i\.i\.d\., the joint conditional Shannon entropy scales linearly,H\(A1:M∣𝒳1:M\)=M⋅H\(A∣𝒳\)H\(A\_\{1:M\}\\mid\\mathcal\{X\}\_\{1:M\}\)=M\\cdot H\(A\\mid\\mathcal\{X\}\)\. Substituting this yields:

𝔼\[K\(A1:M∣𝒳1:M\)\]=M⋅H\(A∣𝒳\)\+O\(K\(𝒟\)\)\.\\mathbb\{E\}\[K\(A\_\{1:M\}\\mid\\mathcal\{X\}\_\{1:M\}\)\]=M\\cdot H\(A\\mid\\mathcal\{X\}\)\+O\(K\(\\mathcal\{D\}\)\)\.\(10\)
The conditional Kolmogorov complexityK⁡\(y∣x\)K\(y\\mid x\)is strictly defined as the length of the shortest program on a universal conditional prefix machine that outputsyywhenxxis provided as input\[[35](https://arxiv.org/html/2610.10857#bib.bib35)\]\. Therefore, for any interpolating hypothesisfZf\_\{Z\}satisfyingfZ​\(𝒳Z,ℓ\)=Aℓf\_\{Z\}\(\\mathcal\{X\}\_\{Z,\\ell\}\)=A\_\{\\ell\}for allℓ\\ell, whereZ∈\{key,bg\}Z\\in\\\{\\mathrm\{key\},\\mathrm\{bg\}\\\}, there exists a fixed evaluator program which, given a description offZf\_\{Z\}and the input sequence𝒳Z,1:M\\mathcal\{X\}\_\{Z,1:M\}, perfectly reconstructs the action sequenceA1:MA\_\{1:M\}\. Hence, the conditional sequence complexity is upper\-bounded by the hypothesis complexity:

K\(A1:M∣𝒳Z,1:M\)≤K\(fZ∗\)\+O\(1\)\.K\(A\_\{1:M\}\\mid\\mathcal\{X\}\_\{Z,1:M\}\)\\leq K\(f\_\{Z\}^\{\*\}\)\+O\(1\)\.Assumption 3 states that, for each input sourceZ∈\{key,bg\}Z\\in\\\{\\mathrm\{key\},\\mathrm\{bg\}\\\}, the minimum\-complexity interpolating hypothesisfZ∗f\_\{Z\}^\{\*\}realizes the corresponding shortest conditional program up to an additive constant:

\|K\(fZ∗\)−K\(A1:M∣𝒳Z,1:M\)\|≤cZ\.\\left\|K\(f\_\{Z\}^\{\*\}\)\-K\(A\_\{1:M\}\\mid\\mathcal\{X\}\_\{Z,1:M\}\)\\right\|\\leq c\_\{Z\}\.Equivalently, unpacking the absolute value gives the two\-sided bound

K\(A1:M∣𝒳Z,1:M\)−cZ≤K\(fZ∗\)≤K\(A1:M∣𝒳Z,1:M\)\+cZ\.K\(A\_\{1:M\}\\mid\\mathcal\{X\}\_\{Z,1:M\}\)\-c\_\{Z\}\\leq K\(f\_\{Z\}^\{\*\}\)\\leq K\(A\_\{1:M\}\\mid\\mathcal\{X\}\_\{Z,1:M\}\)\+c\_\{Z\}\.To analyze the complexity gapK⁡\(fbg∗\)−K⁡\(fkey∗\)K\(f\_\{\\mathrm\{bg\}\}^\{\*\}\)\-K\(f\_\{\\mathrm\{key\}\}^\{\*\}\), we derive a conservative lower bound by using a lower bound for the background hypothesis and an upper bound for the keyframe hypothesis\. This yields

K⁡\(fbg∗\)\\displaystyle K\(f\_\{\\mathrm\{bg\}\}^\{\*\}\)≥K\(A1:M∣𝒳bg,1:M\)−cbg,\\displaystyle\\geq K\(A\_\{1:M\}\\mid\\mathcal\{X\}\_\{\\mathrm\{bg\},1:M\}\)\-c\_\{\\mathrm\{bg\}\},K⁡\(fkey∗\)\\displaystyle K\(f\_\{\\mathrm\{key\}\}^\{\*\}\)≤K\(A1:M∣𝒳key,1:M\)\+ckey\.\\displaystyle\\leq K\(A\_\{1:M\}\\mid\\mathcal\{X\}\_\{\\mathrm\{key\},1:M\}\)\+c\_\{\\mathrm\{key\}\}\.
Subtracting the keyframe upper bound from the background lower bound, and taking the expectation over the data\-generating distribution, yields the expected hypothesis complexity gap:

𝔼\[K\(fbg∗\)\]−𝔼\[K\(fkey∗\)\]≥𝔼\[K\(A1:M∣𝒳bg,1:M\)\]−𝔼\[K\(A1:M∣𝒳key,1:M\)\]−\(ckey\+cbg\)\.\\mathbb\{E\}\[K\(f\_\{\\mathrm\{bg\}\}^\{\*\}\)\]\-\\mathbb\{E\}\[K\(f\_\{\\mathrm\{key\}\}^\{\*\}\)\]\\geq\\mathbb\{E\}\[K\(A\_\{1:M\}\\mid\\mathcal\{X\}\_\{\\mathrm\{bg\},1:M\}\)\]\-\\mathbb\{E\}\[K\(A\_\{1:M\}\\mid\\mathcal\{X\}\_\{\\mathrm\{key\},1:M\}\)\]\-\(c\_\{\\mathrm\{key\}\}\+c\_\{\\mathrm\{bg\}\}\)\.Substituting the sequence complexities from Equation[8](https://arxiv.org/html/2610.10857#A1.E8)and[10](https://arxiv.org/html/2610.10857#A1.E10), we obtain:

𝔼⁡\[K⁡\(fbg∗\)\]−𝔼⁡\[K⁡\(fkey∗\)\]≥M⁡\(I⁡\(A,𝒳key\)−I⁡\(A,𝒳bg\)\)−O⁡\(K⁡\(𝒟\)\)−\(ckey\+cbg\)\.\\mathbb\{E\}\[K\(f\_\{\\mathrm\{bg\}\}^\{\*\}\)\]\-\\mathbb\{E\}\[K\(f\_\{\\mathrm\{key\}\}^\{\*\}\)\]\\geq M\\big\(I\(A;\\mathcal\{X\}\_\{\\mathrm\{key\}\}\)\-I\(A;\\mathcal\{X\}\_\{\\mathrm\{bg\}\}\)\\big\)\-O\(K\(\\mathcal\{D\}\)\)\-\(c\_\{\\mathrm\{key\}\}\+c\_\{\\mathrm\{bg\}\}\)\.Under Assumption 2, the mutual information gapI⁡\(A,𝒳key\)−I⁡\(A,𝒳bg\)I\(A;\\mathcal\{X\}\_\{\\mathrm\{key\}\}\)\-I\(A;\\mathcal\{X\}\_\{\\mathrm\{bg\}\}\)is strictly positive\. Because the sample supportMMis combinatorially massive \(Lemma 1\), this linear scaling term strictly dominates the combined constant overheadsO⁡\(K⁡\(𝒟\)\)\+\(ckey\+cbg\)O\(K\(\\mathcal\{D\}\)\)\+\(c\_\{\\mathrm\{key\}\}\+c\_\{\\mathrm\{bg\}\}\)\. Thus, for sufficiently largeMM, we obtain:

𝔼⁡\[K⁡\(fbg∗\)\]\>𝔼⁡\[K⁡\(fkey∗\)\]\.\\mathbb\{E\}\[K\(f\_\{\\mathrm\{bg\}\}^\{\*\}\)\]\>\\mathbb\{E\}\[K\(f\_\{\\mathrm\{key\}\}^\{\*\}\)\]\.□\\square

Assumption 4 \(Simplicity\-biased SGD approximation\)\.Conditioned on reaching an interpolating solution inℋ∗\\mathcal\{H\}^\{\*\}, SGD is approximated as selecting hypotheses with probability biased toward lower Kolmogorov complexity:

PSGD​\(f∣𝒟\)∝2−K⁡\(f\),f∈ℋ∗\.P\_\{\\mathrm\{SGD\}\}\(f\\mid\\mathcal\{D\}\)\\propto 2^\{\-K\(f\)\},\\qquad f\\in\\mathcal\{H\}^\{\*\}\.This approximation follows the simplicity\-bias view of neural networks, where low\-complexity functions receive larger prior mass\[[26](https://arxiv.org/html/2610.10857#bib.bib26),[38](https://arxiv.org/html/2610.10857#bib.bib38),[27](https://arxiv.org/html/2610.10857#bib.bib27)\]\.

Proposition 1 \(Conditional simplicity\-bias preference\)\.*Under the simplicity\-bias assumption and the complexity gap established in Theorem 1, Stochastic Gradient Descent \(SGD\) is biased toward the invariant hypothesis classℋin\\mathcal\{H\}\_\{\\mathrm\{in\}\}\. Given a masking\-induced sample distribution with support sizeMM\(Lemma 1\), the probability of selecting an invariant hypothesis is approximated by*

PSGD​\(ℋin∣𝒟\)≈11\+2−\[M⁡\(I⁡\(A,𝒳key\)−I⁡\(A,𝒳bg\)\)−𝒪⁡\(1\)\]\.P\_\{\\mathrm\{SGD\}\}\(\\mathcal\{H\}\_\{\\mathrm\{in\}\}\\mid\\mathcal\{D\}\)\\approx\\frac\{1\}\{1\+2^\{\-\\left\[M\\big\(I\(A;\\mathcal\{X\}\_\{\\mathrm\{key\}\}\)\-I\(A;\\mathcal\{X\}\_\{\\mathrm\{bg\}\}\)\\big\)\-\\mathcal\{O\}\(1\)\\right\]\}\}\.*Proof\.*Under the Empirical Risk Minimization \(ERM\) framework\[[55](https://arxiv.org/html/2610.10857#bib.bib55)\], an overparameterized neural network can realize hypotheses in the interpolating setℋ∗\\mathcal\{H\}^\{\*\}that achieve zero empirical risk on𝒟\\mathcal\{D\}\. To analyze which interpolating hypothesis is selected by SGD, we use the simplicity\-bias approximation in Assumption 4, motivated by evidence that neural network function priors and SGD solutions are biased toward low\-complexity functions\[[26](https://arxiv.org/html/2610.10857#bib.bib26),[38](https://arxiv.org/html/2610.10857#bib.bib38),[27](https://arxiv.org/html/2610.10857#bib.bib27)\]\. The probability of selecting an invariant hypothesis is obtained by marginalizing over the invariant class:

PSGD​\(ℋin∣𝒟\)≈∑f∈ℋin2−K⁡\(f\)∑f∈ℋ∗2−K⁡\(f\)\.P\_\{\\mathrm\{SGD\}\}\(\\mathcal\{H\}\_\{\\mathrm\{in\}\}\\mid\\mathcal\{D\}\)\\approx\\frac\{\\sum\_\{f\\in\\mathcal\{H\}\_\{\\mathrm\{in\}\}\}2^\{\-K\(f\)\}\}\{\\sum\_\{f\\in\\mathcal\{H\}^\{\*\}\}2^\{\-K\(f\)\}\}\.Sinceℋ∗\\mathcal\{H\}^\{\*\}is partitioned into the invariant class and the non\-invariant class,

ℋ∗=ℋin∪ℋ¬in,\\mathcal\{H\}^\{\*\}=\\mathcal\{H\}\_\{\\mathrm\{in\}\}\\cup\\mathcal\{H\}\_\{\\neg\\mathrm\{in\}\},we can rewrite this as

PSGD​\(ℋin∣𝒟\)≈∑f∈ℋin2−K⁡\(f\)∑f∈ℋin2−K⁡\(f\)\+∑f∈ℋ¬in2−K⁡\(f\)\.P\_\{\\mathrm\{SGD\}\}\(\\mathcal\{H\}\_\{\\mathrm\{in\}\}\\mid\\mathcal\{D\}\)\\approx\\frac\{\\sum\_\{f\\in\\mathcal\{H\}\_\{\\mathrm\{in\}\}\}2^\{\-K\(f\)\}\}\{\\sum\_\{f\\in\\mathcal\{H\}\_\{\\mathrm\{in\}\}\}2^\{\-K\(f\)\}\+\\sum\_\{f\\in\\mathcal\{H\}\_\{\\neg\\mathrm\{in\}\}\}2^\{\-K\(f\)\}\}\.
Because the probability mass decays exponentially with complexity, we use the standard minimum\-description approximation that each class\-level mass is dominated, up to constant factors independent ofMM, by its minimum\-complexity interpolating hypothesis\. Let

fkey∗=arg⁡minf∈ℋin⁡K⁡\(f\),fbg∗=arg⁡minf∈ℋ¬in⁡K⁡\(f\)\.f\_\{\\mathrm\{key\}\}^\{\*\}=\\arg\\min\_\{f\\in\\mathcal\{H\}\_\{\\mathrm\{in\}\}\}K\(f\),\\qquad f\_\{\\mathrm\{bg\}\}^\{\*\}=\\arg\\min\_\{f\\in\\mathcal\{H\}\_\{\\neg\\mathrm\{in\}\}\}K\(f\)\.Then

∑f∈ℋin2−K⁡\(f\)≈2−K⁡\(fkey∗\),∑f∈ℋ¬in2−K⁡\(f\)≈2−K⁡\(fbg∗\)\.\\sum\_\{f\\in\\mathcal\{H\}\_\{\\mathrm\{in\}\}\}2^\{\-K\(f\)\}\\approx 2^\{\-K\(f\_\{\\mathrm\{key\}\}^\{\*\}\)\},\\qquad\\sum\_\{f\\in\\mathcal\{H\}\_\{\\neg\\mathrm\{in\}\}\}2^\{\-K\(f\)\}\\approx 2^\{\-K\(f\_\{\\mathrm\{bg\}\}^\{\*\}\)\}\.Substituting this approximation gives

PSGD​\(ℋin∣𝒟\)≈2−K⁡\(fkey∗\)2−K⁡\(fkey∗\)\+2−K⁡\(fbg∗\)\.P\_\{\\mathrm\{SGD\}\}\(\\mathcal\{H\}\_\{\\mathrm\{in\}\}\\mid\\mathcal\{D\}\)\\approx\\frac\{2^\{\-K\(f\_\{\\mathrm\{key\}\}^\{\*\}\)\}\}\{2^\{\-K\(f\_\{\\mathrm\{key\}\}^\{\*\}\)\}\+2^\{\-K\(f\_\{\\mathrm\{bg\}\}^\{\*\}\)\}\}\.Dividing numerator and denominator by2−K⁡\(fkey∗\)2^\{\-K\(f\_\{\\mathrm\{key\}\}^\{\*\}\)\}gives

PSGD​\(ℋin∣𝒟\)≈11\+2−\(K⁡\(fbg∗\)−K⁡\(fkey∗\)\)\.P\_\{\\mathrm\{SGD\}\}\(\\mathcal\{H\}\_\{\\mathrm\{in\}\}\\mid\\mathcal\{D\}\)\\approx\\frac\{1\}\{1\+2^\{\-\\left\(K\(f\_\{\\mathrm\{bg\}\}^\{\*\}\)\-K\(f\_\{\\mathrm\{key\}\}^\{\*\}\)\\right\)\}\}\.
By Theorem 1, the expected complexity gap between the non\-invariant/background\-based hypothesis and the invariant/keyframe\-based hypothesis satisfies

𝔼⁡\[K⁡\(fbg∗\)\]−𝔼⁡\[K⁡\(fkey∗\)\]≥M⁡\(I⁡\(A,𝒳key\)−I⁡\(A,𝒳bg\)\)−𝒪⁡\(1\)\.\\mathbb\{E\}\[K\(f\_\{\\mathrm\{bg\}\}^\{\*\}\)\]\-\\mathbb\{E\}\[K\(f\_\{\\mathrm\{key\}\}^\{\*\}\)\]\\geq M\\big\(I\(A;\\mathcal\{X\}\_\{\\mathrm\{key\}\}\)\-I\(A;\\mathcal\{X\}\_\{\\mathrm\{bg\}\}\)\\big\)\-\\mathcal\{O\}\(1\)\.Here, the constant overheadO⁡\(K⁡\(𝒟\)\)\+ckey\+cbgO\(K\(\\mathcal\{D\}\)\)\+c\_\{\\mathrm\{key\}\}\+c\_\{\\mathrm\{bg\}\}is absorbed into the𝒪⁡\(1\)\\mathcal\{O\}\(1\)term\. Therefore, for a typical dataset sampled from the masking\-induced distribution, we use the corresponding expected gap approximation

K⁡\(fbg∗\)−K⁡\(fkey∗\)≈M⁡\(I⁡\(A,𝒳key\)−I⁡\(A,𝒳bg\)\)−𝒪⁡\(1\)\.K\(f\_\{\\mathrm\{bg\}\}^\{\*\}\)\-K\(f\_\{\\mathrm\{key\}\}^\{\*\}\)\\approx M\\big\(I\(A;\\mathcal\{X\}\_\{\\mathrm\{key\}\}\)\-I\(A;\\mathcal\{X\}\_\{\\mathrm\{bg\}\}\)\\big\)\-\\mathcal\{O\}\(1\)\.Substituting this into the class\-probability expression yields

PSGD​\(ℋin∣𝒟\)≈11\+2−\[M⁡\(I⁡\(A,𝒳key\)−I⁡\(A,𝒳bg\)\)−𝒪⁡\(1\)\]\.P\_\{\\mathrm\{SGD\}\}\(\\mathcal\{H\}\_\{\\mathrm\{in\}\}\\mid\\mathcal\{D\}\)\\approx\\frac\{1\}\{1\+2^\{\-\\left\[M\\big\(I\(A;\\mathcal\{X\}\_\{\\mathrm\{key\}\}\)\-I\(A;\\mathcal\{X\}\_\{\\mathrm\{bg\}\}\)\\big\)\-\\mathcal\{O\}\(1\)\\right\]\}\}\.Thus, when the keyframe information advantage is positive and the assumptions hold, the simplicity\-bias approximation predicts a rapid increase in the probability of selecting an invariant hypothesis\. This follows because the exponent scales linearly with the combinatorially large support sizeMM\.444If the assumptions fail, for example, when a lower\-complexity non\-invariant shortcut exists, this preference need not hold\.□\\square

Corollary 1 \(Keyframe recovery from the frozen loss objective\)\.*Let us define a subset\-selection objective function*

Jθ∗​\(ℬ\)=𝔼⁡\[ℒ⁡\(fθ∗​\(ℬ\),A\)\]J\_\{\\theta^\{\*\}\}\(\\mathcal\{B\}\)=\\mathbb\{E\}\\\!\\left\[\\mathcal\{L\}\(f\_\{\\theta^\{\*\}\}\(\\mathcal\{B\}\),A\)\\right\]*over a frozen predictorfθ∗f\_\{\\theta^\{\*\}\}obtained under Proposition 1 after training with negative log\-likelihood lossℒ\\mathcal\{L\}, wherefθ∗f\_\{\\theta^\{\*\}\}represents a near\-optimal invariant hypothesis inℋin\\mathcal\{H\}\_\{\\mathrm\{in\}\}\. Consider candidate buffer subsets𝒞k=\{ℬt⊆ht:\|ℬt\|≤k\}\\mathcal\{C\}\_\{k\}=\\\{\\mathcal\{B\}\_\{t\}\\subseteq h\_\{t\}:\|\\mathcal\{B\}\_\{t\}\|\\leq k\\\}, and letℬopt∈arg⁡minℬ∈𝒞k​Jθ∗​\(ℬ\)\\mathcal\{B\}\_\{\\mathrm\{opt\}\}\\in\\arg\\min\_\{\\mathcal\{B\}\\in\\mathcal\{C\}\_\{k\}\}J\_\{\\theta^\{\*\}\}\(\\mathcal\{B\}\)\. If keyframe subsetsℬ⊂𝒳key\\mathcal\{B\}\\subset\\mathcal\{X\}\_\{\\mathrm\{key\}\}have a positive action\-information margin over all non\-keyframe candidate subsets, and this margin dominates the approximation error of the frozen predictorfθ∗f\_\{\\theta^\{\*\}\}, then the loss objectiveJθ∗​\(ℬ\)J\_\{\\theta^\{\*\}\}\(\\mathcal\{B\}\)recovers the keyframes up to information\-equivalent subsets\. In particular, if𝒳key\\mathcal\{X\}\_\{\\mathrm\{key\}\}is unique under the memory budgetkk, thenℬopt=𝒳key\\mathcal\{B\}\_\{\\mathrm\{opt\}\}=\\mathcal\{X\}\_\{\\mathrm\{key\}\}\.*

*Proof\.*For negative log\-likelihood loss, the Bayes\-optimal expected loss conditioned on an input buffer subsetℬ\\mathcal\{B\}is the conditional entropy, since the logarithmic score is a strictly proper scoring rule and the expected negative log\-likelihood decomposes into conditional entropy plus a nonnegative KL divergence\[[56](https://arxiv.org/html/2610.10857#bib.bib56),[57](https://arxiv.org/html/2610.10857#bib.bib57)\]:

inff𝔼⁡\[ℒ⁡\(fθ∗​\(ℬ\),A\)\]=H⁡\(A∣ℬ\)\.\\inf\_\{f\}\\mathbb\{E\}\[\\mathcal\{L\}\(f\_\{\\theta^\{\*\}\}\(\\mathcal\{B\}\),A\)\]=H\(A\\mid\\mathcal\{B\}\)\.Using the Shannon identity

H⁡\(A∣ℬ\)=H⁡\(A\)−I⁡\(A,ℬ\)H\(A\\mid\\mathcal\{B\}\)=H\(A\)\-I\(A;\\mathcal\{B\}\)minimizing the Bayes\-optimal negative log\-likelihood is equivalent to maximizing the mutual information between the selected subset and the action\[[56](https://arxiv.org/html/2610.10857#bib.bib56)\]\. Define the action\-information margin of the keyframe set over competing candidate subsets as

μk=infℬ∈𝒞k:ℬ≁𝒳key\[I\(A;𝒳key\)−I\(A;ℬ\)\],\\mu\_\{k\}=\\inf\_\{\\mathcal\{B\}\\in\\mathcal\{C\}\_\{k\}:\\,\\mathcal\{B\}\\not\\sim\\mathcal\{X\}\_\{\\mathrm\{key\}\}\}\\left\[I\(A;\\mathcal\{X\}\_\{\\mathrm\{key\}\}\)\-I\(A;\\mathcal\{B\}\)\\right\],whereℬ∼𝒳key\\mathcal\{B\}\\sim\\mathcal\{X\}\_\{\\mathrm\{key\}\}denotes an information\-equivalent subset\. The conditionμk\>0\\mu\_\{k\}\>0extends Assumption 2 from the background subset to the full candidate search space𝒞k\\mathcal\{C\}\_\{k\}, requiring that keyframes have strictly larger action information than any non\-equivalent competing subset\.

Since Proposition 1 biases training toward the invariant class, suppose the frozen modelfθ∗f\_\{\\theta^\{\*\}\}is a near\-optimal keyframe\-conditioned predictor\. Define its approximation error on the keyframe input as

ϵapp=Jθ∗​\(𝒳key\)−H⁡\(A∣𝒳key\)\.\\epsilon\_\{\\mathrm\{app\}\}=J\_\{\\theta^\{\*\}\}\(\\mathcal\{X\}\_\{\\mathrm\{key\}\}\)\-H\(A\\mid\\mathcal\{X\}\_\{\\mathrm\{key\}\}\)\.Since negative log\-likelihood risk is lower\-bounded by conditional entropy,ϵapp≥0\\epsilon\_\{\\mathrm\{app\}\}\\geq 0\. Thus,

Jθ∗​\(𝒳key\)=H⁡\(A∣𝒳key\)\+ϵapp\.J\_\{\\theta^\{\*\}\}\(\\mathcal\{X\}\_\{\\mathrm\{key\}\}\)=H\(A\\mid\\mathcal\{X\}\_\{\\mathrm\{key\}\}\)\+\\epsilon\_\{\\mathrm\{app\}\}\.
For any competing subsetℬ∈𝒞k\\mathcal\{B\}\\in\\mathcal\{C\}\_\{k\}withℬ≁𝒳key\\mathcal\{B\}\\not\\sim\\mathcal\{X\}\_\{\\mathrm\{key\}\}, every predictor using onlyℬ\\mathcal\{B\}has negative log\-likelihood risk at least its Bayes risk by nonnegativity of the KL divergence\[[56](https://arxiv.org/html/2610.10857#bib.bib56),[57](https://arxiv.org/html/2610.10857#bib.bib57)\]:

Jθ∗​\(ℬ\)≥H⁡\(A∣ℬ\)\.J\_\{\\theta^\{\*\}\}\(\\mathcal\{B\}\)\\geq H\(A\\mid\\mathcal\{B\}\)\.By the definition ofμk\\mu\_\{k\},

I⁡\(A,𝒳key\)−I⁡\(A,ℬ\)≥μk,I\(A;\\mathcal\{X\}\_\{\\mathrm\{key\}\}\)\-I\(A;\\mathcal\{B\}\)\\geq\\mu\_\{k\},which is equivalent to

H⁡\(A∣ℬ\)≥H⁡\(A∣𝒳key\)\+μk\.H\(A\\mid\\mathcal\{B\}\)\\geq H\(A\\mid\\mathcal\{X\}\_\{\\mathrm\{key\}\}\)\+\\mu\_\{k\}\.Therefore,

Jθ∗​\(ℬ\)≥H⁡\(A∣𝒳key\)\+μk\.J\_\{\\theta^\{\*\}\}\(\\mathcal\{B\}\)\\geq H\(A\\mid\\mathcal\{X\}\_\{\\mathrm\{key\}\}\)\+\\mu\_\{k\}\.
If the information margin dominates the approximation error,

μk\>ϵapp,\\mu\_\{k\}\>\\epsilon\_\{\\mathrm\{app\}\},then

Jθ∗​\(𝒳key\)=H⁡\(A∣𝒳key\)\+ϵapp<H⁡\(A∣𝒳key\)\+μk≤Jθ∗​\(ℬ\)J\_\{\\theta^\{\*\}\}\(\\mathcal\{X\}\_\{\\mathrm\{key\}\}\)=H\(A\\mid\\mathcal\{X\}\_\{\\mathrm\{key\}\}\)\+\\epsilon\_\{\\mathrm\{app\}\}<H\(A\\mid\\mathcal\{X\}\_\{\\mathrm\{key\}\}\)\+\\mu\_\{k\}\\leq J\_\{\\theta^\{\*\}\}\(\\mathcal\{B\}\)for every non\-equivalent competing subsetℬ\\mathcal\{B\}\. Hence, the minimizer of the frozen loss objective belongs to the keyframe equivalence class\. If𝒳key\\mathcal\{X\}\_\{\\mathrm\{key\}\}is unique under the memory budgetkk, then

ℬopt=𝒳key\.\\mathcal\{B\}\_\{\\mathrm\{opt\}\}=\\mathcal\{X\}\_\{\\mathrm\{key\}\}\.This keyframe recovery guarantee naturally extends to continuous action spaces optimized via Mean Squared Error \(MSE\)\. Under the assumption that the conditional distribution of the actions is Gaussian with a fixed variance, minimizing the MSE is equivalent to minimizing the negative log\-likelihood up to affine constants\[[58](https://arxiv.org/html/2610.10857#bib.bib58),[59](https://arxiv.org/html/2610.10857#bib.bib59)\]\. Consequently, the same argument applies with conditional differential entropy and mutual information replacing their discrete counterparts, so the frozen MSE objective recovers𝒳key\\mathcal\{X\}\_\{\\mathrm\{key\}\}under the same positive\-margin and approximation\-error conditions\.□\\square

Theorem 2 \(Infinite\-horizon keyframe retention\)\.*Assume a POMDP in which the expert behavior strictly satisfies Assumption 1 and 2\. Letπs∗\\pi\_\{\\mathrm\{s\}\}^\{\*\}be an optimal selector minimizing the frozen objectiveJθ∗​\(ℬt\)J\_\{\\theta^\{\*\}\}\(\\mathcal\{B\}\_\{t\}\)\(Corollary 1\), and suppose a unique keyframe set𝒳key\\mathcal\{X\}\_\{\\mathrm\{key\}\}exists that has the highest mutual information with the action and fits within the memory budgetkk\. Then, asT→∞T\\to\\infty,πs∗\\pi\_\{\\mathrm\{s\}\}^\{\*\}assigns higher priority to keyframes than to background frames, ensuring that the priority queue retains the keyframe set for the entire horizonTTwithout eviction, i\.e\.,ℬT=𝒳key\\mathcal\{B\}\_\{T\}=\\mathcal\{X\}\_\{\\mathrm\{key\}\}\.*

*Proof\.*We analyze the behavior of an optimal selector policyπs∗\\pi\_\{\\mathrm\{s\}\}^\{\*\}over the temporal horizonTT\. The reward functionR⁡\(ℬt\)R\(\\mathcal\{B\}\_\{t\}\)is designed to increase as the frozen loss objectiveJθ∗​\(ℬt\)J\_\{\\theta^\{\*\}\}\(\\mathcal\{B\}\_\{t\}\)decreases\. Thus, maximizing expected return requiresπs∗\\pi\_\{\\mathrm\{s\}\}^\{\*\}to assign priority scores that configure the bufferℬt\\mathcal\{B\}\_\{t\}into the subset minimizingJθ∗​\(ℬ\)J\_\{\\theta^\{\*\}\}\(\\mathcal\{B\}\)\.

By Assumption 2 and Corollary 1, the keyframe set𝒳key\\mathcal\{X\}\_\{\\mathrm\{key\}\}has a strictly positive action\-information marginμk\>0\\mu\_\{k\}\>0over all non\-keyframe subsets\. Consider a timestepttat which the keyframe set has already been inserted into the buffer\. Let a new background observationot∈𝒳bgo\_\{t\}\\in\\mathcal\{X\}\_\{\\mathrm\{bg\}\}be evaluated\. Ifπs∗\\pi\_\{\\mathrm\{s\}\}^\{\*\}assigns a priorityptp\_\{t\}high enough to evict a keyframe, then the resulting bufferℬt\\mathcal\{B\}\_\{t\}has strictly lower mutual information with the action, decreasing by at leastμk\\mu\_\{k\}\.

By Corollary 1, this induces a strictly higher expected loss:

Jθ∗​\(ℬt\)≥H⁡\(A∣𝒳key\)\+μk\>Jθ∗​\(𝒳key\),J\_\{\\theta^\{\*\}\}\(\\mathcal\{B\}\_\{t\}\)\\geq H\(A\\mid\\mathcal\{X\}\_\{\\mathrm\{key\}\}\)\+\\mu\_\{k\}\>J\_\{\\theta^\{\*\}\}\(\\mathcal\{X\}\_\{\\mathrm\{key\}\}\),which penalizes the reward\. Therefore, any such eviction is suboptimal, andπs∗\\pi\_\{\\mathrm\{s\}\}^\{\*\}must assign lower priority to background observations than to keyframes\. Once the keyframe set𝒳key\\mathcal\{X\}\_\{\\mathrm\{key\}\}is fully inserted into the buffer, this argument applies recursively\. Hence, under the optimal selector policy, background observations never receive sufficient priority to evict keyframes\. Thus, the priority queue preserves𝒳key\\mathcal\{X\}\_\{\\mathrm\{key\}\}indefinitely, yieldingℬT=𝒳key\\mathcal\{B\}\_\{T\}=\\mathcal\{X\}\_\{\\mathrm\{key\}\}\.□\\square

## Appendix BAlgorithms

Algorithm 2FIFO priority queue — training\-time eviction rule\.1:Parameters:training memory budget

kk, rejection threshold

ηr\\eta\_\{\\mathrm\{r\}\}
2:State:buffer

ℬt←∅\\mathcal\{B\}\_\{t\}\\leftarrow\\emptyset, insertion counter

c←0c\\leftarrow 0
3:procedurePush\(observation

oo, priority

pp\)

4:if

p<ηrp<\\eta\_\{\\mathrm\{r\}\}then

5:return⊳\\trianglerightreject priority below threshold

6:endif

7:

c←c\+1c\\leftarrow c\+1⊳\\trianglerightmonotone insertion timestamp

8:if

\|ℬt\|<k\|\\mathcal\{B\}\_\{t\}\|<kthen

9:

ℬt\+1←ℬt∪\{\(o,p,c\)\}\\mathcal\{B\}\_\{t\+1\}\\leftarrow\\mathcal\{B\}\_\{t\}\\cup\\\{\(o,p,c\)\\\}⊳\\trianglerightfree slot, insert directly

10:else

11:

\(o⋆,p⋆,c⋆\)←arg⁡min\(o′,p′,c′\)∈ℬt⁡\(p′,c′\)\(o^\{\\star\},p^\{\\star\},c^\{\\star\}\)\\leftarrow\\arg\\min\_\{\(o^\{\\prime\},p^\{\\prime\},c^\{\\prime\}\)\\in\\mathcal\{B\}\_\{t\}\}\(p^\{\\prime\},\\,c^\{\\prime\}\)⊳\\trianglerightlowest priority;*oldest*on a tie

12:if

p≥p⋆p\\geq p^\{\\star\}then

13:

ℬt\+1←\(ℬt∖\{\(o⋆,p⋆,c⋆\)\}\)∪\{\(o,p,c\)\}\\mathcal\{B\}\_\{t\+1\}\\leftarrow\\big\(\\mathcal\{B\}\_\{t\}\\setminus\\\{\(o^\{\\star\},p^\{\\star\},c^\{\\star\}\)\\\}\\big\)\\cup\\\{\(o,p,c\)\\\}
14:endif

15:endif

16:endprocedure

Algorithm 3LIFO priority queue — inference\-time eviction rule\.1:Parameters:inference memory budget

kk, rejection threshold

ηr\\eta\_\{\\mathrm\{r\}\}, no\-repeat threshold

ηn\\eta\_\{\\mathrm\{n\}\}
2:State:buffer

ℬt←∅\\mathcal\{B\}\_\{t\}\\leftarrow\\emptyset, insertion counter

c←0c\\leftarrow 0, last priority

plast←0p\_\{\\mathrm\{last\}\}\\leftarrow 0
3:procedurePush\(observation

oo, priority

pp\)

4:if

p<ηrp<\\eta\_\{\\mathrm\{r\}\}or

\|p−plast\|<ηn\|p\-p\_\{\\mathrm\{last\}\}\|<\\eta\_\{\\mathrm\{n\}\}then

5:return⊳\\trianglerightreject low or quasi\-repeated priority

6:endif

7:

c←c\+1c\\leftarrow c\+1
8:if

\|ℬt\|<k\|\\mathcal\{B\}\_\{t\}\|<kthen

9:

ℬt\+1←ℬt∪\{\(o,p,c\)\}\\mathcal\{B\}\_\{t\+1\}\\leftarrow\\mathcal\{B\}\_\{t\}\\cup\\\{\(o,p,c\)\\\}
10:else

11:

\(o⋆,p⋆,c⋆\)←arg⁡min\(o′,p′,c′\)∈ℬt⁡\(p′,−c′\)\(o^\{\\star\},p^\{\\star\},c^\{\\star\}\)\\leftarrow\\arg\\min\_\{\(o^\{\\prime\},p^\{\\prime\},c^\{\\prime\}\)\\in\\mathcal\{B\}\_\{t\}\}\(p^\{\\prime\},\\,\-c^\{\\prime\}\)⊳\\trianglerightlowest priority;*newest*on a tie

12:if

p≥p⋆p\\geq p^\{\\star\}then

13:

ℬt\+1←\(ℬt∖\{\(o⋆,p⋆,c⋆\)\}\)∪\{\(o,p,c\)\}\\mathcal\{B\}\_\{t\+1\}\\leftarrow\\big\(\\mathcal\{B\}\_\{t\}\\setminus\\\{\(o^\{\\star\},p^\{\\star\},c^\{\\star\}\)\\\}\\big\)\\cup\\\{\(o,p,c\)\\\}
14:endif

15:endif

16:

plast←pp\_\{\\mathrm\{last\}\}\\leftarrow p⊳\\trianglerightupdate even if no eviction occurred

17:endprocedure

### B\.1Buffer priority queue

The memory bufferℬt\\mathcal\{B\}\_\{t\}introduced in Section[2\.2](https://arxiv.org/html/2610.10857#S2.SS2)is realised by a bounded priority queue of capacitykk\. The buffer design supports two key behaviors: \(i\)overflow forgetting, where consistently assigning high priorities causes queue overflow and removes keyframes from the buffer, and \(ii\)background rejection, where assigning priorities below a rejection threshold actively ignores background observations\. Both training and inference queue use the same insertion semantics \- the selector emits a priorityp∈\[0,1\]p\\in\[0,1\]per observation, and the queue accepts or rejects based onpp; the two phases differ*only*in how ties on the lowest priority are broken at eviction time\. Both preserve the temporal ordering of frames in the buffer\. Algorithms[2](https://arxiv.org/html/2610.10857#alg2)and[3](https://arxiv.org/html/2610.10857#alg3)state the two rules\. We useηr=0\.5\\eta\_\{\\mathrm\{r\}\}=0\.5as the rejection threshold throughout, andηn=0\.05\\eta\_\{\\mathrm\{n\}\}=0\.05as the no\-repeat threshold for the inference queue\.

Why FIFO at training time\.The training rule \(Algorithm[2](https://arxiv.org/html/2610.10857#alg2)\) breaks ties in favour of evicting the*oldest*entry\. Under this rule, a frame that has been assigned a uniformly high priority will still age out as new high\-priority frames arrive, so the selector cannot rely on saturating priorities to retain information\. This sharpens the gradient signal drivingπs\\pi\_\{\\mathrm\{s\}\}towards genuinely discriminative scores rather than constant\-1 outputs\.

Why LIFO at inference time\.The inference rule \(Algorithm[3](https://arxiv.org/html/2610.10857#alg3)\) breaks ties by evicting the*newest*entry\. Thus, initial frames assigned high priority by the selector are preserved, enabling robust keyframe retention and maintaining the keyframe set over arbitrarily long horizons\. The no\-repeat thresholdηn\\eta\_\{\\mathrm\{n\}\}further prevents near\-tied priority observations from entering the queue, ensuring that it retains diversity\. This especially helps with retaining keyframes when there exists multiple information\-equivalent keyframe subsets in the demonstration dataset\.

Note that under a near\-optimal selector and a sufficient memory budgetkk\(Theorem 2\), such that the buffer can store all true keyframes, both queues exhibit identical behavior\. We observed this empirically in synthetic domains, where the discovered mnemonics remained consistent regardless of the queue strategy used during inference\.

### B\.2Proxy policy training

Section[2\.1](https://arxiv.org/html/2610.10857#S2.SS1)introduces the proxy policyfθ:ℬ→𝒜f\_\{\\theta\}:\\mathcal\{B\}\\to\\mathcal\{A\}trained by masked\-sampling of past observations \(Algorithm[1](https://arxiv.org/html/2610.10857#alg1)\)\. In practice, we implement two equivalent realizations of this training procedure, selected based on the scale of the masked\-sampling supportMM\(Lemma 1\)\.

#### Online masked\-sampling \(robot manipulation\)\.

ForMIKASA\-Robotasks we draw a fresh maskΨk\\Psi\_\{k\}at every gradient step, as stated in Algorithm[1](https://arxiv.org/html/2610.10857#alg1): a trajectoryτ∼𝒟\\tau\\sim\\mathcal\{D\}and timestept∈\{1,…,Tτ\}t\\in\\\{1,\\dots,T\_\{\\tau\}\\\}are sampled, the masked\-sampling distributionn∼min⁡\(Binomial⁡\(t,0\.5\),k\)n\\sim\\min\(\\mathrm\{Binomial\}\(t,0\.5\),k\)selects a memory bufferℬt=\{oi∈ht∣Ψk​\(i\)=1\}\\mathcal\{B\}\_\{t\}=\\\{o\_\{i\}\\in h\_\{t\}\\mid\\Psi\_\{k\}\(i\)=1\\\}of cardinality\|ℬt\|≤k\\lvert\\mathcal\{B\}\_\{t\}\\rvert\\leq k, and the proxy regresses the expert actionata\_\{t\}under negative log\-likelihood lossℒ\\mathcal\{L\}\.

#### Offline masked\-sampling rollouts \(synthetic and grid domains\)\.

For the synthetic and grid domains \(Add, Scattered\-Copy, T\-Maze, MiniGrid\-Memory, LTMB\-Hallway\), the support sizeMMis small enough to precompute masked samples\. We therefore run multiple offline passes over𝒟\\mathcal\{D\}and cache the random buffer\-action pairs on disk, rather than resampling masks at every gradient step\. In each pass, for each trajectoryτi∈𝒟\\tau\_\{i\}\\in\\mathcal\{D\}, we iterate overt=1,…,Tit=1,\\dots,T\_\{i\}, and execute the masked\-sampling procedure \(Algorithm[1](https://arxiv.org/html/2610.10857#alg1)\) for each timestep\. The resulting pairs\(ℬt,at\)\(\\mathcal\{B\}\_\{t\},a\_\{t\}\)are written to the cache, and the proxyfθf\_\{\\theta\}is trained for multiple epochs on the cached pairs\. The number of passes trades off cache size against Monte Carlo coverage of the masked\-sampling support and is chosen empirically so that the total number of cached pairs exceeds the sufficient support sizeMMrequired by Theorem 1\. Although online masked\-sampling provides the maximum possible cardinality supportMM, this offline counterpart helped us with more controlled experiments onMMto empirically verify Theorem 1 as shown in Figure[2](https://arxiv.org/html/2610.10857#S3.F2)\. See Appendix[C\.3](https://arxiv.org/html/2610.10857#A3.SS3)for more details\.

#### Lemma 2 \(Equivalence to uniform selector priority queue\)\.

*Let𝒫mask\\mathcal\{P\}\_\{\\mathrm\{mask\}\}denote the distribution over buffer statesℬ\\mathcal\{B\}induced by masked\-sampling \(Algorithm[1](https://arxiv.org/html/2610.10857#alg1)\), and let𝒫queue\\mathcal\{P\}\_\{\\mathrm\{queue\}\}denote the distribution induced by using a random selectorπrand\\pi\_\{\\mathrm\{rand\}\}that assigns uniform prioritiesp∈\[0,1\]p\\in\[0,1\]to observations retained in a priority queue\. Both procedures induce the same distribution over buffer states\.*

*Proof\.*Under𝒫mask\\mathcal\{P\}\_\{\\mathrm\{mask\}\}, the buffer is drawn by picking a sizen∼min⁡\(Binomial⁡\(t,0\.5\),k\)n\\sim\\min\(\\mathrm\{Binomial\}\(t,0\.5\),k\)uniformly at random and then selectingnnindices uniformly without replacement from\{1,…,t\}\\\{1,\\dots,t\\\}\(Lemma 1\)\. Under𝒫queue\\mathcal\{P\}\_\{\\mathrm\{queue\}\}, each past observationoi∈hto\_\{i\}\\in h\_\{t\}receives an i\.i\.d\. prioritypi∼Unif⁡\(0,1\)p\_\{i\}\\sim\\mathrm\{Unif\}\(0,1\), and the queue retains the top\-kkpriorities exceeding the rejection thresholdηr\\eta\_\{r\}\. Because the priorities are exchangeable and independent ofii, everykk\-subset ofhth\_\{t\}is selected with equal probability conditional on its cardinality, and the cardinality distribution ismin\(Binomial\(t,Pr\[p\>ηr\]\),k\)\\min\(\\mathrm\{Binomial\}\(t,\\Pr\[p\>\\eta\_\{r\}\]\),k\)\. Withηr=0\.5\\eta\_\{r\}=0\.5we havePr\[p\>ηr\]=0\.5\\Pr\[p\>\\eta\_\{r\}\]=0\.5, recovering the masked\-sampling cardinality distribution of Algorithm[1](https://arxiv.org/html/2610.10857#alg1)\. Hence𝒫queue=𝒫mask\\mathcal\{P\}\_\{\\mathrm\{queue\}\}=\\mathcal\{P\}\_\{\\mathrm\{mask\}\}at the level of marginal overℬt\\mathcal\{B\}\_\{t\}\.□\\square

The above result establishes an equivalence between the masked\-sampling procedure \(Algorithm[1](https://arxiv.org/html/2610.10857#alg1)\) and selector\-driven buffer construction under a random selector policy\. Intuitively, this shows that the selector optimization \(Section[2\.2](https://arxiv.org/html/2610.10857#S2.SS2)\) operates over the same combinatorial space of buffer states explored during masked\-sampling training\. Consequently, the learned objectiveJθ∗J\_\{\\theta^\{\*\}\}remains aligned with the optimization problem solved by the selector policy\. BecauseJθ∗J\_\{\\theta^\{\*\}\}wraps a function that learns a keyframe\-conditioned hypothesis favored by Theorem 1, optimizing the selector is naturally biased toward recovering keyframe subsets\.

### B\.3Selector RL environment

We provide the algorithmic specification of the deterministic MDP⟨𝒪,𝒫,𝒦,R,γ⟩\\langle\\mathcal\{O\},\\mathcal\{P\},\\mathcal\{K\},R,\\gamma\\rangledescribed in Section[2\.2](https://arxiv.org/html/2610.10857#S2.SS2), on which the selector policyπs\\pi\_\{\\mathrm\{s\}\}is trained with PPO\. Algorithm[4](https://arxiv.org/html/2610.10857#alg4)states the per\-step transition: the policy observesot∈𝒪o\_\{t\}\\in\\mathcal\{O\}, emits a prioritypt∈𝒫=\[0,1\]p\_\{t\}\\in\\mathcal\{P\}=\[0,1\], and the environment updates the priority queueℬt\\mathcal\{B\}\_\{t\}via the transition rule𝒦\\mathcal\{K\}\(Algorithm[2](https://arxiv.org/html/2610.10857#alg2)\)\. The post\-update buffer is then scored by the frozen proxyfθ∗f\_\{\\theta^\{\\ast\}\}to compute the surprise\-weighted log\-rewardR⁡\(ℬt\)R\(\\mathcal\{B\}\_\{t\}\)from Equation[5](https://arxiv.org/html/2610.10857#S2.E5)\. The episode terminates afterTτT\_\{\\tau\}steps and the next trajectory is loaded from𝒟\\mathcal\{D\}\.

Algorithm 4Selector RL environment stepRequire:dataset𝒟\\mathcal\{D\}, frozen proxyfθ∗f\_\{\\theta^\{\\ast\}\}, memory budgetkk, normalisation constantλ\\lambda, surprise coefficients\{βτ,t\}\\\{\\beta\_\{\\tau,t\}\\\}, queue eviction rulePush\(Algorithm[2](https://arxiv.org/html/2610.10857#alg2)\) State:current trajectoryτ∈𝒟\\tau\\in\\mathcal\{D\}, step countertt, bufferℬt−1\\mathcal\{B\}\_\{t\-1\}

1:procedureStep\(

ot,pto\_\{t\},p\_\{t\}\)⊳\\trianglerightpt∼πs\(⋅∣ot\)p\_\{t\}\\sim\\pi\_\{\\mathrm\{s\}\}\(\\cdot\\mid o\_\{t\}\)

2:

at←τ\.Action​\(t\)a\_\{t\}\\leftarrow\\tau\.\\textsc\{Action\}\(t\)⊳\\trianglerightexpert action from𝒟\\mathcal\{D\}

3:

ℬt←Push​\(ℬt−1,ot,pt\)\\mathcal\{B\}\_\{t\}\\leftarrow\\textsc\{Push\}\(\\mathcal\{B\}\_\{t\-1\},o\_\{t\},p\_\{t\}\)⊳\\trianglerightpriority\-queue update

4:

ℓt←ℒ⁡\(fθ∗​\(ℬt\),at\)\\ell\_\{t\}\\leftarrow\\mathcal\{L\}\\bigl\(f\_\{\\theta^\{\\ast\}\}\(\\mathcal\{B\}\_\{t\}\),a\_\{t\}\\bigr\)⊳\\trianglerightfrozen proxy objectiveJθ∗​\(ℬt\)J\_\{\\theta^\{\*\}\}\(\\mathcal\{B\}\_\{t\}\)

5:

rt←βτ,t⋅log⁡\(ℓt\+λ\)log⁡\(λ\)r\_\{t\}\\leftarrow\\beta\_\{\\tau,t\}\\cdot\\dfrac\{\\log\(\\ell\_\{t\}\+\\lambda\)\}\{\\log\(\\lambda\)\}⊳\\trianglerightEquation[5](https://arxiv.org/html/2610.10857#S2.E5)

6:

t←t\+1t\\leftarrow t\+1
7:if

t=Tτt=T\_\{\\tau\}then⊳\\trianglerightepisode terminated

8:

τ∼𝒟\\tau\\sim\\mathcal\{D\}⊳\\trianglerightload next trajectory

9:

ℬt−1←∅\\mathcal\{B\}\_\{t\-1\}\\leftarrow\\emptyset,

t←1t\\leftarrow 1
10:

done←True\\mathrm\{done\}\\leftarrow\\textsc\{True\}
11:else

12:

done←False\\mathrm\{done\}\\leftarrow\\textsc\{False\}
13:endif

14:

ot\+1←τ\.Obs​\(t\)o\_\{t\+1\}\\leftarrow\\tau\.\\textsc\{Obs\}\(t\)
15:return

\(ot\+1,rt,done\)\(o\_\{t\+1\},r\_\{t\},\\mathrm\{done\}\)
16:endprocedure

The surprise coefficientβτ,t\\beta\_\{\\tau,t\}in Equation[5](https://arxiv.org/html/2610.10857#S2.E5)is precomputed once before PPO training asβτ,t=Tτ⋅softmaxt​\(Jθ∗τ,t/κ\)\\beta\_\{\\tau,t\}=T\_\{\\tau\}\\cdot\\mathrm\{softmax\}\_\{t\}\(J\_\{\\theta^\{\\ast\}\}^\{\\tau,t\}/\\kappa\), whereJθ∗τ,t=𝔼Ψk​\[ℒ⁡\(fθ∗​\(ℬt\),at\)\]J\_\{\\theta^\{\\ast\}\}^\{\\tau,t\}=\\mathbb\{E\}\_\{\\Psi\_\{k\}\}\[\\mathcal\{L\}\(f\_\{\\theta^\{\\ast\}\}\(\\mathcal\{B\}\_\{t\}\),a\_\{t\}\)\]is the expected masked\-sampling loss evaluated at\(τ,t\)\(\\tau,t\)\. Surprise scaling is applied on a per\-domain basis: we enable it for synthetic and grid domains, where the loss landscape overttis sharply peaked at decision\-critical timesteps, and disable it \(βτ,t≡1\\beta\_\{\\tau,t\}\\\!\\equiv\\\!1\) for robot manipulation domains, where the unscaled reward in Equation[5](https://arxiv.org/html/2610.10857#S2.E5)empirically provides a more stable PPO training signal\. At training time, the environment is vectorized acrossnenvsn\_\{\\mathrm\{envs\}\}parallel trajectories drawn from𝒟\\mathcal\{D\}, and the proxy is evaluated on batches of buffer states for scalable RL training\.

### B\.4End\-to\-end training and inference

Algorithm[5](https://arxiv.org/html/2610.10857#alg5)describes the end\-to\-end training procedure introduced in Sections[2\.1](https://arxiv.org/html/2610.10857#S2.SS1)–[2\.3](https://arxiv.org/html/2610.10857#S2.SS3)\. The proxyfθf\_\{\\theta\}is trained using the masked\-sampling algorithm \(Algorithm[1](https://arxiv.org/html/2610.10857#alg1)\), then frozen and used to score the rewardR⁡\(ℬt\)R\(\\mathcal\{B\}\_\{t\}\)for selector RL \(Algorithm[4](https://arxiv.org/html/2610.10857#alg4)\)\. The trained selector is then used to produce an augmented dataset𝒟†\\mathcal\{D\}^\{\\dagger\}by rollingπs\\pi\_\{\\mathrm\{s\}\}on𝒟\\mathcal\{D\}with the LIFO eviction rule555For synthetic domains where Theorem 2 holds strongly, we empirically observe that either FIFO or LIFO eviction can be used at inference time without affecting the discovered mnemonics; the LIFO rule is preferred for grid and robot manipulation domains where the assumptions hold less tightly\., which is then used to train the BC policyπBC​\(at∣ot,ℬt\)\\pi\_\{\\mathrm\{BC\}\}\(a\_\{t\}\\mid o\_\{t\},\\mathcal\{B\}\_\{t\}\)\. See Appendix[C](https://arxiv.org/html/2610.10857#A3)and[D](https://arxiv.org/html/2610.10857#A4)for more details\.

Algorithm 5Keyframe Mnemonics trainingRequire:dataset𝒟\\mathcal\{D\}, training memory budgetkk, proxyfθf\_\{\\theta\}, selectorπs\\pi\_\{\\mathrm\{s\}\}, BC policyπBC\\pi\_\{\\mathrm\{BC\}\}, training epochsEf,Es,EBCE\_\{f\},E\_\{\\mathrm\{s\}\},E\_\{\\mathrm\{BC\}\} Returns:trained\(fθ,πs,πBC\)\(f\_\{\\theta\},\\pi\_\{\\mathrm\{s\}\},\\pi\_\{\\mathrm\{BC\}\}\)

1:// Stage 1 \-\- proxy policy training \(Section[2\.1](https://arxiv.org/html/2610.10857#S2.SS1)\)

2:for

e=1,…,Efe=1,\\dots,E\_\{f\}do

3:update

θ\\thetaon a batch sampled by Algorithm[1](https://arxiv.org/html/2610.10857#alg1)on

𝒟\\mathcal\{D\}
4:endfor

5:freeze

θ∗←θ\\theta^\{\\ast\}\\leftarrow\\theta; define

Jθ∗​\(ℬ\)=𝔼⁡\[ℒ⁡\(fθ∗​\(ℬ\),a\)\]J\_\{\\theta^\{\\ast\}\}\(\\mathcal\{B\}\)=\\mathbb\{E\}\[\\mathcal\{L\}\(f\_\{\\theta^\{\\ast\}\}\(\\mathcal\{B\}\),a\)\]
6:// Stage 2 \-\- selector policy training \(Section[2\.2](https://arxiv.org/html/2610.10857#S2.SS2)\)

7:precompute surprise coefficients

\{βτ,t\}\\\{\\beta\_\{\\tau,t\}\\\}from

Jθ∗J\_\{\\theta^\{\\ast\}\}
8:for

e=1,…,Ese=1,\\dots,E\_\{\\mathrm\{s\}\}do

9:roll out the selector MDP \(Algorithm[4](https://arxiv.org/html/2610.10857#alg4)\) with FIFO eviction

10:update

πs\\pi\_\{\\mathrm\{s\}\}by PPO on the surprise\-weighted reward

R⁡\(ℬt\)R\(\\mathcal\{B\}\_\{t\}\)
11:endfor

12:// Stage 3 \-\- BC policy training \(Section[2\.3](https://arxiv.org/html/2610.10857#S2.SS3)\)

13:

𝒟†←∅\\mathcal\{D\}^\{\\dagger\}\\leftarrow\\emptyset
14:for

τ∈𝒟\\tau\\in\\mathcal\{D\}do

15:roll

πs\\pi\_\{\\mathrm\{s\}\}on

τ\\tauwith LIFO eviction \(Algorithm[3](https://arxiv.org/html/2610.10857#alg3)\)

16:append

\{\(ℬt,ot,at\)\}t=1Tτ\\\{\(\\mathcal\{B\}\_\{t\},o\_\{t\},a\_\{t\}\)\\\}\_\{t=1\}^\{T\_\{\\tau\}\}to

𝒟†\\mathcal\{D\}^\{\\dagger\}
17:endfor

18:for

e=1,…,EBCe=1,\\dots,E\_\{\\mathrm\{BC\}\}do

19:update

πBC\\pi\_\{\\mathrm\{BC\}\}on a batch from

𝒟†\\mathcal\{D\}^\{\\dagger\}under the BC loss

20:endfor

21:return

\(fθ∗,πs,πBC\)\(f\_\{\\theta^\{\*\}\},\\pi\_\{\\mathrm\{s\}\},\\pi\_\{\\mathrm\{BC\}\}\)

Algorithm 6Keyframe Mnemonics inferenceRequire:environmentℰ\\mathcal\{E\}, trained selectorπs\\pi\_\{\\mathrm\{s\}\}, trained BC policyπBC\\pi\_\{\\mathrm\{BC\}\}, training memory budgetkk, max rollout stepsTmaxT\_\{\\max\}, queue eviction rulePush\(Algorithm[3](https://arxiv.org/html/2610.10857#alg3)\)

1:

o1←ℰ\.Reset​\(\)o\_\{1\}\\leftarrow\\mathcal\{E\}\.\\textsc\{Reset\}\(\)
2:

ℬ0←∅\\mathcal\{B\}\_\{0\}\\leftarrow\\emptyset
3:for

t=1,…,Tmaxt=1,\\dots,T\_\{\\max\}do

4:

pt←πs​\(ot\)p\_\{t\}\\leftarrow\\pi\_\{\\mathrm\{s\}\}\(o\_\{t\}\)⊳\\trianglerightassign priority

5:

ℬt←Push​\(ℬt−1,ot,pt\)\\mathcal\{B\}\_\{t\}\\leftarrow\\textsc\{Push\}\(\\mathcal\{B\}\_\{t\-1\},o\_\{t\},p\_\{t\}\)⊳\\trianglerightLIFO buffer update

6:

at←πBC​\(ℬt,ot\)a\_\{t\}\\leftarrow\\pi\_\{\\mathrm\{BC\}\}\(\\mathcal\{B\}\_\{t\},o\_\{t\}\)
7:

ot\+1←ℰ\.Step​\(at\)o\_\{t\+1\}\\leftarrow\\mathcal\{E\}\.\\textsc\{Step\}\(a\_\{t\}\)
8:endfor

At inference time the trained\(πs,πBC\)\(\\pi\_\{\\mathrm\{s\}\},\\pi\_\{\\mathrm\{BC\}\}\)pair is rolled out in the target environment as specified in Algorithm[6](https://arxiv.org/html/2610.10857#alg6)\. At each step the selector emits a prioritypt=πs​\(ot\)p\_\{t\}=\\pi\_\{\\mathrm\{s\}\}\(o\_\{t\}\)that updates the bufferℬt\\mathcal\{B\}\_\{t\}under the LIFO eviction rule of Algorithm[3](https://arxiv.org/html/2610.10857#alg3)\(matching the rule used to construct𝒟†\\mathcal\{D\}^\{\\dagger\}\), and the BC policy emits an actionat=πBC​\(ℬt,ot\)a\_\{t\}=\\pi\_\{\\mathrm\{BC\}\}\(\\mathcal\{B\}\_\{t\},o\_\{t\}\)that is applied to the environment for at mostTmaxT\_\{\\max\}steps\.

## Appendix CSynthetic and Grid Domain Details

### C\.1Synthetic environments

In this section we show expert trajectories for the three synthetic memory benchmarks\. Cell colors group conceptually related observations:

/

/

mark the*keyframes*that an oracle selector must retain,

marks background/noise observations that an optimal selector should reject, and

marks the decision\-critical query timestep at which the policy emits its action\.

#### Add\[[28](https://arxiv.org/html/2610.10857#bib.bib28)\]\.

A sequence ofTTnoisy real\-valued observations\. Two positionst1,t2t\_\{1\},t\_\{2\}such thatt1<t2t\_\{1\}<t\_\{2\}are marked with binary flags \(the “first\(1,0\)\(1,0\)” and “second\(0,1\)\(0,1\)” summands\); at the final timestept=T−1t=T\{\-\}1both flags are simultaneously set\(1,1\)\(1,1\), signalling the policy to emit the sum of the two marked values\. The per\-step observation is the tripleot=\(vt,1\[t=t1\],1\[t=t2\]\)o\_\{t\}=\(v\_\{t\},\\;\\mathbf\{1\}\[t=t\_\{1\}\],\\;\\mathbf\{1\}\[t=t\_\{2\}\]\)withvt∼𝒰⁡\[0,1\]v\_\{t\}\\\!\\sim\\\!\\mathcal\{U\}\[0,1\], and the target action is the scalarat=vt1\+vt2a\_\{t\}=v\_\{t\_\{1\}\}\+v\_\{t\_\{2\}\}att=T−1t=T\{\-\}1andat=0a\_\{t\}=0elsewhere\. Example withT=6T=6,t1=1t\_\{1\}=1,t2=3t\_\{2\}=3:

The three keyframes𝒳key=\{o1,o3,o5\}\\mathcal\{X\}\_\{\\mathrm\{key\}\}=\\\{o\_\{1\},o\_\{3\},o\_\{5\}\\\}comprise the two flag\-marked summands \(

,

\) and the emit step \(

\); the latter is itself a keyframe because the policy must use the emit token to know when to emit\. The supervised target at the emit step isa5=vt1\+vt2=0\.83\+0\.65=1\.48a\_\{5\}=v\_\{t\_\{1\}\}\+v\_\{t\_\{2\}\}=0\.83\+0\.65=1\.48; all other timesteps carry unmarked\(vt,0,0\)\(v\_\{t\},0,0\)observations and are background frames\. Evaluation is done on sequences with largerTTobtained by increasing the number of unmarked\(vt,0,0\)\(v\_\{t\},0,0\)background observations in the sequence\.

#### Scattered\-Copy\[[22](https://arxiv.org/html/2610.10857#bib.bib22)\]\.

A subsequence ofnrelevantn\_\{\\mathrm\{relevant\}\}relevant tokens is scattered amongnnoisen\_\{\\mathrm\{noise\}\}noise tokens to form a source segment of lengthLsrc=nrelevant\+nnoiseL\_\{\\mathrm\{src\}\}=n\_\{\\mathrm\{relevant\}\}\+n\_\{\\mathrm\{noise\}\}; the source is followed by a recall segment ofnrelevantn\_\{\\mathrm\{relevant\}\}delimiter tokensEscattered amongnblankn\_\{\\mathrm\{blank\}\}blank tokens□\\square\(lengthLrec=nrelevant\+nblankL\_\{\\mathrm\{rec\}\}=n\_\{\\mathrm\{relevant\}\}\+n\_\{\\mathrm\{blank\}\}\)\. At each delimiter the policy must emit the next relevant token in the order it originally appeared, and a no\-op token0at every other position\. The set of tokens for relevant and noise are disjoint and remains fixed for the task during both training and evaluation\. Example withnrelevant=3n\_\{\\mathrm\{relevant\}\}=3,nnoise=3n\_\{\\mathrm\{noise\}\}=3,nblank=5n\_\{\\mathrm\{blank\}\}=5:

The six keyframes𝒳key=\{o1,o3,o5,o8,o10,o13\}\\mathcal\{X\}\_\{\\mathrm\{key\}\}=\\\{o\_\{1\},o\_\{3\},o\_\{5\},o\_\{8\},o\_\{10\},o\_\{13\}\\\}comprise the three relevant source tokens \(

/

/

\) and the three delimiter tokens \(

\); the delimiters are themselves keyframes because the policy must use anEtoken to know it is at an emit position\. The noise tokensNiin the source and blank tokens in the recall are background frames\. Evaluation is done by increasingnnoisen\_\{\\mathrm\{noise\}\}andnblankn\_\{\\mathrm\{blank\}\}in the sequence\.

#### Active T\-Maze\[[29](https://arxiv.org/html/2610.10857#bib.bib29)\]\.

The agent is placed at the start positionx=1x=1in a corridor of lengthLLwith the oracle atx=0x=0and the T\-junction atx=L\+1x=L\+1\. The goalg∈\{\+1,−1\}\\texttt\{g\}\\in\\\{\+1,\-1\\\}is observable only at the oracle position \(cue observed atx=0→\(0,g\)x=0\\rightarrow\(0,\\texttt\{g\}\)\); elsewhere along the corridor the observation is the ambiguous\(0,0\)\(0,0\)\. At the T\-junction the observation is\(1,0\)\(1,0\)and the agent must choose to turn up or down based on the remembered cue\. The action is the 2D displacementat∈\{\(1,0\),\(−1,0\),\(0,1\),\(0,−1\)\}a\_\{t\}\\in\\\{\(1,0\),\(\-1,0\),\(0,1\),\(0,\-1\)\\\}\(right/left/up/down\)\.

1\+1\{\+\}g=±1\{\\pm\}1ss⋅\\cdot⋅\\cdot⋅\\cdotJ1−1\{\-\}wheressis the start,Jthe T\-junction,\{1\+,1−\}\\\{1\+,1\-\\\}the two goal arms, and the three dots betweenssandJare a schematic ellipsis denoting a variable\-length corridor ofLLambiguous\(0,0\)\(0,0\)cells; the agent must traverses→g→s→⋯→J→1±s\\\!\\to\\\!\\texttt\{g\}\\\!\\to\\\!s\\\!\\to\\\!\\cdots\\\!\\to\\\!\\texttt\{J\}\\\!\\to\\\!1\{\\pm\}correctly to succeed\. Example withL=4L=4andg=\+1\\texttt\{g\}=\+1is

The two keyframes𝒳key=\{o1,o6\}\\mathcal\{X\}\_\{\\mathrm\{key\}\}=\\\{o\_\{1\},o\_\{6\}\\\}are the cue exposure att=1t=1\(

\) and the T\-junction observation att=6t=6\(

\); the junction is itself a keyframe because the policy must use the\(1,0\)\(1,0\)token to know when to turn into a goal arm\. At the junction the agent must emit\(0,\+1\)\(0,\+1\)ifg=\+1\\texttt\{g\}=\+1and\(0,−1\)\(0,\-1\)ifg=−1\\texttt\{g\}=\-1\. The ambiguous\(0,0\)\(0,0\)corridor observations at all other timesteps \(

\) are background frames\. Evaluation is done by stretching the gap of\(0,0\)\(0,0\)corridor observations between the cuegand the junctionJ\.

### C\.2Grid environments

Both grid benchmarks share the MiniGrid\[[30](https://arxiv.org/html/2610.10857#bib.bib30)\]interface\.

![Refer to caption](https://arxiv.org/html/2610.10857v1/minigridmem_env.png)Figure 8:MiniGrid\-Memory environment\.The agent \(red triangle\) observes only a local \(7×77\\times 7\) view of the grid environment \(dotted red box\)\. It must navigate to the left room to observe the cue \(green ball\), then make a U\-turn and traverse the corridor to the junction, where it must decide whether to move up or down\. To succeed, the agent must move toward the same object observed in the left room\.![Refer to caption](https://arxiv.org/html/2610.10857v1/ltmb_env.png)Figure 9:LTMB\-Hallway environment\.The agent \(red triangle\) observes only a local \(7×77\\times 7\) view of the grid environment \(dotted red box\)\. It must remember the cue in the left room and then traverse the hallway while matching the remembered cue to the objects behind the locked doors\. To succeed, it must open and enter the door containing the same object as the one in the left room\. In the example above, the correct target is highlighted with a green circle\.#### MiniGrid\-Memory\[[30](https://arxiv.org/html/2610.10857#bib.bib30)\]\.

A greenKeyorBallis placed in a start room attached to a long corridor; the corridor ends in a T\-junction with oneKeyarm and oneBallarm \(order randomised\) \(Figure[8](https://arxiv.org/html/2610.10857#A3.F8)\)\. The agent spawns at a random cell along the corridor and must reach the arm matching the start\-room object; stepping into the wrong arm terminates with task failure\. The keyframe is the start\-room object observation \(cue\); all corridor observations are background frames\. Evaluation is done stretching the corridor length\.

#### LTMB\-Hallway\[[31](https://arxiv.org/html/2610.10857#bib.bib31)\]\.

A colored target object \(Key,Ball, orBoxin one of six colors\) is placed in the start room \(Figure[9](https://arxiv.org/html/2610.10857#A3.F9)\)\. A horizontal corridor leads past several branching vertical hallways, each containing random objects behind colored doors\. Exactly one branch object matches the target in both shape and color, and the agent must open and enter the corresponding hallway door to succeed\. Entering any other door results in task failure\. The keyframe is the start\-room target observation \(cue\), which must be remembered and matched with hallway targets, while the other corridor observations are background frames\. Evaluation is performed by increasing the number of distractor hallways\.

### C\.3Training dataset

All training datasets contain expert demonstrations only, with no failure trajectories\. This matches the requirements of Keyframe Mnemonics, which needs an expert dataset for training\. For each baseline we trained two variants per benchmark, differing only in the training\-distribution scale: a*small*variant matching the original demonstration distribution of𝒟\\mathcal\{D\}used for training Keyframe Mnemonics, and a*large*variant in which the trajectory horizon and the number of variations are scaled up\. Training sequence lengths and number of demonstrations \(\|𝒟\|\|\\mathcal\{D\}\|,NNin Lemma 1\) used in both two variants are summarized in Table[1](https://arxiv.org/html/2610.10857#A3.T1)\. Policies trained on both variants are evaluated on the same out\-of\-distribution test distribution, and the variant that achieves the higher test success rate \(or lower test MSE for Add\) is considered for benchmarking\. Some baselines benefit from the larger distribution and from more diverse training data \(e\.g\. DT and TrXL on Scattered\-Copy\), while others struggle on the*large*variant and generalize better when trained on the*small*dataset \(e\.g\. LSTM\-BPTT on T\-Maze\)\. See Appendix[C\.7](https://arxiv.org/html/2610.10857#A3.SS7)for more details\.

As discussed in Appendix[B\.2](https://arxiv.org/html/2610.10857#A2.SS2), we run multiple offline passes over the dataset𝒟\\mathcal\{D\}during masked\-sampling training of the proxy policy\. The number of passes is denoted bypassms\\mathrm\{pass\}\_\{\\mathrm\{ms\}\}in Table[1](https://arxiv.org/html/2610.10857#A3.T1)\. We determine this value by starting from a small constant and increasing it by orders of1010until the requirement onMMin Theorem 1 is satisfied\. The x\-axis multipliers in Figure[2](https://arxiv.org/html/2610.10857#S3.F2)scalepassms\\mathrm\{pass\}\_\{\\mathrm\{ms\}\}; for example,M=0\.1×M=0\.1\\timesfor the Add domain means thatpassms=0\.1×300=30\\mathrm\{pass\}\_\{\\mathrm\{ms\}\}=0\.1\\times 300=30passes were used to train the proxy policy\.

Table 1:Training and test sequence lengths and number of demonstrations\|𝒟\|\|\\mathcal\{D\}\|per dataset variant\. MiniGrid\-Memory and LTMB\-Hallway use a single dataset \(no*large*variant\)\. In the “Seq len” columns, “aa–bb” denotes a range \(lower–upper bound\) and “∼n\{\\sim\}n” denotes the maximum sequence length\. The last columnpassms\\mathrm\{pass\}\_\{\\mathrm\{ms\}\}reports the number of offline rollout passes used to create the masked\-sampling cache for proxy training \(Appendix[B\.2](https://arxiv.org/html/2610.10857#A2.SS2)\)\.
### C\.4Training Keyframe Mnemonics policies

Keyframe Mnemonics \(KM\) uses three networks per benchmark \(Algorithm[5](https://arxiv.org/html/2610.10857#alg5)\): a proxy policyfθf\_\{\\theta\}trained by masked\-sampling, a selectorπs\\pi\_\{\\mathrm\{s\}\}policy trained with PPO, and a BC policyπBC\\pi\_\{\\mathrm\{BC\}\}\. Per\-benchmark architecture and PPO hyperparameters are summarised in Tables[2](https://arxiv.org/html/2610.10857#A3.T2)and[3](https://arxiv.org/html/2610.10857#A3.T3)\.

Table 2:KM module architectures per benchmark\.kk: memory budget;dfeatd\_\{\\mathrm\{feat\}\}: selector feature dimension;dhidd\_\{\\mathrm\{hid\}\}: proxy / BC hidden width;HH: action\-chunk horizon;LL: proxy / BC hidden layers\. Cells with two valuesa/ba/bare proxy / BC policy\. Architecture of proxy and BC policy can be different per\-domain basis\.Table 3:KM selector optimization settings per benchmark\. PPO uses Adam with the SB3\[[60](https://arxiv.org/html/2610.10857#bib.bib60)\]defaults except where shown \(nenvsn\_\{\\mathrm\{envs\}\}parallel selector environments, total environment stepsppo, rollout lengthnstepsn\_\{\\mathrm\{steps\}\}, batch size,nepochsn\_\{\\mathrm\{epochs\}\}per update, learning rateηπ\\eta\_\{\\pi\}, and reward normalisation constantλ\\lambdafrom Equation[5](https://arxiv.org/html/2610.10857#S2.E5)\)\. The value ofλ\\lambdadepends on the type of loss function used for reward objective\. The proxy is trained with AdamW \(learning rateηf\\eta\_\{f\}, batch size\)\.In Table[2](https://arxiv.org/html/2610.10857#A3.T2), note that our formulation allows flexibility in the memory budgetkkused for proxy and BC policy training, since the inference buffer size can be adjusted after selector training to store a different number of keyframes than what was used during proxy training\. The memory budget for the proxy and BC policies is selected empirically for each benchmark, guided by the expected number of task\-relevant keyframes\. We try a small set of candidate values around this estimate and choose the smallestkkthat yields stable proxy policy convergence and produces qualitatively valid mnemonics after selector training\. See Appendix[G](https://arxiv.org/html/2610.10857#A7)for detailed analysis\.

Also note that in Table[2](https://arxiv.org/html/2610.10857#A3.T2), the proxy policy uses different network architectures across benchmarks\. We find that Theorem 1 holds generally, irrespective of network architecture; for instance, the synthetic domains use MLPs whereas grid\-world models use CNNs and Transformers\. Our choice of ReLU MLPs in synthetic domains is motivated by prior empirical evidence showing that such models exhibit a stronger simplicity bias\[[61](https://arxiv.org/html/2610.10857#bib.bib61)\]\.

### C\.5Baselines and benchmarking details

The five benchmarks \(Add, Scattered\-Copy, T\-Maze, MiniGrid\-Memory, LTMB\-Hallway\) share a common trainer pipeline; only the per\-baseline hyperparameters and per\-task input/output dimensions differ\. All baselines are trained with AdamW, a linear warm\-up cosine learning\-rate schedule \(10% of total steps warmup, decaying to1%1\\%of the peak rate\), and gradient clipping\. Our baseline design and implementation choices are derived from the open\-source code of RATE\[[45](https://arxiv.org/html/2610.10857#bib.bib45)\]\. Additionally, we tuned the training recipe to obtain the best possible baseline results on our benchmarking domains\.

Table 4:Trainable parameter counts per benchmark\. KM totals sum the proxy, selector and BC policy modules\. The baseline architectural choices that determine these counts \(layer widths, head counts, embedding dimensions\) are inherited from the RATE\[[45](https://arxiv.org/html/2610.10857#bib.bib45)\]open\-source implementation\. Units: K =10310^\{3\}, M =10610^\{6\}\.Table 5:Decomposition of the Keyframe Mnemonics parameter budget into the proxyfθf\_\{\\theta\}, the selector policyπs\\pi\_\{\\mathrm\{s\}\}, and the BC policyπBC\\pi\_\{\\mathrm\{BC\}\}\.
### C\.6Parameter\-budget comparison

Table[4](https://arxiv.org/html/2610.10857#A3.T4)compares the total trainable parameter count of every method per benchmark\. For Keyframe Mnemonics we report the sum of the proxy, selector, and BC policy parameters\. Despite using three networks, the KM total remains well below most of the baselines demonstrating that the architectural decomposition does not come at a parameter cost\. Table[5](https://arxiv.org/html/2610.10857#A3.T5)breaks the KM total down by module\. The selector policy is compact \(under1%1\\%of the total on the synthetic tasks and roughly15%15\\%on the grid tasks\), since it only encodes a single observation at a time; most of the budget sits in the proxy and, the BC policy\.

### C\.7Benchmark result analysis

We report per\-benchmark in\-distribution and out\-of\-distribution performance for every method in Table[6](https://arxiv.org/html/2610.10857#A3.T6)\. For each baseline we report the better of policy performance trained on*small*/*large*dataset variants \(Appendix[C\.3](https://arxiv.org/html/2610.10857#A3.SS3)\); Keyframe Mnemonics is trained only on the*small*variant \(except Scattered\-Copy that uses*large*666For Scattered\-Copy, the proxy policy \(Stage 1\) and selector policy \(Stage 2\) are trained on the*small*variant, while the BC policy \(Stage 3\) is trained on the*large*variant\.\)\. The selected variant per \(benchmark, baseline\) pair is as follows\. On Add, the*small*variant is used for LSTM\(BPTT\), DLSTM, and the*large*variant for LSTM\(TBPTT\), DT, DMamba, TrXL, and ELMUR\. On Scattered\-Copy, the*small*variant is used for LSTM\(BPTT\) and LSTM\(TBPTT\), and the*large*variant for all remaining baselines\. On T\-Maze, the*large*variant is used for LSTM\(TBPTT\), DT, and TrXL, and the*small*variant for all remaining baselines\. The grid benchmarks \(MiniGrid\-Memory and LTMB\-Hallway\) use a single dataset\.

Table 6:Aggregated benchmark results across all five tasks\. Cells show mean±\\pmstd across55seeds for Add, Scattered\-Copy, and T\-Maze, and33seeds for MiniGrid\-Memory and LTMB\-Hallway\. For Add, the metric is final\-step MSE \(↓\\downarrowbetter\); for all others, it is success rate \(↑\\uparrowbetter\)\. “—” marks the LSTM \(BPTT\) variant on the grid tasks, which we did not train because it required significantly more compute than the other baselines; only TBPTT was trained\. See Table[1](https://arxiv.org/html/2610.10857#A3.T1)for train and test sequence lengths and the number of demonstrations\|𝒟\|\|\\mathcal\{D\}\|used for training\.Random baseline omitted; its test values are±0\.0610\.499\\\!\\pm\\\!0\.061test MSE on Add,±0\.0000\.000\\\!\\pm\\\!0\.000test copy SR on Scattered\-Copy,±0\.0000\.000\\\!\\pm\\\!0\.000test SR on T\-Maze,±0\.0100\.030\\\!\\pm\\\!0\.010test SR on MiniGrid\-Memory, and±0\.0000\.000\\\!\\pm\\\!0\.000test SR on LTMB\-Hallway\.

#### Memory blur issue\.

In T\-Maze, sequence models such as LSTM \(BPTT\) and TrXL fit the training distribution well and achieve a train SR of1\.01\.0, but collapse to near\-chance performance on longer test horizons\. We observe a similar trend in the Add domain\. Recurrent memory baselines struggle in these domains because their hidden states or external memories are repeatedly updated over long horizons\. Keyframe Mnemonics avoids these failure modes by retaining only keyframes which are relevant for the task which remain unaffected to change in horizon length\.

A more severe case is Scattered\-Copy, which requires remembering multiple numbers and emitting them in the order they were observed\. This task is particularly difficult for baselines that repeatedly update hidden states or external memories, as such updates blur information over time and fail to generalize to longer OOD sequences\. In contrast, the KM selector learns to identify relevant keyframes during Stage 2 of training and stores only these keyframes in the buffer, making it easier for the BC policy to learn the copy operation over buffer elements while ignoring noise tokens in the sequence\. This highlights a key advantage of our method; since the buffer stores individual tokens, it avoids information blur\.

#### Non\-memory baseline\.

Among the baselines, Decision Transformer \(DT\) and DMamba do not include an explicit memory mechanism\. We include them for completeness in our benchmark evaluation\. The observed results are expected, as these models are limited by the context window length used during training; once the sequence length exceeds this window, relevant information falls out of context and the models can no longer predict the correct action\. Consequently, they perform well on the training distribution but degrade to near\-chance performance on the test distribution\.

### C\.8Hardware and wall\-clock

All training runs were executed on a single NVIDIA RTX 3090 \(2424GB\) GPU\. The three\-stage Keyframe Mnemonics pipeline \(proxy training, selector PPO, BC policy training\) completes across all five benchmarks in∼3\\sim 3h per seed, with the grid benchmarks accounting for the majority of the wall\-clock time\. The cumulative training time for all baselines on a single synthetic benchmark is under∼30\\sim 30minutes per seed\. The cumulative baseline training time is∼4\\sim 4h per seed on MiniGrid\-Memory and∼6\\sim 6h per seed on LTMB\-Hallway\.

## Appendix DRobot Manipulation Domain Details

This section provides details on the hyperparameters and architectural choices used to produce theMIKASA\-Roboresults in Section[3](https://arxiv.org/html/2610.10857#S3)\. For theRobomimicdomain experiments \(Figure[5\(b\)](https://arxiv.org/html/2610.10857#S3.F5.sf2)\), we use the same configuration asMIKASA\-Robo\. Values not stated here take the standard defaults of the underlying libraries\.

### D\.1MIKASA\-Roboenvironments

A set of six representative tasks from the benchmark is shown in Figure[10](https://arxiv.org/html/2610.10857#A4.F10)\.

![Refer to caption](https://arxiv.org/html/2610.10857v1/mikasarobo_env.png)Figure 10:MIKASA\-Robo Environments\.A collage of six representative tasks from theMIKASA\-Robomemory\-intensive robot manipulation benchmark\. The benchmark evaluates robot policies across comprehensive memory categories, including object memory, spatial memory, sequential memory, and memory capacity\. More details can be found in the reference paper\[[7](https://arxiv.org/html/2610.10857#bib.bib7)\]\.
### D\.2Pipeline overview

EachMIKASA\-Robotask is trained in three stages \(Algorithm[5](https://arxiv.org/html/2610.10857#alg5)\), following Sections[2\.1](https://arxiv.org/html/2610.10857#S2.SS1)–[2\.3](https://arxiv.org/html/2610.10857#S2.SS3)of the main paper\.\(i\)The*proxy policy*fθf\_\{\\theta\}is trained under masked\-sampling of past observations, after which its frozen prediction loss serves as the keyframe\-discovery objectiveJθ∗​\(ℬ\)J\_\{\\theta^\{\*\}\}\(\\mathcal\{B\}\)\.\(ii\)The*selector policy*πs\\pi\_\{\\mathrm\{s\}\}is trained with PPO against the reward derived fromJθ∗J\_\{\\theta^\{\*\}\}, learning to assign high priority to keyframes\.\(iii\)The*BC policy*πBC​\(at∣ot,ℬt\)\\pi\_\{\\mathrm\{BC\}\}\(a\_\{t\}\\mid o\_\{t\},\\mathcal\{B\}\_\{t\}\)is trained on the augmented dataset𝒟†\\mathcal\{D\}^\{\\dagger\}, in which every observationoto\_\{t\}is paired with the priority queueℬt\\mathcal\{B\}\_\{t\}produced byπs\\pi\_\{\\mathrm\{s\}\}\. All three stages use only the offline expert demonstrations𝒟\\mathcal\{D\}; the simulator is never queried during proxy or selector training\. The memory budget is fixed atk=3/2k=3/2\(proxy / BC policy\) throughout, so thatπBC\\pi\_\{\\mathrm\{BC\}\}sees the current observation alongside two mnemonics drawn from the queue\.

### D\.3Architectures and parameter counts

All modules share aResNet\-18 visual encoder, initialised fromImageNetweights and then trained end\-to\-end\. Camera frames are128×128128\{\\times\}128RGB; the proprioceptive state concatenates the TCP pose, joint positions, and joint velocities into a2525\-dimensional vector\. Both overhead and wrist cameras are used\. Table[7](https://arxiv.org/html/2610.10857#A4.T7)summarises the architecture and parameter count of each module\.

Table 7:Module architectures, dimensions and parameter counts\.dd: hidden width,hh: attention heads,LL: transformer layers,HH: action\-chunk length\. The selector and proxy are training\-only; only one BC\-policy variant is deployed per task\.The single\-step regression head \(πBCSST\\pi\_\{\\mathrm\{BC\}\}^\{\\text\{SST\}\}\) is the default BC\-policy\. We substitute a flow\-matching transformer \(πBCFM\\pi\_\{\\mathrm\{BC\}\}^\{\\text\{FM\}\}\) only on the fourIntercept\*dynamics tasks \(InterceptMediumandInterceptGrab\{Slow,Medium,Fast\}\), where the higher\-frequency control profile benefits from action chunking of lengthH=8H=8\. The flow\-matching backbone is a cross\-attention transformer in which action tokens self\-attend and then cross\-attend to visual and proprioceptive context tokens\.

### D\.4Optimizer and schedule

The proxy and BC policies are trained withAdamW\[[62](https://arxiv.org/html/2610.10857#bib.bib62)\]; the selector is trained withPPO\[[39](https://arxiv.org/html/2610.10857#bib.bib39)\]using Adam \(Table[8](https://arxiv.org/html/2610.10857#A4.T8)\)\. Actions are predicted directly in the simulator’s\[−1,1\]\[\-1,\\,1\]range without output normalization\.

Table 8:Optimizer settings for the three training stages\.
### D\.5Stage\-1: Proxy policyfθf\_\{\\theta\}

The proxy policy is trained for6060epochs over the demonstration dataset𝒟\\mathcal\{D\}with batch size128128\. We use online masked\-sampling \(Appendix[B\.2](https://arxiv.org/html/2610.10857#A2.SS2)\) and at every gradient step we sample a random binary maskΨk\\Psi\_\{k\}over the past observationshth\_\{t\}with cardinality bounded bykk, forming the masked memory bufferℬt=\{oi∈ht∣Ψk​\(i\)=1\}\\mathcal\{B\}\_\{t\}=\\\{o\_\{i\}\\in h\_\{t\}\\mid\\Psi\_\{k\}\(i\)=1\\\}\(Section[2\.1](https://arxiv.org/html/2610.10857#S2.SS1)\)\. Conditioned onℬt\\mathcal\{B\}\_\{t\}, the proxy is trained to reconstruct the expert action chunk at horizonH=8H=8under L1 loss\.

### D\.6Stage\-2: Selector policyπs\\pi\_\{\\mathrm\{s\}\}

The selector policy is a PPO actor\-critic with the hyperparameters of Table[9](https://arxiv.org/html/2610.10857#A4.T9)\. The reward at every step is the normalised log objective defined in Section[2\.2](https://arxiv.org/html/2610.10857#S2.SS2), evaluated on the post\-update memory bufferℬt\\mathcal\{B\}\_\{t\}\. We do not use surprise coefficient for robot manipulation domains\. We clip the proxy loss inside the log\-reward atλ=10−5\\lambda=10^\{\-5\}to prevent unbounded rewards near the proxy’s convergence\. The selector observes the same RGB and proprioceptive state input as the proxy policy, and emits a continuous priority scorept∈\[0,1\]p\_\{t\}\\in\[0,1\]per observation; the priority queue uses this score to updateℬt\\mathcal\{B\}\_\{t\}fromℬt−1\\mathcal\{B\}\_\{t\-1\}\(Algorithm[2](https://arxiv.org/html/2610.10857#alg2)\)\. We train the selector for5×1065\\,\{\\times\}\\,10^\{6\}environment steps acrossnenvs=32n\_\{\\mathrm\{envs\}\}=32parallel selector environments\.

Table 9:PPO hyperparameters used to train the selector policyπs\\pi\_\{\\mathrm\{s\}\}\.Table 10:Success rate \(mean±\\pmsem over 3 seeds, 100 episodes per seed\) on the MIKASA\-Robo benchmark\.Boldmarks every method whose mean is within one standard deviation of the best per row\.
### D\.7Stage\-3: BC policyπBC\\pi\_\{\\mathrm\{BC\}\}

Optimization budget\.The regression head \(πBCSST\\pi\_\{\\mathrm\{BC\}\}^\{\\text\{SST\}\}\) is trained for15,00015\{,\}000gradient steps; the flow\-matching variant \(πBCFM\\pi\_\{\\mathrm\{BC\}\}^\{\\text\{FM\}\}\) is trained for100,000100\{,\}000gradient steps\. Batch size is128128throughout, and the final checkpoint is used for evaluation\.

Flow\-matching scheduler\.We adopt the Euler discrete flow\-matching scheduler\[[63](https://arxiv.org/html/2610.10857#bib.bib63)\]with200200training timesteps sampled uniformly at random\. At inference we run55Euler steps; we found this sufficient on everyIntercept\*task\.

Closed\-loop control\.The BC policy emits an action chunk of lengthHHat every environment step, but only the first action of the chunk is executed before re\-querying\.

### D\.8Camera\-image data augmentation

We apply augmentation to all camera images in the buffer during BC policy training\. Each of the following operations is applied independently with probabilityp=0\.5p=0\.5: brightness jitter\[0\.8,1\.2\]\[0\.8,\\,1\.2\], contrast jitter\[0\.8,1\.2\]\[0\.8,\\,1\.2\], saturation jitter\[0\.5,1\.5\]\[0\.5,\\,1\.5\], hue jitter\[−0\.05,0\.05\]\[\-0\.05,\\,0\.05\], sharpness adjustment \(factor1\.51\.5\), and a random affine transform with up to10∘10^\{\\circ\}rotation,±10%\\pm 10\\%translation, and±10%\\pm 10\\%scaling\. The proprioceptive state vector is not augmented\. We find this augmentation crucial for training the BC policy to avoid causal confusion\[[15](https://arxiv.org/html/2610.10857#bib.bib15)\]induced by the larger context size\.

### D\.9MIKASA\-Roboall results

Averaged over the 23 non\-zero tasks, Keyframe Mnemonics outperforms Decision Transformer by\+40\.4%\+40\.4\\%, Conservative Q\-Learning by\+42\.5%\+42\.5\\%, Behavior Cloning by\+37\.1%\+37\.1\\%,RATEby\+30\.6%\+30\.6\\%, the diffusion\-policy baseline by\+36\.6%\+36\.6\\%, andELMURby\+13\.9%\+13\.9\\%absolute success rate\. See Table[10](https://arxiv.org/html/2610.10857#A4.T10)for complete results\.

### D\.10Hardware and wall\-clock

Training runs were conducted on a mix ofNVIDIAA4040\(4848GB\) and L4040S \(4646GB\) GPU instances\. Per\-stage wall\-clock, averaged across the2323non\-zeroMIKASA\-Robotasks and rounded, is approximately1\.51\.5h for the proxyfθf\_\{\\theta\}\(6060epochs\),∼10\\sim\\\!10h for the selectorπs\\pi\_\{\\mathrm\{s\}\}\(5×1065\\,\{\\times\}\\,10^\{6\}environment steps\), and∼1\.5\\sim\\\!1\.5h for the BC policyπBCSST\\pi\_\{\\mathrm\{BC\}\}^\{\\text\{SST\}\}\(15,00015\{,\}000gradient steps\)\. End\-to\-end training therefore takes roughly half a day per SST task \(∼13\\sim\\\!13h\)\. The flow\-matching variantπBCFM\\pi\_\{\\mathrm\{BC\}\}^\{\\text\{FM\}\}trains for100,000100\{,\}000gradient steps and takes proportionally longer at the BC stage \(∼10\\sim\\\!10h\), bringing the fourIntercept\*flow\-matching tasks to roughly a full day each\.

## Appendix EExperimental Details

Table 11:Horizon\-invariance results across synthetic and grid benchmarks\. Cells report mean±~\\pm~std of the relevant metric: final\-step MSE for Add \(↓\\downarrowbetter\), Copy SR for Scattered\-Copy and Episode SR for T\-Maze, MiniGrid\-Memory and LTMB\-Hallway \(↑\\uparrowbetter\)\. Each benchmark lists the horizons \(or environment size for MiniGrid\-Memory\) evaluated\. “—” marks cells for baselines in extreme\-horizon experiments as we run it only for Keyframe Mnemonics, or for the LSTM\(BPTT\) variant on grid tasks, which were trained only with LSTM\(TBPTT\)\.Random baseline omitted; its test SR remains low across every \(benchmark, horizon\) cell, consistent with performance shown in Table[6](https://arxiv.org/html/2610.10857#A3.T6)\.

### E\.1Horizon\-invariance experiments on synthetic and grid domains

Table[11](https://arxiv.org/html/2610.10857#A5.T11)reports the consolidated horizon\-invariance results underlying Figure[3](https://arxiv.org/html/2610.10857#S3.F3)\. We evaluated the trained checkpoints from Appendix[C\.7](https://arxiv.org/html/2610.10857#A3.SS7), by instantiating the test environment with a target horizonhhtaken from a pre\-defined set per benchmark: Add, Scattered\-Copy, T\-Maze episode lengthh∈\{10,20,30,1000,10000\}h\\in\\\{10,20,30,1000,10000\\\}; Keyframe Mnemonics is additionally evaluated ath∈\{105,106\}h\\in\\\{10^\{5\},10^\{6\}\\\}to test horizon\-invariance behavior\. MiniGrid\-Memory environment max sizeh∈\{17,51,101,501,1001\}h\\in\\\{17,51,101,501,1001\\\}\. LTMB\-Hallway corridor lengthh∈\{8,50,100,500,1000\}h\\in\\\{8,50,100,500,1000\\\}\. Per\-cell metrics are mean±~\\pm~std across55random seeds for the synthetic tasks and33random seeds for the grid tasks\. For each horizon, every method is evaluated on55episodes for the synthetic tasks \(Add, Scattered\-Copy, T\-Maze\) and1010episodes for the grid tasks \(MiniGrid\-Memory, LTMB\-Hallway\) per seed\.

Within the training horizon range, several baselines achieve strong performance; for example, LSTM\(BPTT\), DT, and TrXL reach1\.01\.0SR on T\-Maze up toh=30h=30\. However, baseline methods collapse once the test horizon exceeds the training horizon by even one order of magnitude\. In contrast, Keyframe Mnemonics consistently exhibits horizon\-invariant performance\. In particular, on synthetic tasks, it generalizes up to five orders of magnitude beyond the training horizon, confirming the horizon\-invariance guarantee of Theorem 2 in regimes where the underlying assumptions hold strongly\.

### E\.2Horizon\-invariance experiment onRememberColor3\-Δ​T\\Delta T

![Refer to caption](https://arxiv.org/html/2610.10857v1/images/hover.png)Figure 11:Hover time is varied across evaluations resulting in catastrophic failure for the baselines, while KM maintains it’s performance even at250250timesteps of hover\. Training data had 15 timesteps of hover\.The horizon\-invariance study \(Figure[5\(a\)](https://arxiv.org/html/2610.10857#S3.F5.sf1)\) trains every method at a fixed cue\-to\-action delayΔ​T=15\\Delta T=15and evaluates the trained policies, without retraining, at delaysΔ​T∈\{15,35,55,75,95,115,135,150,200,250\}\\Delta T\\in\\\{15,35,55,75,95,115,135,150,200,250\\\}\. The training set is built from the releasedRememberColor3\-v0demonstrations, whose default cue\-to\-action delay isΔ​Torig=5\\Delta T\_\{\\text\{orig\}\}=5\.

Hover\-augmented data\.We re\-replay each successful expert demonstration in a liveManiSkillsimulator with the cue\-to\-action delay extended toΔ​T=15\\Delta T=15and a1010\-step*extra hover*phase spliced into the original action tape immediately after the cue disappears \(Figure[11](https://arxiv.org/html/2610.10857#A5.F11)\)\. During the hover phase the gripper is held and small net\-zero noise is added to the arm for the entire delay time\.

Training on the hover dataset\.We run the standard three\-stage KM pipeline of Appendix[D\.2](https://arxiv.org/html/2610.10857#A4.SS2)on the hover trajectories\. Every hyperparameter matches the corresponding entries forRememberColor3in Tables[7](https://arxiv.org/html/2610.10857#A4.T7)\(k=3k=3, proxy chunkH=8H=8, SST policy withH=1H=1,15,00015\{,\}000BC gradient steps\)\.

We trainPTPusing their official implementation on our dataset with prediction horizon of1616, observation length of88and a sub\-sampling rate of88to roughly cover the entire episode of length7070\. This indicates why we see steeper deterioration in the performance ofPTPon and after7575steps of time delay\. We trainELMURon the same trajectories using its publishedRememberColor3recipe, with the context length extended to cover the full7070\-step hover episode\. All other hyperparameters \(architecture, optimizer, schedule, batch size, number of epochs\) are left at the authors’ defaults\.

Horizon\-invariance evaluation\.The trained policy is evaluated at every target delayΔ​T\\Delta Tby re\-instantiating the simulator with the requestedΔ​T\\Delta T; no retraining or fine\-tuning is performed at anyΔ​T≠15\\Delta T\\neq 15\. Following theMIKASA\-Roboconvention, an episode is counted as a success only if the binary task\-completion reward is reached before the time budget elapses\. We report mean±\\pmstandard error across three independent environment seeds,100100episodes per seed\. Results are plotted in Figure[5\(a\)](https://arxiv.org/html/2610.10857#S3.F5.sf1)and details can be found in Table[12](https://arxiv.org/html/2610.10857#A5.T12)\.

Table 12:RememberColor3\-Δ​T\\Delta Tsuccess rate under cue\-to\-action delays\. Training delay isΔ​T=15\\Delta T=15; all larger values are out\-of\-distribution\. Mean±~\\pm~sem over33seeds,100100episodes per seed\.Results in Table[12](https://arxiv.org/html/2610.10857#A5.T12)show that KM preserves horizon\-invariant performance in memory\-intensive robot manipulation with high\-dimensional visual data\. ELMUR matches the other methods at the training delay \(Δ​T=15\\Delta T=15\), but its external memory is overwritten as the delay increases, causing success to collapse to zero for everyΔ​T≥35\\Delta T\\geq 35\. PTP retains its training\-time performance up to roughlyΔ​T=55\\Delta T=55and then degrades monotonically, dropping below10%10\\%byΔ​T=200\\Delta T=200\. In contrast, Keyframe Mnemonics remains nearly flat across the longer horizons, retaining≈0\.65\\approx 0\.65success rate atΔ​T=250\\Delta T=250, i\.e\.,∼⁣×\\sim 17\\\!\\timesthe training delay, without any architectural change or fine\-tuning\.

### E\.3Reproducibility

Released code, per\-task configurations, and trained checkpoints, together with launch scripts, are available at[https://keyframe\-mnemonics\.github\.io](https://keyframe-mnemonics.github.io/)\.

## Appendix FAblation Study

We report ablation results over 5 seeds for the synthetic domains and 3 seeds for the grid and robot domains unless noted, with 100 evaluation episodes per seed\. All domains are evaluated at out\-of\-distribution horizons\.

### F\.1Learned selector against a random selector

We replace the learned selector with a random selectorπrnd\\pi\_\{\\mathrm\{rnd\}\}that assigns prioritiesp∼𝒰⁡\(0,1\)p\\sim\\mathcal\{U\}\(0,1\)and train the BC policy using Stage 3 \(Algorithm[5](https://arxiv.org/html/2610.10857#alg5)\)\. The resulting policy performs worse at every delay interval, demonstrating that the learned selector is critical for the memory task\.

Table 13:Learned selector against the random selectorπrnd\\pi\_\{\\mathrm\{rnd\}\}on MIKASA\-Robo RememberColor3\-Δ​T\\Delta T\. Reported mean±\\pmsem success rate \(%\)↑\\uparrow\.
### F\.2Selector discount factor

We vary the discount factorγ\\gammaof the PPO selector to optimize over a frozen proxy policy\. Atγ=0\\gamma=0the selector greedily optimizes the per\-step proxy loss\. For T\-Maze we report the active \(default\) and passive variations\[[64](https://arxiv.org/html/2610.10857#bib.bib64)\]\. Greedy optimization fails atγ=0\\gamma=0on every domain except active T\-Maze, and performance recovers asγ\\gammaincreases\. In active T\-Maze, the action changes right after the keyframe is observed, so the immediate change in proxy loss is sufficient\. In the other domains, the gap between the keyframe and the decision\-critical action spans multiple steps, and a higherγ\\gammais necessary\. Keyframe identification from the proxy loss is therefore a delayed credit\-assignment problem, which explains our choice of using RL for selector training \(Section[2\.2](https://arxiv.org/html/2610.10857#S2.SS2)\)\.

Table 14:Effect of the PPO discount factorγ\\gamma\. Reported mean±\\pmsd\.
### F\.3Surprise coefficient

We change the temperatureκ\\kappaof the surprise coefficient in Equation[5](https://arxiv.org/html/2610.10857#S2.E5)\. Off in Table[15](https://arxiv.org/html/2610.10857#A6.T15)corresponds to an unscaled reward, which is theκ→∞\\kappa\\to\\inftylimit where surprise coefficient equals 1\. For every task we first train with the unscaled reward and add surprise scaling only if it improves performance\.κ=0\.2\\kappa=0\.2has the widest usable range and is our default for the tasks that need scaling \(Add, Scattered\-Copy and MiniGrid\-Memory\)\.

Table 15:Effect of the surprise coefficient across temperatures\. Bold marks the best setting per domain\. Reported mean±\\pmsd\.
### F\.4Current observation in proxy training

Includingoto\_\{t\}during proxy training collapses MiniGrid\-Memory and RememberColor3 \(Table[16](https://arxiv.org/html/2610.10857#A6.T16)\), as the proxy overfits to the current observation\. This is consistent with Proposition 1, which requires that no low\-complexity non\-invariant shortcut exists in the proxy training data\. The current observation is helpful for learning the final policy, but not for learning a reward function for keyframe identification\.

Table 16:Effect of including the current observationoto\_\{t\}in the proxy input\. Reported mean±\\pmsd\.
### F\.5Queue tie\-break rule

The memory buffer uses a FIFO queue \(Algorithm 2\) during selector training and a LIFO queue \(Algorithm 3\) at inference\. The two differ in which slot is evicted when several observations are tied at the lowest priority\. We vary this rule independently at training and inference\. On the synthetic and grid domains, all four combinations are identical \(Table[17](https://arxiv.org/html/2610.10857#A6.T17)\)\. The rule matters on RememberColor3\-Δ​T\\Delta T, where the default FIFO→\\rightarrowLIFO beats its reverse at 9 of 10 retention intervals \(Table[18](https://arxiv.org/html/2610.10857#A6.T18)\)\.

Table 17:Queue tie\-break rule, train→\\rightarrowinference\. Reported mean±\\pmsd\.Table 18:Queue tie\-break rule on RememberColor3\-Δ​T\\Delta T\. Reported mean±\\pmsem success rate↑\\uparrow\.
### F\.6Memory buffer size

We scale the memory buffer sizekkto0\.5×0\.5\\timesand2×2\\timesthe default value and retrain all three stages\. Halvingkkdegrades performance across all domains, while doubling it helps on some domains and hurts on others\. A largerkkcan make the KM policy harder to train, and our default values serve as a minimum\-capacity floor, which were chosen per domain independently\.

Table 19:Effect of scaling the memory buffer sizekk\. Reported mean±\\pmsd\.

## Appendix GAdditional Analysis

### G\.1Interpretability of discovered keyframes

For a well\-trained selector, themnemonicsbufferℬt\\mathcal\{B\}\_\{t\}should accumulate keyframes that are information\-critical for the task and provide necessary context for the downstream BC policy to generate actions unambiguously\. Figure[6](https://arxiv.org/html/2610.10857#S3.F6)shows the buffer contents for our visual benchmark domains\. We observe that the buffer keyframes qualitatively capture the belief state for each task by retaining observations with task\-relevant cues\. The discovery process is fully self\-supervised, as the selector is trained only from expert demonstration state\-action pairs without any additional temporal keyframe annotation\. These qualitative results highlight the interpretability of Keyframe Mnemonics: the buffer contains prior observations that can be inspected to understand what the BC policy relies on when taking a particular action in a partially observable environment\.

### G\.2Selector priority behavior

In Figure[12](https://arxiv.org/html/2610.10857#A7.F12)we evaluate trained selector policyπs\\pi\_\{\\mathrm\{s\}\}across synthetic domains to qualitatively assess how well the learned priorities align with the oracle keyframes defined in Appendix[C\.1](https://arxiv.org/html/2610.10857#A3.SS1)\. We find that the selector learns a near\-optimal selection function that agrees with the oracle, highlighting that the proxy objective provides a strong signal for keyframe discovery, consistent with Corollary 1\. During Stage 2 RL training \(Algorithm[4](https://arxiv.org/html/2610.10857#alg4)\), the selector explores priority assignments and learns to commit high priorities to decision\-relevant keyframes\. This qualitatively explains the horizon\-invariant results in Table[11](https://arxiv.org/html/2610.10857#A5.T11), where BC policies achieve perfect performance over horizons orders of magnitude beyond training\. Since the selector is Markovian and depends only on the current observation, its behavior remains invariant to horizon length, allowing it to admit only informative keyframes into the buffer at inference time\.

![Refer to caption](https://arxiv.org/html/2610.10857v1/paper_priority_timeline.png)Figure 12:Trained selector recovers the keyframe oracle as a near\-optimal function\.Per\-step selector priorities \(bottom row of each pair\) are compared against the oracle keyframe indicator \(top row of each pair\) on three synthetic memory benchmarks\. Each benchmark is evaluated at three increasing episode horizons, where the longest horizon extends one order of magnitude beyond the training horizon\. For each panel, we render1010evaluation episodes \(rows\) over time \(columns\), where each column corresponds to a single environment step and color encodes either the selector output \(pt∈\[0,1\]p\_\{t\}\\in\[0,1\]\) or the binary oracle indicator \(11for keyframes and00otherwise\)\. The oracle definition is problem\-specific, as discussed in Appendix[C\.1](https://arxiv.org/html/2610.10857#A3.SS1)\. Across all cells, the selector priority agrees with the oracle\. After squashing the raw selector outputs to\[0,1\]\[0,1\], the mean selector priority is exactly1\.001\.00on keyframes and0\.000\.00on background frames\. This empirical result shows that the selector policyπs\\pi\_\{\\mathrm\{s\}\}learns a near\-optimal keyframe selection function for domains where our assumptions hold strongly\.
### G\.3BC Policy activation map

To understand whether the discovered keyframes are useful to the behavior cloning policy, we use Grad\-CAM\[[65](https://arxiv.org/html/2610.10857#bib.bib65),[66](https://arxiv.org/html/2610.10857#bib.bib66)\], an explainable AI technique commonly used to visualize which parts of the input a policy focuses on when generating actions\. In Figure[13](https://arxiv.org/html/2610.10857#A7.F13), we visualize activation maps at decision\-critical timesteps in grid and robot manipulation domains, where the BC policy must rely on memory\-buffer information to generate the correct action\. We observe that the BC policy attends to information cues from both the buffer and the current observation to infer the POMDP belief state and act correctly\. In the grid domain, the heatmap is concentrated around the matching object cue\. InRememberColor3, activations concentrate around the blue colored cube in both the keyframe and the current observation\. InTakeItBack, activations appear around the initial cube location in the first keyframe, while the second keyframe captures temporal task\-progress information just before the robot begins pushing the cube, helping disambiguate the state so that the policy pulls the cube back to its initial location\. These results show that the keyframes discovered by our method are decision\-relevant and provide necessary context for BC policies to disambiguate actions under partial observability, even in high\-dimensional robot visual domains\.

![Refer to caption](https://arxiv.org/html/2610.10857v1/activation_map.png)Figure 13:BC policy activation map on mnemonics\.Grad\-CAM\[[65](https://arxiv.org/html/2610.10857#bib.bib65),[66](https://arxiv.org/html/2610.10857#bib.bib66)\]activation maps of the behavior cloning policyπBC\\pi\_\{\\mathrm\{BC\}\}on buffer images at decision\-critical steps, where hot regions indicate inputs driving the predicted action\. Frames withboundaries represent discovered keyframes, while the frame with aboundary shows the agent’s current observation\. Across both grid and high\-dimensional visual domains, the BC policy uses task\-specific cue information from thediscoveredkeyframes to generate actions\. This highlights the utility of mnemonics for disambiguating policy actions under partial observability\. See[https://keyframe\-mnemonics\.github\.io](https://keyframe-mnemonics.github.io/)for activation map videos over policy rollouts\.
### G\.4Regret on low memory budgetkk

Figure[14](https://arxiv.org/html/2610.10857#A7.F14)analyzes how memory regret \(Section[3\.1](https://arxiv.org/html/2610.10857#S3.SS1)\) changes with buffer capacitykkfor synthetic domains\. In T\-Maze, active regret remains high whenk=1k=1\(⌈0\.5×2⌉\\lceil 0\.5\\times 2\\rceil\), which is below the true number of keyframes \(22\), because the buffer cannot retain all required keyframes\. In Add, active regret becomes negative whenk=2k=2\(⌈0\.5×3⌉\\lceil 0\.5\\times 3\\rceil\), also below the true keyframe count \(33\)\. This occurs because the selector prioritizes the two marked numbers over the emit step, whereas the oracle assigns equal priority to all three keyframes\. Under the low memory budget, the oracle evicts the first marked step, leading to higher proxy loss\. It shows that the selector can learn a non\-uniform priority policy that favors keyframes with higher mutual information with the action\. For Scattered\-Copy, active regret remains close to zero since the oracle and selector perform similarly across memory budgets\. Passive regret remains low when the buffer size is smaller than the true keyframe count because the proxy cannot converge to an invariant hypothesis under constrained memory\. At the2×2\\timesbuffer size, passive regret is lower than at1×1\\times, since a larger buffer increases the probability that a random selector retains keyframes\.

Figure 14:Memory regret as a function of buffer capacitykk\.The1×1\\timessetting corresponds to the proxy training memory budgetkkin Table[2](https://arxiv.org/html/2610.10857#A3.T2)\.\(a\)Active regret decreases as buffer capacity becomes sufficient to retain the true keyframes, with domain\-specific behavior whenkkis below the true keyframe count\.\(b\)Passive regret remains low under insufficient capacity and decreases again for larger buffers, where random selection is more likely to retain keyframes\.
### G\.5PPO reward plots

Figure[15](https://arxiv.org/html/2610.10857#A7.F15)shows the learning curves for training the selector policyπs\\pi\_\{\\mathrm\{s\}\}on the frozen proxy objective \(Section[2\.2](https://arxiv.org/html/2610.10857#S2.SS2)\) using PPO\[[39](https://arxiv.org/html/2610.10857#bib.bib39)\]\. For synthetic domains where our assumptions hold strongly, we observe clear learning dynamics, with the selector converging within200200K environment \(Algorithm[4](https://arxiv.org/html/2610.10857#alg4)\) steps \. For robot manipulation domains, the selector converges within55M steps\. The plot forChainOfColorsillustrates a task that is more difficult to optimize, exhibiting a noisy learning curve\. We observed that such noise typically appears when the proxy policy does not converge properly, either because it is undertrained or because our assumptions do not hold well in the environment\.

![Refer to caption](https://arxiv.org/html/2610.10857v1/reward.png)Figure 15:Selector training PPO learning curves\.The selector policy converges within200200K environment steps for synthetic domains and55M steps for robot manipulation domains\. For tasks where optimization is difficult, e\.g\.,ChainOfColors, the learning curve is noisy and shows modest reward improvement, indicating that some of our assumptions may not hold strongly in the environment\.
### G\.6Limitation discussion

Our Assumptions 1 is restrictive, as it requires keyframes to be individually identifiable from a single observation because the selector policyπs​\(pt∣ot\)\\pi\_\{\\mathrm\{s\}\}\(p\_\{t\}\\mid o\_\{t\}\)is Markovian\. This formulation allows selector training to scale without recurrent modules, thereby avoiding backpropagation through time\. Hence, our method is best suited for sparse, frame\-localizable memory tasks\. However, problems that do not satisfy the assumption may require multiple keyframes to represent the same underlying information, leading to multiple information\-equivalent buffer states that achieve similar objective values under the proxy policy\. In such cases, it becomes challenging to retain only one mnemonic set among several equivalent alternatives\.

Our Assumption 2 requires action ambiguity to be strongly correlated with a sparse set of keyframes\. In other words, the action distribution must exhibit sufficient multimodality that can be disambiguated by the remembered keyframes\. In tasks where actions are mostly local and exhibit low variance, actions may carry little mutual information with the buffer observations\. In such a regime, the mutual information gap in Theorem 1 becomes negligible, and the proxy policy may fail to converge because the invariant hypothesis is difficult to learn from a sparse memory buffer\. We observed this failure mode on the VizDoom\-Two\-Colors\[[22](https://arxiv.org/html/2610.10857#bib.bib22)\]task, where the agent observes a color cue and must later choose the corresponding target, but the discrete navigation actions remain largely local\.

We found that the proxy policy struggles to converge to the true invariant hypothesis when expert actions at decision\-critical timesteps depend simultaneously on multiple unique keyframes\. This occurs because the probability that masked\-sampling selects all relevant keyframes during proxy training decreases as the number of action\-relevant keyframes increases, making it difficult to learn tasks requiring high memory capacity\. A similar issue arises in very long demonstrations with few keyframes, where keyframe sparsity further reduces the probability of sampling the relevant frames\. A straightforward way to address this issue is to increase the training memory budget, which makes sampling keyframes into the buffer more likely and also increases the support sizeMM, thereby enforcing Theorem 1 more effectively\. Another promising direction is to train proxy policies on shorter low\-memory\-capacity tasks, learn specialized selectors for each, and compose them to support high\-memory\-capacity BC policies, which we leave for future work\.

The PPO training time dominates the computational cost in robot\-domain experiments\. Our current implementation uses SB3\[[60](https://arxiv.org/html/2610.10857#bib.bib60)\], which introduces nontrivial CPU\-GPU overhead and becomes the main bottleneck\. The selector training environment \(Algorithm 4\) primarily requires proxy policy inference, which can in principle be executed entirely on the GPU\. We therefore view this as an engineering limitation\. In future work, we plan to use a GPU\-native PPO implementation to accelerate selector training\.

## Appendix HExtended Related Works

Simplicity\-bias in neural networks\.Simplicity\-bias is a well\-studied property of neural networks, reflecting their tendency to prefer simpler patterns over more complex ones\[[25](https://arxiv.org/html/2610.10857#bib.bib25),[26](https://arxiv.org/html/2610.10857#bib.bib26),[67](https://arxiv.org/html/2610.10857#bib.bib67),[68](https://arxiv.org/html/2610.10857#bib.bib68),[27](https://arxiv.org/html/2610.10857#bib.bib27)\]\. Recent work has leveraged simplicity bias to improve RL performance\[[69](https://arxiv.org/html/2610.10857#bib.bib69)\], but this area remains largely underexplored, particularly in how such biases can be effectively utilized\. We present one of the first methods that explicitly leverage this bias for behavior cloning in POMDP domains\.

Masking\-induced invariant feature learning\.Prior works have used masking in self\-supervised learning over language tokens\[[70](https://arxiv.org/html/2610.10857#bib.bib70)\]and spatial image patches\[[71](https://arxiv.org/html/2610.10857#bib.bib71)\]to learn invariant features from input data\. Our keyframe discovery process instead applies masking over the temporal history of individual observations\. We hypothesize that this induces a similar regularization effect: random input sampling increases the difficulty of learning complex, non\-invariant features, thereby biasing the model toward invariant hypotheses that better capture true correlations over spurious ones\.

Language as a proxy for state tracking\.Recent methods have used segment\-level language\-annotated demonstrations to finetune vision\-language models \(VLMs\) that generate autoregressive subgoal commands for policy steering\[[72](https://arxiv.org/html/2610.10857#bib.bib72),[73](https://arxiv.org/html/2610.10857#bib.bib73)\]\. While language provides high\-level semantic guidance, it lacks the spatio\-temporal detail present in visual observations\. In contrast, our approach does not strictly require language annotations for training\. Instead, we view Keyframe Mnemonics as complementary to such methods, with informative keyframes addressing partial observability and language supporting abstract reasoning over past events\.

## NeurIPS Paper Checklist

1. 1\.Claims
2. Question: Do the main claims made in the abstract and introduction accurately reflect the paper’s contributions and scope?
3. Answer:\[Yes\]
4. Justification: We have clearly stated the contributions in the introduction and have ensured that the Methodology, and the Experiments and Results Analysis sections back up the three contributions clearly\.
5. Guidelines: - •The answer\[N/A\]means that the abstract and introduction do not include the claims made in the paper\. - •The abstract and/or introduction should clearly state the claims made, including the contributions made in the paper and important assumptions and limitations\. A\[No\]or\[N/A\]answer to this question will not be perceived well by the reviewers\. - •The claims made should match theoretical and experimental results, and reflect how much the results can be expected to generalize to other settings\. - •It is fine to include aspirational goals as motivation as long as it is clear that these goals are not attained by the paper\.
6. 2\.Limitations
7. Question: Does the paper discuss the limitations of the work performed by the authors?
8. Answer:\[Yes\]
9. Justification: The paper has a limitations section where we have stated clearly the limitations of the current work\.
10. Guidelines: - •The answer\[N/A\]means that the paper has no limitation while the answer\[No\]means that the paper has limitations, but those are not discussed in the paper\. - •The authors are encouraged to create a separate “Limitations” section in their paper\. - •The paper should point out any strong assumptions and how robust the results are to violations of these assumptions \(e\.g\., independence assumptions, noiseless settings, model well\-specification, asymptotic approximations only holding locally\)\. The authors should reflect on how these assumptions might be violated in practice and what the implications would be\. - •The authors should reflect on the scope of the claims made, e\.g\., if the approach was only tested on a few datasets or with a few runs\. In general, empirical results often depend on implicit assumptions, which should be articulated\. - •The authors should reflect on the factors that influence the performance of the approach\. For example, a facial recognition algorithm may perform poorly when image resolution is low or images are taken in low lighting\. Or a speech\-to\-text system might not be used reliably to provide closed captions for online lectures because it fails to handle technical jargon\. - •The authors should discuss the computational efficiency of the proposed algorithms and how they scale with dataset size\. - •If applicable, the authors should discuss possible limitations of their approach to address problems of privacy and fairness\. - •While the authors might fear that complete honesty about limitations might be used by reviewers as grounds for rejection, a worse outcome might be that reviewers discover limitations that aren’t acknowledged in the paper\. The authors should use their best judgment and recognize that individual actions in favor of transparency play an important role in developing norms that preserve the integrity of the community\. Reviewers will be specifically instructed to not penalize honesty concerning limitations\.
11. 3\.Theory assumptions and proofs
12. Question: For each theoretical result, does the paper provide the full set of assumptions and a complete \(and correct\) proof?
13. Answer:\[Yes\]
14. Justification: In the methodology section and the appendix we have attached mathematical proofs of our claims which are correct in our opinion\.
15. Guidelines: - •The answer\[N/A\]means that the paper does not include theoretical results\. - •All the theorems, formulas, and proofs in the paper should be numbered and cross\-referenced\. - •All assumptions should be clearly stated or referenced in the statement of any theorems\. - •The proofs can either appear in the main paper or the supplemental material, but if they appear in the supplemental material, the authors are encouraged to provide a short proof sketch to provide intuition\. - •Inversely, any informal proof provided in the core of the paper should be complemented by formal proofs provided in appendix or supplemental material\. - •Theorems and Lemmas that the proof relies upon should be properly referenced\.
16. 4\.Experimental result reproducibility
17. Question: Does the paper fully disclose all the information needed to reproduce the main experimental results of the paper to the extent that it affects the main claims and/or conclusions of the paper \(regardless of whether the code and data are provided or not\)?
18. Answer:\[Yes\]
19. Justification: We are submitting self contained code to run our framework on two of the domains demonstrated in our results\. We have aimed to provide parameters for our architectures where appropriate\. We have not collected any data as part of this paper\.
20. Guidelines: - •The answer\[N/A\]means that the paper does not include experiments\. - •If the paper includes experiments, a\[No\]answer to this question will not be perceived well by the reviewers: Making the paper reproducible is important, regardless of whether the code and data are provided or not\. - •If the contribution is a dataset and/or model, the authors should describe the steps taken to make their results reproducible or verifiable\. - •Depending on the contribution, reproducibility can be accomplished in various ways\. For example, if the contribution is a novel architecture, describing the architecture fully might suffice, or if the contribution is a specific model and empirical evaluation, it may be necessary to either make it possible for others to replicate the model with the same dataset, or provide access to the model\. In general\. releasing code and data is often one good way to accomplish this, but reproducibility can also be provided via detailed instructions for how to replicate the results, access to a hosted model \(e\.g\., in the case of a large language model\), releasing of a model checkpoint, or other means that are appropriate to the research performed\. - •While NeurIPS does not require releasing code, the conference does require all submissions to provide some reasonable avenue for reproducibility, which may depend on the nature of the contribution\. For example 1. \(a\)If the contribution is primarily a new algorithm, the paper should make it clear how to reproduce that algorithm\. 2. \(b\)If the contribution is primarily a new model architecture, the paper should describe the architecture clearly and fully\. 3. \(c\)If the contribution is a new model \(e\.g\., a large language model\), then there should either be a way to access this model for reproducing the results or a way to reproduce the model \(e\.g\., with an open\-source dataset or instructions for how to construct the dataset\)\. 4. \(d\)We recognize that reproducibility may be tricky in some cases, in which case authors are welcome to describe the particular way they provide for reproducibility\. In the case of closed\-source models, it may be that access to the model is limited in some way \(e\.g\., to registered users\), but it should be possible for other researchers to have some path to reproducing or verifying the results\.
21. 5\.Open access to data and code
22. Question: Does the paper provide open access to the data and code, with sufficient instructions to faithfully reproduce the main experimental results, as described in supplemental material?
23. Answer:\[Yes\]
24. Justification: We have released two self contained domains with our architecture that are runnable\. This should allow the reviewers to verify our claims\. We have also attached our webpage with more information and access to github\.
25. Guidelines: - •The answer\[N/A\]means that paper does not include experiments requiring code\. - • - •While we encourage the release of code and data, we understand that this might not be possible, so\[No\]is an acceptable answer\. Papers cannot be rejected simply for not including code, unless this is central to the contribution \(e\.g\., for a new open\-source benchmark\)\. - •The instructions should contain the exact command and environment needed to run to reproduce the results\. See the NeurIPS code and data submission guidelines \([https://neurips\.cc/public/guides/CodeSubmissionPolicy](https://neurips.cc/public/guides/CodeSubmissionPolicy)\) for more details\. - •The authors should provide instructions on data access and preparation, including how to access the raw data, preprocessed data, intermediate data, and generated data, etc\. - •The authors should provide scripts to reproduce all experimental results for the new proposed method and baselines\. If only a subset of experiments are reproducible, they should state which ones are omitted from the script and why\. - •At submission time, to preserve anonymity, the authors should release anonymized versions \(if applicable\)\. - •Providing as much information as possible in supplemental material \(appended to the paper\) is recommended, but including URLs to data and code is permitted\.
26. 6\.Experimental setting/details
27. Question: Does the paper specify all the training and test details \(e\.g\., data splits, hyperparameters, how they were chosen, type of optimizer\) necessary to understand the results?
28. Answer:\[Yes\]
29. Justification: The results and Appendix B aim to provide complete experimental details for reproducibility\.
30. Guidelines: - •The answer\[N/A\]means that the paper does not include experiments\. - •The experimental setting should be presented in the core of the paper to a level of detail that is necessary to appreciate the results and make sense of them\. - •The full details can be provided either with the code, in appendix, or as supplemental material\.
31. 7\.Experiment statistical significance
32. Question: Does the paper report error bars suitably and correctly defined or other appropriate information about the statistical significance of the experiments?
33. Answer:\[Yes\]
34. Justification: The error bars are visible in each of the figures and claims are made with respect to these error bars\.
35. Guidelines: - •The answer\[N/A\]means that the paper does not include experiments\. - •The authors should answer\[Yes\]if the results are accompanied by error bars, confidence intervals, or statistical significance tests, at least for the experiments that support the main claims of the paper\. - •The factors of variability that the error bars are capturing should be clearly stated \(for example, train/test split, initialization, random drawing of some parameter, or overall run with given experimental conditions\)\. - •The method for calculating the error bars should be explained \(closed form formula, call to a library function, bootstrap, etc\.\) - •The assumptions made should be given \(e\.g\., Normally distributed errors\)\. - •It should be clear whether the error bar is the standard deviation or the standard error of the mean\. - •It is OK to report 1\-sigma error bars, but one should state it\. The authors should preferably report a 2\-sigma error bar than state that they have a 96% CI, if the hypothesis of Normality of errors is not verified\. - •For asymmetric distributions, the authors should be careful not to show in tables or figures symmetric error bars that would yield results that are out of range \(e\.g\., negative error rates\)\. - •If error bars are reported in tables or plots, the authors should explain in the text how they were calculated and reference the corresponding figures or tables in the text\.
36. 8\.Experiments compute resources
37. Question: For each experiment, does the paper provide sufficient information on the computer resources \(type of compute workers, memory, time of execution\) needed to reproduce the experiments?
38. Answer:\[Yes\]
39. Justification: Appendix[C](https://arxiv.org/html/2610.10857#A3)and[D](https://arxiv.org/html/2610.10857#A4)specified compute times for our methods and computational resources used\.
40. Guidelines: - •The answer\[N/A\]means that the paper does not include experiments\. - •The paper should indicate the type of compute workers CPU or GPU, internal cluster, or cloud provider, including relevant memory and storage\. - •The paper should provide the amount of compute required for each of the individual experimental runs as well as estimate the total compute\. - •The paper should disclose whether the full research project required more compute than the experiments reported in the paper \(e\.g\., preliminary or failed experiments that didn’t make it into the paper\)\.
41. 9\.Code of ethics
43. Answer:\[Yes\]
44. Justification: This work has not collected any human data\. We have listed computational costs and released runnable code to demonstrate reproducibility\. Our work is in introductory stages so does not have comparable societal impact to existing generative AI technologies as of now\. The goal is to improve memory of neural networks to eventually perform long horizon tasks on robots which would be more directly useful to our society\.
45. Guidelines: - •The answer\[N/A\]means that the authors have not reviewed the NeurIPS Code of Ethics\. - •If the authors answer\[No\], they should explain the special circumstances that require a deviation from the Code of Ethics\. - •The authors should make sure to preserve anonymity \(e\.g\., if there is a special consideration due to laws or regulations in their jurisdiction\)\.
46. 10\.Broader impacts
47. Question: Does the paper discuss both potential positive societal impacts and negative societal impacts of the work performed?
48. Answer:\[N/A\]
49. Justification: The work on improving memory for neural networks is not practical yet for direct societal interaction\. This work is on artificial benchmark and simulated robot tasks which is not meaningful to society at large in its current state\.
50. Guidelines: - •The answer\[N/A\]means that there is no societal impact of the work performed\. - •If the authors answer\[N/A\]or\[No\], they should explain why their work has no societal impact or why the paper does not address societal impact\. - •Examples of negative societal impacts include potential malicious or unintended uses \(e\.g\., disinformation, generating fake profiles, surveillance\), fairness considerations \(e\.g\., deployment of technologies that could make decisions that unfairly impact specific groups\), privacy considerations, and security considerations\. - •The conference expects that many papers will be foundational research and not tied to particular applications, let alone deployments\. However, if there is a direct path to any negative applications, the authors should point it out\. For example, it is legitimate to point out that an improvement in the quality of generative models could be used to generate Deepfakes for disinformation\. On the other hand, it is not needed to point out that a generic algorithm for optimizing neural networks could enable people to train models that generate Deepfakes faster\. - •The authors should consider possible harms that could arise when the technology is being used as intended and functioning correctly, harms that could arise when the technology is being used as intended but gives incorrect results, and harms following from \(intentional or unintentional\) misuse of the technology\. - •If there are negative societal impacts, the authors could also discuss possible mitigation strategies \(e\.g\., gated release of models, providing defenses in addition to attacks, mechanisms for monitoring misuse, mechanisms to monitor how a system learns from feedback over time, improving the efficiency and accessibility of ML\)\.
51. 11\.Safeguards
52. Question: Does the paper describe safeguards that have been put in place for responsible release of data or models that have a high risk for misuse \(e\.g\., pre\-trained language models, image generators, or scraped datasets\)?
53. Answer:\[N/A\]
54. Justification: The work is too preliminary to have a high risk of misuse in a societal scale\.
55. Guidelines: - •The answer\[N/A\]means that the paper poses no such risks\. - •Released models that have a high risk for misuse or dual\-use should be released with necessary safeguards to allow for controlled use of the model, for example by requiring that users adhere to usage guidelines or restrictions to access the model or implementing safety filters\. - •Datasets that have been scraped from the Internet could pose safety risks\. The authors should describe how they avoided releasing unsafe images\. - •We recognize that providing effective safeguards is challenging, and many papers do not require this, but we encourage authors to take this into account and make a best faith effort\.
56. 12\.Licenses for existing assets
57. Question: Are the creators or original owners of assets \(e\.g\., code, data, models\), used in the paper, properly credited and are the license and terms of use explicitly mentioned and properly respected?
58. Answer:\[Yes\]
59. Justification: We have cited the papers whose code and/or data we have used to run our experiments\. All other code is our own\.
60. Guidelines: - •The answer\[N/A\]means that the paper does not use existing assets\. - •The authors should cite the original paper that produced the code package or dataset\. - •The authors should state which version of the asset is used and, if possible, include a URL\. - •The name of the license \(e\.g\., CC\-BY 4\.0\) should be included for each asset\. - •For scraped data from a particular source \(e\.g\., website\), the copyright and terms of service of that source should be provided\. - •If assets are released, the license, copyright information, and terms of use in the package should be provided\. For popular datasets,[paperswithcode\.com/datasets](https://paperswithcode.com/datasets)has curated licenses for some datasets\. Their licensing guide can help determine the license of a dataset\. - •For existing datasets that are re\-packaged, both the original license and the license of the derived asset \(if it has changed\) should be provided\. - •If this information is not available online, the authors are encouraged to reach out to the asset’s creators\.
61. 13\.New assets
62. Question: Are new assets introduced in the paper well documented and is the documentation provided alongside the assets?
63. Answer:\[Yes\]
64. Justification: We are releasing code as part of the supplementary material and on our webpage\. No human subjects data was collected or used as part of this project\.
65. Guidelines: - •The answer\[N/A\]means that the paper does not release new assets\. - •Researchers should communicate the details of the dataset/code/model as part of their submissions via structured templates\. This includes details about training, license, limitations, etc\. - •The paper should discuss whether and how consent was obtained from people whose asset is used\. - •At submission time, remember to anonymize your assets \(if applicable\)\. You can either create an anonymized URL or include an anonymized zip file\.
66. 14\.Crowdsourcing and research with human subjects
67. Question: For crowdsourcing experiments and research with human subjects, does the paper include the full text of instructions given to participants and screenshots, if applicable, as well as details about compensation \(if any\)?
68. Answer:\[N/A\]
69. Justification: We did not collect any human subjects data\.
70. Guidelines: - •The answer\[N/A\]means that the paper does not involve crowdsourcing nor research with human subjects\. - •Including this information in the supplemental material is fine, but if the main contribution of the paper involves human subjects, then as much detail as possible should be included in the main paper\. - •According to the NeurIPS Code of Ethics, workers involved in data collection, curation, or other labor should be paid at least the minimum wage in the country of the data collector\.
71. 15\.Institutional review board \(IRB\) approvals or equivalent for research with human subjects
72. Question: Does the paper describe potential risks incurred by study participants, whether such risks were disclosed to the subjects, and whether Institutional Review Board \(IRB\) approvals \(or an equivalent approval/review based on the requirements of your country or institution\) were obtained?
73. Answer:\[N/A\]
74. Justification: We did not collect any human subjects data\.
75. Guidelines: - •The answer\[N/A\]means that the paper does not involve crowdsourcing nor research with human subjects\. - •Depending on the country in which research is conducted, IRB approval \(or equivalent\) may be required for any human subjects research\. If you obtained IRB approval, you should clearly state this in the paper\. - •We recognize that the procedures for this may vary significantly between institutions and locations, and we expect authors to adhere to the NeurIPS Code of Ethics and the guidelines for their institution\. - •For initial submissions, do not include any information that would break anonymity \(if applicable\), such as the institution conducting the review\.
76. 16\.Declaration of LLM usage
77. Question: Does the paper describe the usage of LLMs if it is an important, original, or non\-standard component of the core methods in this research? Note that if the LLM is used only for writing, editing, or formatting purposes and does*not*impact the core methodology, scientific rigor, or originality of the research, declaration is not required\.
78. Answer:\[N/A\]
79. Justification: LLMs were not used to develop the core idea or experimental design of this research\.
80. Guidelines: - •The answer\[N/A\]means that the core method development in this research does not involve LLMs as any important, original, or non\-standard components\. - •Please refer to our LLM policy in the NeurIPS handbook for what should or should not be described\.