CC-AOS: Cost- and Horizon-Conditioned Amortized Backward Induction for Finite-Horizon Optimal Stopping

# CC-AOS: Cost- and Horizon-Conditioned Amortized Backward Induction for Finite-Horizon Optimal Stopping
Source: [https://arxiv.org/html/2607.22774](https://arxiv.org/html/2607.22774)
\(Draft v1 – July 23, 2026\)

###### Abstract

Finite\-horizon optimal stopping is a central problem in early time\-series classification, where a system must decide at each sequence prefix whether the expected benefit of another observation justifies its acquisition cost\. Existing data\-driven backward\-induction methods typically solve each cost–horizon operating point separately, so changing operating conditions requires repeated optimization and separate model stacks, making continuous cost adaptation and multi\-horizon deployment inefficient\. We propose CC\-AOS \(Cost\- and Horizon\-Conditioned Amortized Optimal Stopping\), a structured amortized solver for a family of finite\-horizon stopping problems with continuous costs and multiple horizons\. CC\-AOS learns a shared continuation\-value model conditioned on the current state, absolute time, remaining horizon, and acquisition cost through joint amortized fitted backward induction\. We establish that the exact value and continuation functions are nondecreasing, concave, and horizon\-dependently Lipschitz in cost, encode these properties in the model architecture, and derive residual\-based bounds on value and policy errors\. Experiments on controlled Gaussian and time\-varying non\-Gaussian processes and the FordA engine\-noise time\-series benchmark compare CC\-AOS with representative per\-operating\-point backward\-induction solvers and tuned static stopping rules\. At six unseen FordA cost–horizon pairs, one CC\-AOS checkpoint achieved a lower terminal\-risk\-plus\-sampling\-cost objective than independently fitted Convex Function Learning at all six pairs, with an average reduction of 15\.75%, while matching the tuned static thresholds on average\.

Keywords:optimal stopping; early time\-series classification; backward induction; amortized optimization; shape\-constrained neural networks\.

## 1Introduction

### 1\.1Finite\-horizon stopping under multiple operating points

Sequential decision systems often need to act before all potentially available information has been collected\. In early time\-series classification, for example, a prediction can be made from the current prefix or postponed until another observation is acquired\. Acting early saves time, energy, computation, or measurement effort, but may increase the terminal decision error; waiting can improve the prediction, but its benefit must justify both the additional acquisition cost and the delay\[[25](https://arxiv.org/html/2607.22774#bib.bib1),[27](https://arxiv.org/html/2607.22774#bib.bib5)\]\. This tension makes the stopping rule, rather than classification accuracy alone, central to the decision problem\.

The difficulty becomes sharper under a finite observation budget\. Near the beginning of a sequence, several future observations remain available, whereas near the deadline the same current evidence may warrant a different action\. Backward induction captures this stage dependence by comparing the terminal riskg​\(s\)g\(s\)of stopping at statesswith the complete continuation riskλ\+Ch​\(s;λ\)\\lambda\+C\_\{h\}\(s;\\lambda\)\. Here,λ\\lambdais the cost of the next observation andChC\_\{h\}is the expected future optimal risk after that observation, withhhacquisitions remaining\. Thus, the continuation value and the resulting stopping boundary depend jointly on the current state, the remaining horizon, and the sampling cost\[[6](https://arxiv.org/html/2607.22774#bib.bib3),[17](https://arxiv.org/html/2607.22774#bib.bib4)\]\.

In many applications, however, neither the cost nor the available horizon is fixed once and for all\. Resource availability may change, several latency budgets may need to be supported, or a deployed system may need to trade accuracy against delay at different operating conditions\. The relevant object is therefore not a single stopping policy, but a related family of finite\-horizon problems indexed by the sampling cost and horizon\. Accordingly, this work studies a unified model that learns this family coherently and can be queried at cost–horizon pairs that were not solved separately during training\.

### 1\.2Limitations of per\-operating\-point solvers

Backward induction provides a principled solution once an operating point is specified\. In data\-driven settings, its conditional expectations can be estimated by regression, yielding stagewise approximations of the continuation value\[[14](https://arxiv.org/html/2607.22774#bib.bib15),[23](https://arxiv.org/html/2607.22774#bib.bib16)\]\. Recent early\-classification methods, including FIRMBOUND, combine learned sequential statistics with this finite\-horizon recursion\[[10](https://arxiv.org/html/2607.22774#bib.bib14)\]\. Despite differences in their estimators, such solvers generally treat the sampling cost and maximum horizon as fixed problem parameters: a continuation model is fitted for the requested operating point\.

This formulation is effective when only one setting is needed, but it scales poorly to a changing collection of settings\. Each new cost or horizon can require new recursive targets, another optimization run, and another stack of stagewise models\. More importantly, a collection of independently fitted solvers gives values only at the operating points that were explicitly solved\. It neither defines how the continuation value should behave between those points nor ensures that their approximations form one Bellman\-consistent family\.

Parameter conditioning appears to offer an immediate remedy, but an unconstrained shared network leaves an important part of the stopping problem unused\. For a fixed state and remaining horizon, the exact optimal value has a particular geometry as sampling cost varies: it is nondecreasing, concave, and Lipschitz with a constant determined by the number of acquisitions still available\. A useful amortized solver should therefore do more than compress several separately trained models\. It should couple the operating points through a shared Bellman learning procedure while preserving the cost structure of the underlying finite\-horizon problem\.

### 1\.3Structured amortized optimal stopping

These requirements motivate Cost\- and Horizon\-Conditioned Amortized Optimal Stopping \(CC\-AOS\)\. The starting point is that stopping problems at different costs and horizons are not unrelated tasks: they share the same state process, terminal risk, and finite\-horizon recursion\. CC\-AOS exposes this common structure by treating cost and remaining horizon as conditioning variables of a single continuation modelCθ​\(s,t,h,λ\)C\_\{\\theta\}\(s,t,h,\\lambda\), rather than as fixed settings chosen before optimization\.

The model is learned by joint amortized fitted backward induction\. Recursive targets are constructed over continuously sampled costs and transitions drawn from multiple training horizons, so all operating points contribute to one conditional value surface\. To make the learned surface reliable along the continuous cost axis, a normalized soft\-min affine head encodes the monotonicity, concavity, and remaining\-horizon\-dependent Lipschitz structure established for the exact problem\.

The resulting formulation connects representation, learning, and analysis around the same continuation\-value family\. We derive residual\-based bounds that relate Bellman approximation error to value\-function error and the performance loss of the induced stopping policy, and evaluate whether one checkpoint transfers to unseen cost–horizon pairs\. Experiments span controlled Gaussian and time\-varying non\-Gaussian processes as well as the FordA engine\-noise benchmark under causal prefix\-only normalization\. On FordA, CC\-AOS improves over independently fitted Convex Function Learning at all six unseen operating points, with five paired confidence intervals entirely below zero, while remaining essentially tied on average with a strong validation\-tuned static threshold\.

### 1\.4Contributions

The main contribution of this paper is CC\-AOS, a structured amortized solver for a family of finite\-horizon stopping problems with continuous sampling costs and multiple horizons\. The solver is built on three connected contributions:

1. \(1\)Shared cost–horizon continuation model\.CC\-AOS represents the stopping\-problem family with a single continuation modelCθ​\(s,t,h,λ\)C\_\{\\theta\}\(s,t,h,\\lambda\), conditioned on the current state, absolute time, remaining horizon, and sampling cost\. One trained checkpoint can therefore be queried directly to construct stopping policies at multiple and held\-out cost–horizon pairs\.
2. \(2\)Joint amortized training method\.We develop a joint amortized fitted backward\-induction method that constructs recursive Bellman targets over continuously sampled costs and multiple training horizons\. The method learns the shared continuation family within one optimization process while retaining the recursive coupling across decision stages and operating conditions\.
3. \(3\)Structure\-preserving architecture and theoretical analysis\.We establish the monotonicity, concavity, and remaining\-horizon\-dependent Lipschitz properties of the exact value and continuation functions with respect to sampling cost\. We encode these properties in a normalized soft\-min affine continuation head and derive residual\-based bounds for value\-function error and stopping\-policy suboptimality\.

## 2Related Work

CC\-AOS lies at the intersection of three connected lines of work\. Finite\-horizon optimal stopping and early time\-series classification define the underlying risk–acquisition trade\-off\. Data\-driven continuation learning and sequential density\-ratio estimation provide the basis for learning finite\-horizon stopping decisions from trajectories\. Parameter\-conditioned learning and shape\-constrained approximation then provide the tools needed to share a model across operating conditions without discarding known value\-function structure\.

### 2\.1Finite\-horizon optimal stopping and early time\-series classification

Classical sequential analysis formalizes how evidence should be accumulated before a decision is made\[[25](https://arxiv.org/html/2607.22774#bib.bib1)\]\. For simple hypothesis tests, the sequential probability ratio test gives a canonical stopping rule with established optimality properties\[[24](https://arxiv.org/html/2607.22774#bib.bib2)\]\. When observations are limited by a finite deadline, however, the decision becomes stage dependent: the value of another observation changes with the time remaining\. Dynamic programming and backward induction provide the standard formulation for this finite\-horizon setting\[[6](https://arxiv.org/html/2607.22774#bib.bib3),[17](https://arxiv.org/html/2607.22774#bib.bib4)\]\.

Early time\-series classification places the same stopping problem in a predictive setting\. The foundational formulation seeks accurate decisions from short sequence prefixes\[[27](https://arxiv.org/html/2607.22774#bib.bib5),[28](https://arxiv.org/html/2607.22774#bib.bib6)\]\. Non\-myopic approaches explicitly account for the possible benefit of future observations\[[8](https://arxiv.org/html/2607.22774#bib.bib7)\], while later work jointly optimizes accuracy and earliness or learns adaptive halting policies\[[16](https://arxiv.org/html/2607.22774#bib.bib8),[12](https://arxiv.org/html/2607.22774#bib.bib9)\]\. Other methods emphasize reliable early decisions through ensemble agreement or explicit control of the accumulated accuracy gap\[[20](https://arxiv.org/html/2607.22774#bib.bib10),[18](https://arxiv.org/html/2607.22774#bib.bib13)\]\. CC\-AOS follows this cost\-sensitive finite\-horizon view: stopping incurs a terminal prediction risk, whereas continuing incurs an acquisition cost and preserves the option to decide later\.

### 2\.2Data\-driven backward induction and amortized value learning

Finite\-horizon backward induction requires conditional expectations of future optimal values\. Least\-squares Monte Carlo and regression\-based dynamic programming estimate these continuation values stage by stage from simulated trajectories\[[14](https://arxiv.org/html/2607.22774#bib.bib15),[23](https://arxiv.org/html/2607.22774#bib.bib16)\]\. Statistical\-learning analyses further connect continuation regression to optimal\-stopping approximation\[[11](https://arxiv.org/html/2607.22774#bib.bib17)\]\. Neural approaches extend data\-driven stopping to high\-dimensional state spaces\[[5](https://arxiv.org/html/2607.22774#bib.bib18),[4](https://arxiv.org/html/2607.22774#bib.bib19)\], while interpretable policy classes and randomized neural features provide alternative approximations\[[7](https://arxiv.org/html/2607.22774#bib.bib20),[13](https://arxiv.org/html/2607.22774#bib.bib21)\]\.

For early classification, sequential density\-ratio estimation learns class evidence directly from sequence prefixes and optimizes the speed–accuracy trade\-off\[[9](https://arxiv.org/html/2607.22774#bib.bib11),[15](https://arxiv.org/html/2607.22774#bib.bib12)\]\. Recent work further combines data\-derived sequential statistics, posterior terminal risk, and fitted finite\-horizon backward induction\[[10](https://arxiv.org/html/2607.22774#bib.bib14)\]\. Within this data\-driven stopping framework, CC\-AOS changes the granularity of the solver from a continuation stack for one prescribed cost and horizon to a continuation family indexed jointly by both quantities\.

This extension draws on the broader principle of amortized optimization, which learns a parameter\-to\-solution mapping so that related problem instances share computation\[[3](https://arxiv.org/html/2607.22774#bib.bib25)\]\. Universal value\-function approximators similarly condition a value model on task variables\[[21](https://arxiv.org/html/2607.22774#bib.bib22)\], and multi\-objective reinforcement learning conditions policies on changing preference weights\[[1](https://arxiv.org/html/2607.22774#bib.bib23),[29](https://arxiv.org/html/2607.22774#bib.bib24)\]\. These works provide the conceptual basis for treating cost and horizon as model inputs\. CC\-AOS specializes that principle to finite\-horizon continuation learning by constructing recursive targets jointly over continuous costs and multiple training horizons within one fitted backward\-induction method\.

### 2\.3Shape\-constrained approximation for continuation values

Shape\-constrained neural networks provide mechanisms for incorporating known functional structure into a learned approximation\. Monotonic networks enforce prescribed coordinatewise responses\[[22](https://arxiv.org/html/2607.22774#bib.bib26)\], while input\-convex neural networks encode convexity with respect to selected inputs\[[2](https://arxiv.org/html/2607.22774#bib.bib27)\]\. Subsequent architectures broaden the design space for monotonic approximation and improve its flexibility\[[26](https://arxiv.org/html/2607.22774#bib.bib28),[19](https://arxiv.org/html/2607.22774#bib.bib29)\]\.

CC\-AOS builds on these architectural ideas using constraints derived from the finite\-horizon Bellman problem itself\. Withhhacquisitions remaining, the exact value is nondecreasing, concave, andhh\-Lipschitz in sampling cost; the corresponding continuation value is\(h−1\)\(h\-1\)\-Lipschitz\. The normalized soft\-min affine head imposes this horizon\-dependent cost geometry while allowing state, time, and remaining horizon to determine the affine components\. In this way, data\-driven backward induction, amortized parameter conditioning, and shape\-constrained approximation are combined within a single continuation\-value family\.

## 3Problem Formulation

### 3\.1Sequential information and stopping risk

LetY∈\{1,…,K\}Y\\in\\\{1,\\ldots,K\\\}be the class label and letX1:t=\(X1,…,Xt\)X\_\{1:t\}=\(X\_\{1\},\\ldots,X\_\{t\}\)be the sequence prefix observed at timett\. The available information is represented by the filtrationℱt=σ​\(X1:t\)\\mathcal\{F\}\_\{t\}=\\sigma\(X\_\{1:t\}\)\. A causal state estimator maps each prefix to the decision state

St=ϕ​\(X1:t\)∈𝒮,t=1,…,H\.S\_\{t\}=\\phi\(X\_\{1:t\}\)\\in\\mathcal\{S\},\\qquad t=1,\\ldots,H\.\(1\)We assume thatStS\_\{t\}is a Markov state for the stopping problem\. For a time\-inhomogeneous process, absolute time is included in an augmented state; the learned model in Section 4 retainsttexplicitly\. WhenStS\_\{t\}is learned, it is treated as the working decision state rather than claimed to be an exact sufficient statistic\.

Let𝒜\\mathcal\{A\}be the terminal action space and letℓ​\(a,Y\)\\ell\(a,Y\)be the loss incurred by actionaa\. The risk of stopping at statessand issuing the Bayes action is

g​\(s\)=mina∈𝒜⁡𝔼​\[ℓ​\(a,Y\)∣St=s\]\.g\(s\)=\\min\_\{a\\in\\mathcal\{A\}\}\\mathbb\{E\}\\\!\\left\[\\ell\(a,Y\)\\mid S\_\{t\}=s\\right\]\.\(2\)For zero–one classification loss, this reduces tog​\(s\)=1−maxy⁡Pr⁡\(Y=y∣St=s\)g\(s\)=1\-\\max\_\{y\}\\Pr\(Y=y\\mid S\_\{t\}=s\)\. We assume0≤g​\(s\)≤B<∞0\\leq g\(s\)\\leq B<\\infty\. Acquiring one additional observation incurs a constant costλ≥0\\lambda\\geq 0\.

### 3\.2Finite\-horizon objective and Bellman recursion

The maximum sequence horizon isHH\. At timett, the number of additional observations still available is

Let𝔗t,h\\mathfrak\{T\}\_\{t,h\}be the set of\{ℱt\+k\}k=0h\\\{\\mathcal\{F\}\_\{t\+k\}\\\}\_\{k=0\}^\{h\}\-adapted stopping times that count additional acquisitions, so everyτ∈𝔗t,h\\tau\\in\\mathfrak\{T\}\_\{t,h\}takes values in\{0,…,h\}\\\{0,\\ldots,h\\\}\. Conditional onSt=sS\_\{t\}=s, the finite\-horizon optimal value is

Vh​\(s;λ\)=infτ∈𝔗t,h𝔼​\[g​\(St\+τ\)\+λ​τ∣St=s\]\.V\_\{h\}\(s;\\lambda\)=\\inf\_\{\\tau\\in\\mathfrak\{T\}\_\{t,h\}\}\\mathbb\{E\}\\\!\\left\[g\(S\_\{t\+\\tau\}\)\+\\lambda\\tau\\mid S\_\{t\}=s\\right\]\.\(4\)Thus,τ=0\\tau=0represents immediate stopping, whileτ=k\\tau=kincurs the cost ofkkadditional observations\. We assume thatg​\(St\+τ\)g\(S\_\{t\+\\tau\}\)is integrable and that the state transition law is independent ofλ\\lambda\.

For any bounded measurable functionff, define the one\-step transition operator

\(P​f\)​\(s\)=𝔼​\[f​\(St\+1\)∣St=s\]\.\(Pf\)\(s\)=\\mathbb\{E\}\\\!\\left\[f\(S\_\{t\+1\}\)\\mid S\_\{t\}=s\\right\]\.\(5\)The post\-acquisition continuation value forh≥1h\\geq 1is then

Ch​\(s;λ\)=\(P​Vh−1\)​\(s;λ\)=𝔼​\[Vh−1​\(St\+1;λ\)∣St=s\]\.C\_\{h\}\(s;\\lambda\)=\(PV\_\{h\-1\}\)\(s;\\lambda\)=\\mathbb\{E\}\\\!\\left\[V\_\{h\-1\}\(S\_\{t\+1\};\\lambda\)\\mid S\_\{t\}=s\\right\]\.\(6\)The current acquisition cost is excluded fromChC\_\{h\}; the complete continuation risk isλ\+Ch\\lambda\+C\_\{h\}\. Backward induction gives

V0​\(s;λ\)\\displaystyle V\_\{0\}\(s;\\lambda\)=g​\(s\),\\displaystyle=g\(s\),\(7\)Vh​\(s;λ\)\\displaystyle V\_\{h\}\(s;\\lambda\)=min⁡\{g​\(s\),λ\+Ch​\(s;λ\)\},h≥1\.\\displaystyle=\\min\\\!\\left\\\{g\(s\),\\,\\lambda\+C\_\{h\}\(s;\\lambda\)\\right\\\},\\qquad h\\geq 1\.\(8\)Accordingly, the Bayes\-optimal stopping rule is

πh⋆​\(s;λ\)=\{stop,g​\(s\)≤λ\+Ch​\(s;λ\),continue,g​\(s\)\>λ\+Ch​\(s;λ\),h≥1,\\pi\_\{h\}^\{\\star\}\(s;\\lambda\)=\\begin\{cases\}\\mathrm\{stop\},&g\(s\)\\leq\\lambda\+C\_\{h\}\(s;\\lambda\),\\\\ \\mathrm\{continue\},&g\(s\)\>\\lambda\+C\_\{h\}\(s;\\lambda\),\\end\{cases\}\\qquad h\\geq 1,\(9\)with mandatory stopping ath=0h=0\.

### 3\.3The cost–horizon continuation family

An operating point isω=\(λ,H\)\\omega=\(\\lambda,H\), whereλ∈Λ\\lambda\\in\\LambdaandHHbelongs to a set of admissible finite horizons\. For a fixedω\\omega, conventional backward induction produces the stack

\{Ch​\(⋅;λ\)\}h=1H−1\.\\bigl\\\{C\_\{h\}\(\\,\\cdot\\,;\\lambda\)\\bigr\\\}\_\{h=1\}^\{H\-1\}\.Across operating points, these stacks solve the same recursion under different cost and horizon conditions\. Our approximation target is therefore one conditional function

Cθ:𝒮×ℕ×ℕ×Λ⟶\[0,B\],Cθ​\(s,t,h,λ\)≈Ch​\(s;λ\),C\_\{\\theta\}:\\mathcal\{S\}\\times\\mathbb\{N\}\\times\\mathbb\{N\}\\times\\Lambda\\longrightarrow\[0,B\],\\qquad C\_\{\\theta\}\(s,t,h,\\lambda\)\\approx C\_\{h\}\(s;\\lambda\),\(10\)wherettis the absolute time,h=H−th=H\-tis the remaining horizon, and the parametersθ\\thetaare shared over the complete family\. Section 4 specifies how this function is learned and converted into a stopping policy\.

## 4Method

### 4\.1CC\-AOS overview

CC\-AOS receives a causal decision stateStS\_\{t\}and an operating condition\(λ,H\)\(\\lambda,H\), and returns a finite\-horizon stopping policy\. The solver has three connected components: a continuation model shared over cost and remaining horizon, a joint amortized fitted backward\-induction method, and a structure\-preserving cost head\. Figure[1](https://arxiv.org/html/2607.22774#S4.F1)separates the training recursion from deployment with the resulting checkpoint\.

TrainingState transitions\(St,St\+1\)\(S\_\{t\},S\_\{t\+1\}\)multi\-horizon trajectoriesBackward inductionfrom small remaining horizonf​r​o​z​e​n​m​o​d​e​l→B​e​l​l​m​a​n​t​a​r​g​e​t​sfrozen\\ model\\rightarrow Bellman\\ targetsShared model updateCθ​\(S,t,h,λ\)C\_\{\\theta\}\(S,t,h,\\lambda\)continuous cost replaySave checkpointCθC\_\{\\theta\}DeploymentCurrent state query\(St,t,h,λ\)\(S\_\{t\},t,h,\\lambda\)Load sharedcheckpointCθC\_\{\\theta\}Evaluate continuationλ\+Cθ​\(St,t,h,λ\)\\lambda\+C\_\{\\theta\}\(S\_\{t\},t,h,\\lambda\)Compare withterminal riskg​\(St\)g\(S\_\{t\}\)Stop or continue

Figure 1:CC\-AOS training and deployment framework\. During training, multi\-horizon trajectories are used to generate recursive Bellman targets from frozen models\. A shared structured continuation modelCθC\_\{\\theta\}is updated through continuous\-cost replay and stored as a checkpoint\. During deployment, the checkpoint is queried with the current operating condition to determine the stop\-or\-continue decision\.
### 4\.2Shared cost–horizon continuation model

CC\-AOS parameterizes the continuation term in \([8](https://arxiv.org/html/2607.22774#S3.E8)\) byCθ​\(s,t,h,λ\)C\_\{\\theta\}\(s,t,h,\\lambda\)\. The corresponding approximate value is

V^0​\(s;λ\)\\displaystyle\\widehat\{V\}\_\{0\}\(s;\\lambda\)=g​\(s\),\\displaystyle=g\(s\),\(11\)V^h​\(s,t;λ\)\\displaystyle\\widehat\{V\}\_\{h\}\(s,t;\\lambda\)=min⁡\{g​\(s\),λ\+Cθ​\(s,t,h,λ\)\},h≥1\.\\displaystyle=\\min\\\!\\left\\\{g\(s\),\\,\\lambda\+C\_\{\\theta\}\(s,t,h,\\lambda\)\\right\\\},\\qquad h\\geq 1\.\(12\)A shared feature network maps\(s,t,h\)\(s,t,h\)to the parameters of the cost head described in Section[4\.4](https://arxiv.org/html/2607.22774#S4.SS4)\. The statessrepresents the observed prefix,ttcaptures absolute\-time effects, andh=H−th=H\-tspecifies the depth of the remaining Bellman recursion\. Becauset\+h=Ht\+h=H, the pair\(t,h\)\(t,h\)identifies both the current decision stage and its total finite horizon\.

### 4\.3Joint amortized fitted backward induction

The identity in \([6](https://arxiv.org/html/2607.22774#S3.E6)\) turns continuation learning into conditional regression\. Let\(st,st\+1\)\(s\_\{t\},s\_\{t\+1\}\)be a sampled transition, leth≥1h\\geq 1, and drawλ\\lambdafrom a training distributionqqonΛ\\Lambda\. Given a frozen parameter snapshotθ¯\\bar\{\\theta\}, define the recursive target

yh=\{g​\(st\+1\),h=1,min⁡\{g​\(st\+1\),λ\+Cθ¯​\(st\+1,t\+1,h−1,λ\)\},h\>1\.y\_\{h\}=\\begin\{cases\}g\(s\_\{t\+1\}\),&h=1,\\\\\[5\.69054pt\] \\min\\\!\\left\\\{g\(s\_\{t\+1\}\),\\lambda\+C\_\{\\bar\{\\theta\}\}\(s\_\{t\+1\},t\+1,h\-1,\\lambda\)\\right\\\},&h\>1\.\\end\{cases\}\(13\)Conditioning onSt=sS\_\{t\}=sand averaging over successor states makesyhy\_\{h\}an estimate ofP​V^h−1P\\widehat\{V\}\_\{h\-1\}\. The shared model is fitted by

ℒ​\(θ\)=𝔼\(St,St\+1\)∼𝒟h∼ℋtrain,λ∼q​\[ℓreg​\(Cθ​\(st,t,h,λ\),yh\)\]\.\\mathcal\{L\}\(\\theta\)=\\mathbb\{E\}\_\{\\begin\{subarray\}\{c\}\(S\_\{t\},S\_\{t\+1\}\)\\sim\\mathcal\{D\}\\\\ h\\sim\\mathcal\{H\}\_\{\\mathrm\{train\}\},\\;\\lambda\\sim q\\end\{subarray\}\}\\\!\\left\[\\ell\_\{\\mathrm\{reg\}\}\\\!\\left\(C\_\{\\theta\}\(s\_\{t\},t,h,\\lambda\),y\_\{h\}\\right\)\\right\]\.\(14\)Figure[2](https://arxiv.org/html/2607.22774#S4.F2)details the resulting layerwise training process\. Its inputs are the state trajectories𝒟\\mathcal\{D\}, the set of training horizonsℋtrain\\mathcal\{H\}\_\{\\mathrm\{train\}\}, the cost intervalΛ\\Lambda, and the terminal riskgg\. Starting fromh=1h=1, the method initializes a shared continuation model and an empty cumulative replay buffer\. Before targets are constructed at each remaining\-horizon layer, the current parameters are frozen asθ¯\\bar\{\\theta\}\.

For every training horizon compatible with the currenthh, the corresponding transition time ist=H−ht=H\-h\. Transitions\(st,st\+1\)\(s\_\{t\},s\_\{t\+1\}\)are paired with stratified cost samplesλ∼q\\lambda\\sim qonΛ\\Lambda, and the frozen model generates the detached target in \([13](https://arxiv.org/html/2607.22774#S4.E13)\)\. The resulting tuples\(st,t,h,λ,yh\)\(s\_\{t\},t,h,\\lambda,y\_\{h\}\)are added to the replay buffer\. The current parameters are then updated on all layers accumulated so far using \([14](https://arxiv.org/html/2607.22774#S4.E14)\), after whichhhis incremented and the same cycle is repeated\. The final output is one continuation\-model checkpoint shared across continuous costs, total horizons, and Bellman layers\.

![Refer to caption](https://arxiv.org/html/2607.22774v1/x1.png)Figure 2:Detailed training workflow of CC\-AOS\. Backward induction proceeds from small to large remaining horizons\. At each layer, a frozen model generates Bellman regression targets for all compatible training horizons and stratified costs sampled overΛ\\Lambda; the resulting tuples are accumulated in a replay buffer and used to update the shared continuation model before advancing to the next layer\. A single cost\- and horizon\-conditioned checkpoint is retained after the final layer\.
### 4\.4Structure\-preserving continuation head

For fixed\(s,t,h\)\(s,t,h\), the feature network outputsJJintercept logits andJJslope logits\. They are transformed as

aj​\(s,t,h\)\\displaystyle a\_\{j\}\(s,t,h\)=B​σ​\(a~j​\(s,t,h\)\),\\displaystyle=B\\,\\sigma\\\!\\left\(\\widetilde\{a\}\_\{j\}\(s,t,h\)\\right\),\(15\)bj​\(s,t,h\)\\displaystyle b\_\{j\}\(s,t,h\)=\(h−1\)​σ​\(b~j​\(s,t,h\)\),\\displaystyle=\(h\-1\)\\,\\sigma\\\!\\left\(\\widetilde\{b\}\_\{j\}\(s,t,h\)\\right\),\(16\)so0≤aj≤B0\\leq a\_\{j\}\\leq Band0≤bj≤h−10\\leq b\_\{j\}\\leq h\-1\. The continuation head is the normalized soft minimum

C~θ​\(s,t,h,λ\)=−ρ​log⁡\[1J​∑j=1Jexp⁡\(−aj​\(s,t,h\)\+bj​\(s,t,h\)​λρ\)\],\\widetilde\{C\}\_\{\\theta\}\(s,t,h,\\lambda\)=\-\\rho\\log\\left\[\\frac\{1\}\{J\}\\sum\_\{j=1\}^\{J\}\\exp\\left\(\-\\frac\{a\_\{j\}\(s,t,h\)\+b\_\{j\}\(s,t,h\)\\lambda\}\{\\rho\}\\right\)\\right\],\(17\)whereρ\>0\\rho\>0is the temperature\. The factor1/J1/Jensures that identical affine components return their common value\. The final estimate is capped at the terminal\-risk upper bound,

Cθ​\(s,t,h,λ\)=min⁡\{B,C~θ​\(s,t,h,λ\)\}\.C\_\{\\theta\}\(s,t,h,\\lambda\)=\\min\\\{B,\\widetilde\{C\}\_\{\\theta\}\(s,t,h,\\lambda\)\\\}\.\(18\)For binary zero–one loss,B=0\.5B=0\.5\. The slope limith−1h\-1matches the maximum number of acquisitions remaining after the current observation is obtained\. The explicit cost in \([12](https://arxiv.org/html/2607.22774#S4.E12)\) supplies the final unit of cost sensitivity\. Section 5 proves the resulting monotonicity, concavity, and Lipschitz properties\.

### 4\.5Stopping policy and amortized deployment

Given a query\(s,t,h,λ\)\(s,t,h,\\lambda\), CC\-AOS compares the stopping and continuation risks:

π^h​\(s,t;λ\)=\{stop,g​\(s\)≤λ\+Cθ​\(s,t,h,λ\),continue,otherwise\.\\widehat\{\\pi\}\_\{h\}\(s,t;\\lambda\)=\\begin\{cases\}\\text\{stop\},&g\(s\)\\leq\\lambda\+C\_\{\\theta\}\(s,t,h,\\lambda\),\\\\ \\text\{continue\},&\\text\{otherwise\}\.\\end\{cases\}\(19\)If the policy continues, one observation is acquired and\(t,h\)\(t,h\)becomes\(t\+1,h−1\)\(t\+1,h\-1\); stopping is mandatory ath=0h=0\. Becauseλ\\lambdaandhhare query variables, the same checkpoint constructs stopping policies for all operating conditions represented by the learned continuation family\.

## 5Theoretical Analysis

This section first characterizes the cost geometry of the exact finite\-horizon problem, then proves that the structured head preserves it\. We finally connect fitted Bellman errors to value and policy performance\. Throughout, the horizon is finite,g​\(Sτ\)g\(S\_\{\\tau\}\)is integrable, and the state transition law is independent ofλ\\lambda\.

### 5\.1Cost geometry of exact values and continuations

###### Theorem 5\.1\(Exact value geometry\)\.

For every statessand finite remaining horizonhh,Vh​\(s;λ\)V\_\{h\}\(s;\\lambda\)is nondecreasing and concave inλ≥0\\lambda\\geq 0\. Moreover, for allλ1,λ2≥0\\lambda\_\{1\},\\lambda\_\{2\}\\geq 0,

\|Vh​\(s;λ2\)−Vh​\(s;λ1\)\|≤h​\|λ2−λ1\|\.\\left\|V\_\{h\}\(s;\\lambda\_\{2\}\)\-V\_\{h\}\(s;\\lambda\_\{1\}\)\\right\|\\leq h\|\\lambda\_\{2\}\-\\lambda\_\{1\}\|\.\(20\)

###### Proof\.

For an admissible stopping timeτ≤h\\tau\\leq h, define

Lτ​\(λ\)=𝔼s​\[g​\(Sτ\)\]\+λ​𝔼s​\[τ\]\.L\_\{\\tau\}\(\\lambda\)=\\mathbb\{E\}\_\{s\}\[g\(S\_\{\\tau\}\)\]\+\\lambda\\,\\mathbb\{E\}\_\{s\}\[\\tau\]\.This is affine inλ\\lambdawith slope𝔼s​\[τ\]∈\[0,h\]\\mathbb\{E\}\_\{s\}\[\\tau\]\\in\[0,h\]\. Equation \([4](https://arxiv.org/html/2607.22774#S3.E4)\) is the pointwise infimum of the family\{Lτ\}\\\{L\_\{\\tau\}\\\}, hence is concave\. Since every member is nondecreasing, the infimum is nondecreasing\. Forλ2≥λ1\\lambda\_\{2\}\\geq\\lambda\_\{1\}, evaluating anε\\varepsilon\-optimal rule forλ1\\lambda\_\{1\}atλ2\\lambda\_\{2\}gives

Vh​\(s;λ2\)−Vh​\(s;λ1\)≤h​\(λ2−λ1\)\+ε\.V\_\{h\}\(s;\\lambda\_\{2\}\)\-V\_\{h\}\(s;\\lambda\_\{1\}\)\\leq h\(\\lambda\_\{2\}\-\\lambda\_\{1\}\)\+\\varepsilon\.Lettingε↓0\\varepsilon\\downarrow 0, and using nondecreasingness for the other direction, proves \([20](https://arxiv.org/html/2607.22774#S5.E20)\)\. ∎

###### Corollary 5\.2\(Exact continuation geometry\)\.

Forh≥1h\\geq 1,Ch​\(s;λ\)=P​Vh−1​\(s;λ\)C\_\{h\}\(s;\\lambda\)=PV\_\{h\-1\}\(s;\\lambda\)is nondecreasing, concave, and\(h−1\)\(h\-1\)\-Lipschitz inλ\\lambda\. Consequently, the complete continuation riskλ\+Ch​\(s;λ\)\\lambda\+C\_\{h\}\(s;\\lambda\)has cost slopes in\[1,h\]\[1,h\]wherever it is differentiable\.

###### Proof\.

Conditional expectation preserves nondecreasingness and concavity\. Applying \([20](https://arxiv.org/html/2607.22774#S5.E20)\) toVh−1V\_\{h\-1\}and then taking conditional expectation gives the\(h−1\)\(h\-1\)\-Lipschitz bound\. Addingλ\\lambdashifts every slope by one\. ∎

### 5\.2Structure preservation by the conditional head

###### Theorem 5\.4\(Structure\-preserving approximation\)\.

Assumeλ≥0\\lambda\\geq 0,0≤aj≤B0\\leq a\_\{j\}\\leq B, and0≤bj≤h−10\\leq b\_\{j\}\\leq h\-1for every component in \([17](https://arxiv.org/html/2607.22774#S4.E17)\)\. For everyh≥1h\\geq 1,Cθ​\(s,t,h,⋅\)C\_\{\\theta\}\(s,t,h,\\cdot\)is nondecreasing, concave, and\(h−1\)\(h\-1\)\-Lipschitz, with0≤Cθ≤B0\\leq C\_\{\\theta\}\\leq B\. The induced valueV^h​\(s,t;⋅\)\\widehat\{V\}\_\{h\}\(s,t;\\cdot\)in \([12](https://arxiv.org/html/2607.22774#S4.E12)\) is nondecreasing, concave, andhh\-Lipschitz\. The terminal boundaryV^0=g\\widehat\{V\}\_\{0\}=gholds exactly\.

###### Proof\.

Letzj​\(λ\)=aj\+bj​λz\_\{j\}\(\\lambda\)=a\_\{j\}\+b\_\{j\}\\lambda\. Differentiating the uncapped head yields

∂λC~θ=∑j=1Jwj​\(λ\)​bj,wj​\(λ\)=exp⁡\[−zj​\(λ\)/ρ\]∑kexp⁡\[−zk​\(λ\)/ρ\]\.\\partial\_\{\\lambda\}\\widetilde\{C\}\_\{\\theta\}=\\sum\_\{j=1\}^\{J\}w\_\{j\}\(\\lambda\)b\_\{j\},\\qquad w\_\{j\}\(\\lambda\)=\\frac\{\\exp\[\-z\_\{j\}\(\\lambda\)/\\rho\]\}\{\\sum\_\{k\}\\exp\[\-z\_\{k\}\(\\lambda\)/\\rho\]\}\.\(21\)The derivative is a convex combination of slopes in\[0,h−1\]\[0,h\-1\]\. The negative log\-sum\-exp of negative affine functions is concave, while the factor1/J1/Jadds only a constant\. Because everyzj≥0z\_\{j\}\\geq 0, the normalized soft minimum is nonnegative\. Taking the minimum with the constantBBpreserves nondecreasingness and concavity and gives0≤Cθ≤B0\\leq C\_\{\\theta\}\\leq B\. ThusCθC\_\{\\theta\}is\(h−1\)\(h\-1\)\-Lipschitz\.

The functionλ\+Cθ\\lambda\+C\_\{\\theta\}is nondecreasing and concave with slope bounded byhh\. Taking its pointwise minimum with the constantg​\(s\)g\(s\)preserves these properties\. The terminal statement follows from \([11](https://arxiv.org/html/2607.22774#S4.E11)\)\. ∎

### 5\.3Uniform Bellman residual\-to\-value bound

Define the Bellman operator

\(Tλ​f\)​\(s\)=min⁡\{g​\(s\),λ\+\(P​f\)​\(s\)\}\.\(T\_\{\\lambda\}f\)\(s\)=\\min\\\{g\(s\),\\lambda\+\(Pf\)\(s\)\\\}\.\(22\)For bounded functionsffandqq,

‖Tλ​f−Tλ​q‖∞≤‖f−q‖∞,\\left\\lVert T\_\{\\lambda\}f\-T\_\{\\lambda\}q\\right\\rVert\_\{\\infty\}\\leq\\left\\lVert f\-q\\right\\rVert\_\{\\infty\},\(23\)because bothPPand the scalar minimum are non\-expansive in the supremum norm\. Define the layerwise population Bellman residual

ϵh​\(λ\)=‖V^h−Tλ​V^h−1‖∞\.\\epsilon\_\{h\}\(\\lambda\)=\\left\\lVert\\widehat\{V\}\_\{h\}\-T\_\{\\lambda\}\\widehat\{V\}\_\{h\-1\}\\right\\rVert\_\{\\infty\}\.\(24\)
###### Theorem 5\.5\(Residual\-to\-value bound\)\.

IfV^0=V0=g\\widehat\{V\}\_\{0\}=V\_\{0\}=g, then for everyh≥1h\\geq 1,

‖V^h−Vh‖∞≤∑k=1hϵk​\(λ\)\.\\left\\lVert\\widehat\{V\}\_\{h\}\-V\_\{h\}\\right\\rVert\_\{\\infty\}\\leq\\sum\_\{k=1\}^\{h\}\\epsilon\_\{k\}\(\\lambda\)\.\(25\)Ifϵk​\(λ\)≤ϵ¯k\\epsilon\_\{k\}\(\\lambda\)\\leq\\bar\{\\epsilon\}\_\{k\}uniformly forλ∈Λ\\lambda\\in\\Lambda, then

supλ∈Λ‖V^h​\(⋅;λ\)−Vh​\(⋅;λ\)‖∞≤∑k=1hϵ¯k\.\\sup\_\{\\lambda\\in\\Lambda\}\\left\\lVert\\widehat\{V\}\_\{h\}\(\\cdot;\\lambda\)\-V\_\{h\}\(\\cdot;\\lambda\)\\right\\rVert\_\{\\infty\}\\leq\\sum\_\{k=1\}^\{h\}\\bar\{\\epsilon\}\_\{k\}\.\(26\)

###### Proof\.

Adding and subtractingTλ​V^h−1T\_\{\\lambda\}\\widehat\{V\}\_\{h\-1\}, then applying \([23](https://arxiv.org/html/2607.22774#S5.E23)\), gives

‖V^h−Vh‖∞≤ϵh​\(λ\)\+‖V^h−1−Vh−1‖∞\.\\left\\lVert\\widehat\{V\}\_\{h\}\-V\_\{h\}\\right\\rVert\_\{\\infty\}\\leq\\epsilon\_\{h\}\(\\lambda\)\+\\left\\lVert\\widehat\{V\}\_\{h\-1\}\-V\_\{h\-1\}\\right\\rVert\_\{\\infty\}\.Iterating from the exact terminal boundary proves \([25](https://arxiv.org/html/2607.22774#S5.E25)\); taking the supremum overΛ\\Lambdaproves \([26](https://arxiv.org/html/2607.22774#S5.E26)\)\. ∎

### 5\.4Greedy\-policy performance from continuation residuals

The deployed policy usesCθ,hC\_\{\\theta,h\}directly\. Define its continuation residual by

δh​\(λ\)=‖Cθ,h−P​V^h−1‖∞\.\\delta\_\{h\}\(\\lambda\)=\\left\\lVert C\_\{\\theta,h\}\-P\\widehat\{V\}\_\{h\-1\}\\right\\rVert\_\{\\infty\}\.\(27\)The minimum operator is 1\-Lipschitz, soϵh​\(λ\)≤δh​\(λ\)\\epsilon\_\{h\}\(\\lambda\)\\leq\\delta\_\{h\}\(\\lambda\)\. LetVhπ^V\_\{h\}^\{\\widehat\{\\pi\}\}be the value obtained by the greedy policy in \([19](https://arxiv.org/html/2607.22774#S4.E19)\)\.

###### Theorem 5\.6\(Greedy\-policy performance\)\.

For everyss,hh, andλ\\lambda,

0≤Vhπ^​\(s;λ\)−Vh​\(s;λ\)≤2​∑k=1hδk​\(λ\)\.0\\leq V\_\{h\}^\{\\widehat\{\\pi\}\}\(s;\\lambda\)\-V\_\{h\}\(s;\\lambda\)\\leq 2\\sum\_\{k=1\}^\{h\}\\delta\_\{k\}\(\\lambda\)\.\(28\)The same statement holds uniformly overΛ\\Lambdawhen the continuation residuals admit uniform bounds on that interval\.

###### Proof\.

LetDh=‖Vhπ^−V^h‖∞D\_\{h\}=\\left\\lVert V\_\{h\}^\{\\widehat\{\\pi\}\}\-\\widehat\{V\}\_\{h\}\\right\\rVert\_\{\\infty\}\. If the greedy rule stops, both quantities equalgg\. If it continues, the policy recursion and \([27](https://arxiv.org/html/2607.22774#S5.E27)\) give

Dh≤Dh−1\+δh​\(λ\)\.D\_\{h\}\\leq D\_\{h\-1\}\+\\delta\_\{h\}\(\\lambda\)\.SinceD0=0D\_\{0\}=0,Dh≤∑k=1hδk​\(λ\)D\_\{h\}\\leq\\sum\_\{k=1\}^\{h\}\\delta\_\{k\}\(\\lambda\)\. Theorem[5\.5](https://arxiv.org/html/2607.22774#S5.Thmtheorem5)andϵk≤δk\\epsilon\_\{k\}\\leq\\delta\_\{k\}also give

‖V^h−Vh‖∞≤∑k=1hδk​\(λ\)\.\\left\\lVert\\widehat\{V\}\_\{h\}\-V\_\{h\}\\right\\rVert\_\{\\infty\}\\leq\\sum\_\{k=1\}^\{h\}\\delta\_\{k\}\(\\lambda\)\.Finally,Vh≤Vhπ^V\_\{h\}\\leq V\_\{h\}^\{\\widehat\{\\pi\}\}becauseVhV\_\{h\}is optimal\. Combining the last two inequalities with the triangle inequality proves \([28](https://arxiv.org/html/2607.22774#S5.E28)\)\. ∎

### 5\.5Theoretical implications

The results establish four linked properties\. First, the structured head is reliable along the continuous cost axis by construction: its continuation values and induced full values satisfy the exact monotonicity, concavity, and remaining\-horizon Lipschitz pattern\. Second, the terminal boundary is exact and the continuation slope range matches the number of future acquisitions still available\. Third, uniformly small Bellman residuals control the value approximation accumulated over a finite horizon\. Fourth, uniformly small continuation residuals connect fitted continuation learning to an explicit performance bound for the deployed greedy policy\. The cost\-geometry audit, held\-out Bellman diagnostics, and direct policy evaluation in Section[6](https://arxiv.org/html/2607.22774#S6)are organized around these quantities\.

## 6Experiments

### 6\.1Experimental setup

#### Evaluation settings\.

Three complementary settings are considered\. The Gaussian/DRE setting is a controlled binary sequential problem in which a frozen density\-ratio estimator supplies the log\-likelihood\-ratio state used by every stopping solver\. It evaluates transfer to six held\-out cost–horizon pairs under an estimated state representation\. The time\-varying non\-Gaussian mixture changes its weights, means, and variances over time and uses an oracle log\-likelihood ratio, thereby separating stopping approximation from state\-estimation error\. It evaluates held\-out operating points and includes one independently fitted CFL anchor\.

FordA is the primary real time\-series benchmark and uses its standard train/test split\. Its frozen posterior model contains only causal convolutions and per\-time\-step LayerNorm, so every stopping decision uses the observed prefix alone\. All stopping methods reuse the same leakage\-free state trajectories\. Test prefix accuracy rises from 58\.71% at the first block to 76\.36%, 84\.24%, 90\.00%, 94\.09%, and 95\.61% at blocks 5, 10, 20, 35, and 50, respectively, yielding a nontrivial accuracy–earliness trade\-off\.

#### Compared methods\.

We compare CC\-AOS with three references\. Per\-setting FIRMBOUND with Convex Function Learning \(CFL\) fits an independent finite\-horizon solver at each requested operating point\. The per\-horizon reference retains the continuation architecture and continuous\-cost training used by CC\-AOS but fits a separate model for each horizon, isolating the effect of horizon amortization\. The static reference selects a stopping threshold on validation data separately for every\(λ,H\)\(\\lambda,H\)\. Within each setting, all methods use the same state trajectories and terminal\-risk definition\.

#### Metrics and statistics\.

LetTiT\_\{i\}be the stopping time selected for test trajectoryii, and letτi=Ti−t0\\tau\_\{i\}=T\_\{i\}\-t\_\{0\}be the number of samples acquired after the initial decision timet0t\_\{0\}\. The primary metric is the empirical posterior objective

J^​\(λ,H\)=1N​∑i=1N\[g​\(STi\(i\)\)\+λ​τi\],\\widehat\{J\}\(\\lambda,H\)=\\frac\{1\}\{N\}\\sum\_\{i=1\}^\{N\}\\left\[g\\\!\\left\(S\_\{T\_\{i\}\}^\{\(i\)\}\\right\)\+\\lambda\\tau\_\{i\}\\right\],\(29\)which combines terminal posterior risk and sampling cost; lower values are better\. Classification error and the expected number of additional samples are retained as diagnostic quantities\. Main pairwise comparisons use paired bootstrap confidence intervals over the common FordA test trajectories\. The reported FordA checkpoint comes from one frozen training seed, so these intervals quantify test\-trajectory variation rather than variation across training runs\.

### 6\.2Generalization to unseen operating points

#### Primary FordA comparison\.

One CC\-AOS checkpoint is trained onH=\{20,30,40,50\}H=\\\{20,30,40,50\\\}andλ∈\[0\.0005,0\.008\]\\lambda\\in\[0\.0005,0\.008\], then evaluated directly at the six interleaved pairs formed byH=\{25,35,45\}H=\\\{25,35,45\\\}andλ=\{0\.002375,0\.006125\}\\lambda=\\\{0\.002375,0\.006125\\\}\. Table[1](https://arxiv.org/html/2607.22774#S6.T1)compares the posterior objectives, and Table[2](https://arxiv.org/html/2607.22774#S6.T2)reports the paired trajectory\-level differences from CFL\. CC\-AOS has a lower posterior objective than per\-setting CFL at all six pairs\. Relative reductions range from 0\.95% to 31\.29%, with a mean of 15\.75%\. The paired\-bootstrap 95% interval lies entirely below zero at five pairs; the interval atH=25,λ=0\.006125H=25,\\lambda=0\.006125crosses zero slightly\.

Table 1:Posterior objective of CC\-AOS and baselines at six unseen FordA operating points\.Posterior objective↓\\downarrowHHλ\\lambdaCC\-AOSCFLStatic250\.0023750\.115650\.124720\.11659250\.0061250\.162660\.164230\.16445350\.0023750\.101290\.130410\.10268350\.0061250\.156250\.192200\.15311450\.0023750\.094320\.137260\.09126450\.0061250\.153090\.177980\.14839Six\-pair mean0\.130540\.154470\.12941
- •Lower is better\. Bold and underline mark the best and second\-best objective in each row, respectively\. All values use the causal per\-time\-step LayerNorm pipeline\.

Table 2:Paired trajectory\-level comparison between CC\-AOS and per\-setting CFL\.HHλ\\lambdaΔ↓\\Delta\\downarrow95% CIRelative reduction \(%\)↑\\uparrow250\.002375\-0\.00907\[−0\.01148,−0\.00646\-0\.01148,\-0\.00646\]7\.27250\.006125\-0\.00156\[−0\.00334,0\.00018\-0\.00334,\\phantom\{\-\}0\.00018\]0\.95350\.002375\-0\.02912\[−0\.03162,−0\.02669\-0\.03162,\-0\.02669\]22\.33350\.006125\-0\.03595\[−0\.03967,−0\.03225\-0\.03967,\-0\.03225\]18\.70450\.002375\-0\.04295\[−0\.04596,−0\.04007\-0\.04596,\-0\.04007\]31\.29450\.006125\-0\.02489\[−0\.02891,−0\.02114\-0\.02891,\-0\.02114\]13\.98Six\-pair mean\-0\.02392—15\.75
- •Δ=JCC​\-​AOS−JCFL\\Delta=J\_\{\\mathrm\{CC\\text\{\-\}AOS\}\}\-J\_\{\\mathrm\{CFL\}\}; negative values favor CC\-AOS\. Confidence intervals are paired bootstrap intervals over the common FordA test trajectories\.

Against the same\-architecture per\-horizon reference, the mean positive posterior\-objective gap is 0\.49% and the maximum is 1\.66%, indicating a small empirical cost for sharing across horizons\. Against the static threshold, CC\-AOS wins at three of six pairs and has a six\-pair mean of 0\.13054 versus 0\.12941\. The two methods are therefore essentially tied on the six\-pair average\.

#### Controlled\-process validation\.

The controlled settings test whether the same conclusion persists under different state and transition mechanisms\. In the Gaussian/DRE setting, the shared model improves on per\-setting CFL at all six held\-out pairs, with an average objective reduction of 10\.99%\. In the time\-varying non\-Gaussian mixture, the mean positive gap to same\-architecture per\-horizon models is 0\.55% and the maximum is 1\.53%\. At the anchorH=35,λ=0\.00325H=35,\\lambda=0\.00325, CC\-AOS obtains0\.25041, compared with0\.28250for CFL, an 11\.36% reduction\.

#### Bellman\-fit diagnostic\.

As a separate diagnostic of the joint fitted recursion, we evaluate held\-out one\-step full\-value residuals using the target definition in Section 4\.3 on FordA states excluded from fitting\. Across the audited operating points, the residual MAE ranges from0\.002630\.00263to0\.011020\.01102, the RMSE from0\.009750\.00975to0\.023130\.02313, and the 95th percentile from0\.008010\.00801to0\.045060\.04506\. These finite\-sample quantities provide empirical counterparts to the layerwise approximation errors in Theorem[5\.5](https://arxiv.org/html/2607.22774#S5.Thmtheorem5)and diagnose how closely the learned recursion matches its held\-out Bellman targets\.

### 6\.3Ablation of the structure\-preserving head

To isolate the contribution of the cost\-structure constraints, we compare the normalized soft\-min affine head with a parameter\-matched unstructured head on the controlled DRE task\. The two variants use the same state representation, recursive targets, and training protocol; only the continuation head is changed\. Table[3](https://arxiv.org/html/2607.22774#S6.T3)reports both task performance and violations of the cost\-dependent properties established in Section 5\.

Table 3:Parameter\-matched ablation of the structure\-preserving continuation head on the controlled DRE task\.MetricStructured headParameter\-matchedunstructured headMean task objective↓\\downarrow0\.0595010\.059223Monotonicity violations↓\\downarrow07\.10%Continuation\-concavity violations↓\\downarrow01\.91%Value\-concavity violations↓\\downarrow01\.69%Value\-Lipschitz violations↓\\downarrow00\.38%
- •Lower is better for every metric\. Bold and underline mark the best and second\-best values, respectively\. Violation rates are proportions of audited cost cells\.

The unstructured head has the lower mean task objective by0\.0002780\.000278, whereas the structured head eliminates all audited monotonicity, continuation\-concavity, value\-concavity, and value\-Lipschitz violations\. The parameter\-matched comparison therefore attributes the structural behavior to the proposed head rather than to model size, with only a small observed difference in task objective\. A complementary FordA audit over 21 costs, three unseen horizons, and 256 trajectories examines 2,962,080 slope cells and detects no violation by the structured model\.

### 6\.4Computational efficiency

Table[4](https://arxiv.org/html/2607.22774#S6.T4)reports the observed training cost and number of independently fitted solver instances for the evaluated FordA configurations\. CC\-AOS trains one shared model in 18\.04 s\. Training three same\-architecture per\-horizon references takes 33\.48 s in total, while fitting CFL independently at the six evaluated operating points takes approximately 53 min\.

Table 4:Observed training and model\-management cost on FordA\.MethodTraining scopeIndependent fits↓\\downarrowTotal wall\-clocktime↓\\downarrowCC\-AOSContinuous costs and multiple horizons118\.04 sPer\-horizon referenceContinuous costs; one model per horizon333\.48 sPer\-setting CFLOne solver per\(λ,H\)\(\\lambda,H\)6≈53\\approx 53min
- •Lower is better\. Bold and underline mark the best and second\-best values, respectively\. Wall\-clock values are observed measurements for the evaluated configurations\.

The timing comparison follows the deployment granularity of the methods: CC\-AOS produces one checkpoint for the complete queried family, the per\-horizon reference produces three checkpoints, and CFL produces six operating\-point\-specific solver instances\. This observed comparison directly measures the amortization achieved in the evaluated setting\.

## 7Discussion

CC\-AOS converts a collection of operating\-point\-specific finite\-horizon solvers into one cost\- and horizon\-conditioned fitted stopping model\. Across the leakage\-free FordA evaluation, one checkpoint serves unseen cost–horizon combinations and improves on per\-setting CFL at every tested pair\. The controlled Gaussian/DRE and time\-varying mixture results provide complementary evidence under estimated and oracle state representations\.

The practical value is not that CC\-AOS produces the lowest objective under every comparison\. The strong static threshold is competitive and is marginally better on the six\-pair FordA average\. Rather, CC\-AOS supplies a dynamic, state\-dependent stopping boundary, a shared continuation surface, and direct multi\-operating\-point deployment\. A static threshold does not represent a general finite\-horizon Bellman boundary or expose a continuation value that can be queried across cost and remaining horizon\.

CC\-AOS retains the data\-driven state interface used by FIRMBOUND while changing the granularity of the stopping solver\. Instead of fitting a continuation stack for one requested operating point, it fits a conditional family through shared recursive targets\. The structure\-preserving head gives this family verifiable behavior along the continuous cost axis\. The observed training times and one\-checkpoint representation further support its use when several latency or acquisition\-cost configurations must be maintained\.

### 7\.1Limitations and future work

The current real\-data evidence is concentrated on binary FordA, and the primary checkpoint comes from one frozen training seed; paired bootstrap intervals quantify variation across common test trajectories rather than training randomness\. The held\-out operating points mainly test cost interpolation within the modeled interval and interleaved\-horizon generalization\. In addition, the stopping solver depends on a fixed upstream causal state estimator, and the present formulation assumes a constant per\-observation cost\. Natural extensions include multiclass or higher\-dimensional belief states, larger real sequential benchmarks, online cost calibration, time\- or state\-dependent acquisition costs, and verifiable structure along the horizon axis\.

## 8Conclusion

Finite\-horizon optimal stopping is commonly re\-solved when the sampling cost or horizon changes, limiting efficient multi\-operating\-point deployment\. CC\-AOS represents a family of such problems with one state\-, time\-, horizon\-, and cost\-conditioned continuation model learned by joint amortized fitted backward induction\. Its normalized soft\-min affine head translates exact monotonicity, concavity, and remaining\-horizon Lipschitz geometry into by\-construction guarantees, while Bellman and continuation residuals connect approximation quality to value and greedy\-policy performance\. Across controlled Gaussian/DRE, time\-varying non\-Gaussian, and leakage\-free FordA experiments, one checkpoint serves unseen cost–horizon combinations; on FordA it improves on per\-setting CFL at all six pairs and remains essentially tied with a strong static threshold on average\. The results support structured amortization as a practical route from repeated finite\-horizon solver stacks to one geometrically well\-behaved model that can be queried directly across operating points\.

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