无需重训，只需混搭：基于混合超网络的网页智能体适配方法

# Don’t Retrain, Remix: Adaptation of Web Agents using Mixture Hypernetworks
Source: [https://arxiv.org/html/2610.22539](https://arxiv.org/html/2610.22539)
###### Abstract

The web is not static or uniform\. Websites that serve the same purpose can look and navigate very differently\. Even within a single site, successive redesigns can change how familiar features are presented and accessed\. Humans adapt to this by recognizing familiar interaction patterns and reusing what they already know\. We give web agents a similar ability withWebMix, a mixture hypernetwork that adapts them to unseen interfaces by composing previously trained low\-rank adapters\.WebMixbegins with a bank of adapters trained on source interfaces\. At deployment, a scout agent explores the target interface and summarizes its observations into a compact manual\. A lightweight router uses this manual to predict mixing coefficients over the adapter bank, which combine the existing adapters into a new one for inference\. Adapting to a new interface takes only a scouting run and a forward pass, without gradient updates on the target\. With Qwen and Llama models, we evaluatedomain transferon TimeWarp and WebArena\-Lite\-v2, andversion transferon TimeWarp\. In all three settings,WebMixmatches or outperforms in\-context learning and LoRA baselines, often matching the average success of specialists trained directly on those interfaces\. On TimeWarp domain transfer, it gains about 8 percentage points on average over the best baseline with both models\. These gains do not come at the cost of seen interfaces, whereWebMixkeeps its lead, and extend to version transfer\. Adapting to a new interface can therefore rely on exploration and composition instead of retraining\.

![Refer to caption](https://arxiv.org/html/2610.22539v1/teaserV1.0.3.png)Figure 1:WebMixat a glance\.We train a bank of interface\-specific LoRA adapters, each KL\-anchored to the base model on the other source interfaces\. At deployment, a scout explores the unseen interface and writes a short manual\. A router reads the encoded manual and mixes the bank into one adapter for the web agent\. We test transfer to unseen domains and unseen versions\.## 1Introduction

Humans compose and transfer skills to unfamiliar interfaces with little effort\. A person who has booked flights on American Airlines’ website can book the same flight through their credit card’s travel portal, following the same search, filter, and checkout flow in a layout they have never seen\. The same holds across time: when a site is redesigned, its regular users are slowed down for a few visits, not retaught\. A fine\-tuned web agent can do neither\. Mind2Web\([Deng et al\., 2023](https://arxiv.org/html/2610.22539#bib.bib37)\)reports a steady drop in performance from its cross\-task split to its cross\-website and cross\-domain splits\. TimeWarp\([Ishmam and Marino, 2026](https://arxiv.org/html/2610.22539#bib.bib32)\), which renders the same sites in the styles of six eras of web design, shows that an agent trained on one era degrades on the next\. Frontier models reduce this brittleness through scale\([Zheng et al\., 2024](https://arxiv.org/html/2610.22539#bib.bib23);[He et al\., 2024](https://arxiv.org/html/2610.22539#bib.bib25);[Anthropic, 2024](https://arxiv.org/html/2610.22539#bib.bib29);[AI, 2025](https://arxiv.org/html/2610.22539#bib.bib30);[Müller and Žunič, 2024](https://arxiv.org/html/2610.22539#bib.bib31)\), but their closed weights and serving cost raise privacy and deployment concerns\([Qi et al\., 2024](https://arxiv.org/html/2610.22539#bib.bib26);[Zhang et al\., 2025](https://arxiv.org/html/2610.22539#bib.bib24)\)\.

Small open\-weight models that run on commodity hardware are the practical alternative\([Grattafiori et al\., 2024](https://arxiv.org/html/2610.22539#bib.bib27);[Qwen Team, 2026](https://arxiv.org/html/2610.22539#bib.bib28)\), and this brittleness hits them hardest\. They become competent web agents only after fine\-tuning on demonstrations, typically through a low\-rank adapter \(LoRA\)\([Hu et al\., 2022](https://arxiv.org/html/2610.22539#bib.bib34)\)on a frozen base, and every new site or redesign calls for a new corpus and a new adapter\. Their performance is therefore bounded by the demonstrations they are trained on, and web demonstrations are expensive to collect: each one requires a human operating a live site, and the site may be redesigned before the corpus is large enough\. We therefore ask whether an agent can get more out of the demonstrations it already has\. Facebook has altered its interface repeatedly since 2004, but it has always had a search bar\. Amazon and Wikipedia serve different purposes entirely, yet browsing a catalog of products and browsing an encyclopedia of articles are the same operation: filtering, paging, and selecting from a list\. If an adapter trained on one interface has already learned these operations, that knowledge ought to be transferable to the next\.

Recent work adapts small web agents with less supervision\. It reduces target\-interface demonstrations to one or two examples\([Verma et al\., 2025](https://arxiv.org/html/2610.22539#bib.bib22)\), synthesizes trajectories through autonomous exploration of the target\([Murty et al\., 2024](https://arxiv.org/html/2610.22539#bib.bib19)\), transfers reusable action routines across websites\([Wang et al\., 2024](https://arxiv.org/html/2610.22539#bib.bib18)\), or learns lightweight alignment vectors from deployment\-time interaction\([Chen et al\., 2025](https://arxiv.org/html/2610.22539#bib.bib17)\)\. All of these still treat each new interface as a separate adaptation problem\. They reuse knowledge about websites, but none of them learns how to adapt, so the next interface starts from scratch again\. A hypernetwork can learn such an adaptation rule, because it generates an adapter from a description of the target\([Ha et al\., 2017](https://arxiv.org/html/2610.22539#bib.bib36);[Charakorn et al\., 2025](https://arxiv.org/html/2610.22539#bib.bib35)\)\. For web interfaces, however, generating adapter weights directly is hard\. A family of interfaces provides only a handful of conditionings from which to regress millions of weights, and LoRA factors are not identifiable, so adapters that behave alike can lie far apart in weight space\([Charakorn et al\., 2025](https://arxiv.org/html/2610.22539#bib.bib35)\)\.

We introduceWebMix, a mixture hypernetwork\([Qu et al\., 2023](https://arxiv.org/html/2610.22539#bib.bib2)\)that learns to adapt web agents by composing LoRA adapters trained on familiar interfaces\. A scout briefly explores a new interface and summarizes its observations in a compact manual\. Conditioned on this description, a small router predicts one mixing coefficient per adapter and combines their weight updates into a single adapter for the frozen base policy\. We train the router on source demonstrations through the composed policy and keep the adapter bank fixed\. Adaptation then requires only scouting and one forward pass, with no further gradient updates\.

On TimeWarp\([Ishmam and Marino, 2026](https://arxiv.org/html/2610.22539#bib.bib32)\),WebMixachieves 56\.1% success on the unseen News domain with Qwen\-3\.5 4B, exceeding transferred LoRA by 10\.6 percentage points\. Across held\-out domains, average gains reach 8\.3 points for Qwen and 7\.5 points for Llama\-3\.1 8B, nearly doubling Llama’s success and matching the average performance of target\-trained specialists for both models\. Improvements extend to held\-out TimeWarp versions and WebArena\-Lite\-v2 domains, where Qwen gains 3\.2 and 4\.6 points, respectively\. The benefits also extend to seen interfaces: with target domains included in training, Qwen achieves 42\.4% overall success on WebArena\-Lite\-v2, outperforming LoRA trained on all domains by 4\.3 points\.

Our contributions are the following:

- ✥We introduceWebMix, a mixture hypernetwork that learns to compose existing LoRA adapters from a scouted interface description\. The learned adaptation rule transfers to new websites and redesigns without parameter optimization on the target interface \(§[3](https://arxiv.org/html/2610.22539#S3)\)\.
- ✥We evaluate domain transfer on TimeWarp and WebArena\-Lite\-v2 and version transfer on TimeWarp with two open\-weight models \(§[4](https://arxiv.org/html/2610.22539#S4)\)\. On held\-out TimeWarp domains,WebMixmatches target\-trained specialists on average for both models, reaches 56\.1% success on unseen News with Qwen \(\+10\.6 percentage points over transferred LoRA\), and nearly doubles Llama’s average domain\-transfer success\.
- ✥We decompose held\-out imitation risk into bank approximation and routing terms, clarifying the limits of adapter composition \(§[D](https://arxiv.org/html/2610.22539#A4)\)\. Ablations compare composition with direct adapter generation and examine how scouting and adapter regularization contribute to transfer \(§[4\.2](https://arxiv.org/html/2610.22539#S4.SS2)\)\.

## 2Related Work

##### Web Agents and Interface Generalization\.

Web\-agent benchmarks test generalization to unseen interfaces along different axes of variation\. Along the cross\-site axis, Mind2Web evaluates agents on static demonstrations split by task, website, and domain\([Deng et al\., 2023](https://arxiv.org/html/2610.22539#bib.bib37)\), while WebArena and WorkArena provide self\-hosted, fully functional websites\([Zhou et al\., 2024](https://arxiv.org/html/2610.22539#bib.bib38);[Drouin et al\., 2024](https://arxiv.org/html/2610.22539#bib.bib39)\), and WebVoyager and Online\-Mind2Web move evaluation to live websites\([He et al\., 2024](https://arxiv.org/html/2610.22539#bib.bib25);[Xue et al\., 2025](https://arxiv.org/html/2610.22539#bib.bib12)\)\. Along the temporal axis, TimeWarp evaluates agents on successive eras of web design\([Ishmam and Marino, 2026](https://arxiv.org/html/2610.22539#bib.bib32)\)\. Complementary diagnostic suites isolate individual factors by perturbing the layout, interaction semantics, visual attributes, or network events of otherwise fixed tasks\([Bai et al\., 2026](https://arxiv.org/html/2610.22539#bib.bib5);[Kara et al\., 2025](https://arxiv.org/html/2610.22539#bib.bib6)\)\. On all of these benchmarks, agents lose accuracy on interfaces they were not trained on\. Mind2Web reports substantially lower performance for fine\-tuned agents on unseen websites and domains than on its cross\-task split\([Deng et al\., 2023](https://arxiv.org/html/2610.22539#bib.bib37)\), and TimeWarp finds that text\-based agents become brittle under interface changes after fine\-tuning on a single version\([Ishmam and Marino, 2026](https://arxiv.org/html/2610.22539#bib.bib32)\)\. A practical web agent should therefore adapt at test time to visually and functionally unfamiliar interfaces without relying on extensive demonstrations\. We evaluateWebMixalong both axes\. TimeWarp provides both domain shifts and version shifts, and WebArena\-Lite\-v2\([Liu et al\., 2024](https://arxiv.org/html/2610.22539#bib.bib3);[Liu et al\., 2026](https://arxiv.org/html/2610.22539#bib.bib20)\)provides domain shifts\.

##### Adapting Web Agents to Unseen Interfaces\.

An agent facing an unfamiliar interface must learn something about it before it can act well\. Prior work differs in where that knowledge comes from and where it is kept\. The most direct source is a human: AdaptAgent adapts multimodal agents to new websites from up to two human demonstrations, using in\-context demonstrations for proprietary models and meta\-learned adaptation for open\-weight models\([Verma et al\., 2025](https://arxiv.org/html/2610.22539#bib.bib22)\)\. Demonstrations, however, are costly to collect, so a second line of work lets the agent gather its own experience through guided or unguided exploration\([Murty et al\., 2024](https://arxiv.org/html/2610.22539#bib.bib19);[Su et al\., 2025](https://arxiv.org/html/2610.22539#bib.bib11);[Pahuja et al\., 2025](https://arxiv.org/html/2610.22539#bib.bib14);[Sun et al\., 2024](https://arxiv.org/html/2610.22539#bib.bib13)\)\. Beyond demonstrations and raw experience, an agent can also carry forward explicit knowledge\. Non\-parametric methods keep the weights fixed and store what the agent learns outside them: as reusable workflows\([Wang et al\., 2024](https://arxiv.org/html/2610.22539#bib.bib18)\)or episodic memories\([Sarch et al\., 2025](https://arxiv.org/html/2610.22539#bib.bib1)\), as an instruction manual written through interaction\([Chen et al\., 2024](https://arxiv.org/html/2610.22539#bib.bib4)\), as skills distilled from practice\([Zheng et al\., 2025](https://arxiv.org/html/2610.22539#bib.bib15);[Wang et al\., 2026b](https://arxiv.org/html/2610.22539#bib.bib16)\), or as summaries of environment dynamics learned at deployment time\([Chen et al\., 2025](https://arxiv.org/html/2610.22539#bib.bib17)\)\. Like AutoManual\([Chen et al\., 2024](https://arxiv.org/html/2610.22539#bib.bib4)\),WebMixconstructs an explicit description of the environment from interaction\. AutoManual uses this description to condition planning, whereas we use it to predict a composition of existing adapters\. More generally, prior methods treat each new interface as a separate adaptation problem\.WebMixinstead learns an interface\-conditioned rule for composing adapters on the source interfaces and applies it unchanged to the target interface\. Target\-side exploration supplies the conditioning information, without target demonstrations or test\-time gradient updates\.

##### Adapter Composition and Generation\.

Low\-rank adaptation\([Hu et al\., 2022](https://arxiv.org/html/2610.22539#bib.bib34)\)lets many adapters share one frozen base model\. A growing body of work reuses such collections instead of training a new model for each task\([Yadav et al\., 2024](https://arxiv.org/html/2610.22539#bib.bib7)\)\. The simplest form of reuse combines adapters in weight space\. Merging methods add or average updates arithmetically\([Ilharco et al\., 2023](https://arxiv.org/html/2610.22539#bib.bib41);[Yadav et al\., 2023](https://arxiv.org/html/2610.22539#bib.bib42);[Wortsman et al\., 2022](https://arxiv.org/html/2610.22539#bib.bib40)\)\. Other approaches optimize composition coefficients on target examples, as in LoraHub\([Huang et al\., 2024](https://arxiv.org/html/2610.22539#bib.bib43)\), or compose adapters dynamically during inference through layer\-wise gating in MoLE\([Wu et al\., 2024](https://arxiv.org/html/2610.22539#bib.bib44)\)and token\-level routing in Arrow and PHATGOOSE\([Ostapenko et al\., 2024](https://arxiv.org/html/2610.22539#bib.bib45);[Muqeeth et al\., 2024](https://arxiv.org/html/2610.22539#bib.bib46)\)\. A more ambitious alternative skips the bank altogether and generates the adapter with a hypernetwork\([Ha et al\., 2017](https://arxiv.org/html/2610.22539#bib.bib36)\), conditioned on contexts and documents\([Charakorn et al\., 2026](https://arxiv.org/html/2610.22539#bib.bib8)\), task descriptions and prompts\([Charakorn et al\., 2025](https://arxiv.org/html/2610.22539#bib.bib35);[Liang et al\., 2025](https://arxiv.org/html/2610.22539#bib.bib47)\), whole code repositories\([Hotsko et al\., 2026](https://arxiv.org/html/2610.22539#bib.bib9)\), or per\-sample memories in continual learning\([Wang et al\., 2026a](https://arxiv.org/html/2610.22539#bib.bib10)\)\. Generation is flexible, but it is hard to learn\. The generator must emit every entry of every LoRA factor, and because LoRA factors are not identifiable, many different factors encode the same update\. This is manageable with abundant conditionings but not with few: Drag\-and\-Drop LLMs, for example, improve little over their training adapters when trained on only a few datasets\([Liang et al\., 2025](https://arxiv.org/html/2610.22539#bib.bib47)\), and a family of web interfaces provides only a handful\.WebMixsits between these two lines\. Like a hypernetwork, it maps a description of the target to an adapter in one forward pass\. Like merging, it builds that adapter from a frozen bank\. It predicts only one coefficient per bank adapter and emits a single standard LoRA adapter that an unmodified inference stack can serve\. We compare against Text\-to\-LoRA conditioned on the same interface representation, following[Hotsko et al\. \(2026\)](https://arxiv.org/html/2610.22539#bib.bib9), and analyze an idealization of it in the Appendix §[D\.2](https://arxiv.org/html/2610.22539#A4.SS2)\.

![Refer to caption](https://arxiv.org/html/2610.22539v1/methFinalV1.0.png)Figure 2:Overview ofWebMix\.A scout explores the target website\. Its Explorer interacts with the environment, and its Writer turns the observations into a compact manualmm\. An encoder mapsmmto an embeddinghh, from which a router predicts mixing weightsc1,…,c5c\_\{1\},\\dots,c\_\{5\}over a frozen bank of KL\-anchored LoRA adapters\. Their weighted sum is a site\-specific adapterΔ​W\\Delta W, added to the base weightsWθW\_\{\\theta\}of the agentπθ\\pi\_\{\\theta\}, which reads the accessibility tree and emits grounded actions \(e\.g\.,click \[31\]\)\. Websites vary across versions \(v1v\_\{1\}–v5v\_\{5\}\) and domains\.

## 3WebMix: Adaptation through Composition

Figure[2](https://arxiv.org/html/2610.22539#S2.F2)gives an overview\. §[3\.1](https://arxiv.org/html/2610.22539#S3.SS1)states the problem, §[3\.2](https://arxiv.org/html/2610.22539#S3.SS2)describes the mixture hypernetwork, and §[3\.3](https://arxiv.org/html/2610.22539#S3.SS3)describes its training and its adaptation to an unseen interface\.

### 3\.1Preliminaries

A web agent is a language model that completes a task on a website through a browser\. It reads a natural\-language goal and the current page, and it emits one browser action at a time, such as a click, a typed string, or a scroll, until the task ends\. A task succeeds if a verifier accepts the final state of the website\. A web environment consists of several*interfaces*\. Each interface is either a website domain, such as a wiki or an online shop, or a version,i\.e\., a design of the website\. Interfaces differ in presentation, navigation structure, and content\. Web agents are often trained using parameter\-efficient methods, such as Low Rank Adaptation \(LoRA\)\([Hu et al\., 2022](https://arxiv.org/html/2610.22539#bib.bib34)\)\. We now formalize the web environment, agents, LoRA, and our problem setting in the following paragraphs\.

##### Web environment and agents\.

We model the agent’s interaction with a web interface as a partially observable Markov decision process \(POMDP\)⟨𝒮,𝒜,𝒪,𝕋,𝕋0,𝕆,Rg⟩\\langle\\mathcal\{S\},\\mathcal\{A\},\\mathcal\{O\},\\mathbb\{T\},\\mathbb\{T\}\_\{0\},\\mathbb\{O\},R\_\{g\}\\rangle\([Kaelbling et al\., 1998](https://arxiv.org/html/2610.22539#bib.bib33)\)\. A states∈𝒮s\\in\\mathcal\{S\}comprises the website’s front\-end \(external\) and back\-end \(internal\) states\. An episode starts froms0∼𝕋0s\_\{0\}\\sim\\mathbb\{T\}\_\{0\}, and browser actionsa∈𝒜a\\in\\mathcal\{A\}change the state according tost\+1∼𝕋\(⋅∣st,at\)s\_\{t\+1\}\\sim\\mathbb\{T\}\(\\cdot\\mid s\_\{t\},a\_\{t\}\)\. The agent receives only an observationot∼𝕆\(⋅∣st\)∈𝒪o\_\{t\}\\sim\\mathbb\{O\}\(\\cdot\\mid s\_\{t\}\)\\in\\mathcal\{O\}, in the form of an accessibility tree\([Zhou et al\., 2024](https://arxiv.org/html/2610.22539#bib.bib38)\)\. A task has a natural\-language goalggand a verifierVg:𝒮→\{0,1\}V\_\{g\}:\\mathcal\{S\}\\rightarrow\\\{0,1\\\}, which defines the terminal rewardRg​\(sT\)=Vg​\(sT\)R\_\{g\}\(s\_\{T\}\)=V\_\{g\}\(s\_\{T\}\)at the final stepTT\. The agent is a policyπθ\\pi\_\{\\theta\}with parametersθ\\thetathat selects each action from the interaction history,at∼πθ\(⋅∣Ht\)a\_\{t\}\\sim\\pi\_\{\\theta\}\(\\cdot\\mid H\_\{t\}\), whereHt=\(g,o0:t,a0:t−1\)H\_\{t\}=\(g,o\_\{0:t\},a\_\{0:t\-1\}\)\. Letℰ\\mathcal\{E\}be the set of interfaces of the environment\. We index the source interfaces asℰsrc=\{v1,…,vK\}⊂ℰ\\mathcal\{E\}\_\{\\mathrm\{src\}\}=\\\{v\_\{1\},\\dots,v\_\{K\}\\\}\\subset\\mathcal\{E\}, soK=\|ℰsrc\|K=\|\\mathcal\{E\}\_\{\\mathrm\{src\}\}\|, and write an unseen interface asu∈ℰ∖ℰsrcu\\in\\mathcal\{E\}\\setminus\\mathcal\{E\}\_\{\\mathrm\{src\}\}\. Each source interfacevvhas a demonstration corpus𝒟v=\{\(x,y\)\}\\mathcal\{D\}\_\{v\}=\\\{\(x,y\)\\\}, and we write𝒟k=𝒟vk\\mathcal\{D\}\_\{k\}=\\mathcal\{D\}\_\{v\_\{k\}\}\. Each inputxxis a historyHtH\_\{t\}, rendered as a prompt that contains the goal, the accessibility tree of the current page, and a record of the earlier steps\. Each targety=\(y1,…,y\|y\|\)y=\(y\_\{1\},\\dots,y\_\{\|y\|\}\)is the expert’s response, a token sequence that contains its reasoning and one browser action\.

##### Low\-rank adaptation \(LoRA\)\.

Fine\-tuning all ofθ\\thetafor each interface would store one copy of the model per interface\. Low\-rank adaptation \(LoRA\)\([Hu et al\., 2022](https://arxiv.org/html/2610.22539#bib.bib34)\)keepsθ\\thetafrozen and learns a low\-rank update to selected weight matrices\. LetΩ\\Omegaindex the adapted weight matrices layer\-wise, and letWθ,ℓ∈ℝdℓout×dℓinW\_\{\\theta,\\ell\}\\in\\mathbb\{R\}^\{d\_\{\\ell\}^\{\\mathrm\{out\}\}\\times d\_\{\\ell\}^\{\\mathrm\{in\}\}\}be the frozen matrix atℓ∈Ω\\ell\\in\\Omega\. LoRA replaces it with

Wℓ=Wθ,ℓ\+Δ​Wℓ,Δ​Wℓ=αr​Bℓ​Aℓ,W\_\{\\ell\}=W\_\{\\theta,\\ell\}\+\\Delta W\_\{\\ell\},\\qquad\\Delta W\_\{\\ell\}=\\frac\{\\alpha\}\{r\}\\,B\_\{\\ell\}A\_\{\\ell\},\(1\)whereAℓ∈ℝr×dℓinA\_\{\\ell\}\\in\\mathbb\{R\}^\{r\\times d\_\{\\ell\}^\{\\mathrm\{in\}\}\}andBℓ∈ℝdℓout×rB\_\{\\ell\}\\in\\mathbb\{R\}^\{d\_\{\\ell\}^\{\\mathrm\{out\}\}\\times r\}are trainable factors,rris the adaptation rank, and the scaleα\\alphasets the size of the update independently ofrr\.AℓA\_\{\\ell\}is initialized randomly andBℓB\_\{\\ell\}is set to zero, so training starts from the base policy\. We call the collection of updatesΔ​θ=\(Δ​Wℓ\)ℓ∈Ω\\Delta\\theta=\(\\Delta W\_\{\\ell\}\)\_\{\\ell\\in\\Omega\}an*adapter*, and writeπθ\+Δ​θ\\pi\_\{\\theta\+\\Delta\\theta\}for the policy with the adapter installed\. Adapters can be held separate from the base weights and applied at serving time, so many adapters can share one frozen base model\. We adapt the attention projections \(query, key, value, output\) and the MLP projections \(gate, up, down\), withr=16r=16andα=32\\alpha=32\.

##### Problem setting\.

We start from a frozen base model and a set of*source*interfaces, each with a corpus of expert demonstrations\. Our goal is for the agent to succeed on an*unseen*interface\. In*domain transfer*, it is an unseen domain, and in*version transfer*, it is an unseen version\. The unseen interface has no demonstrations, and no parameter is optimized on it\. Before the agent starts its tasks, it may only make a brief, bounded visit to the unseen interface, which we call*scouting*\. We ask whether the parameters the agent needs on the unseen interface can be assembled from what it learned on the source interfaces, using this visit alone\.

### 3\.2The Mixture Hypernetwork

A hypernetwork\([Ha et al\., 2017](https://arxiv.org/html/2610.22539#bib.bib36)\)is a network that outputs the parameters of another network\.WebMixuses a hypernetworkFϕF\_\{\\phi\}with trainable parametersϕ\\phito produce the adapter for an interface from a bounded exploration record of it\. For an unseen interfaceuuwith recordEuE\_\{u\},FϕF\_\{\\phi\}outputs one update per adapted matrix, which is added to the frozen base weights,

\{Δ​Wu,ℓ\}ℓ∈Ω=Fϕ​\(Eu\),Wu,ℓ=Wθ,ℓ\+Δ​Wu,ℓ\.\\\{\\Delta W\_\{u,\\ell\}\\\}\_\{\\ell\\in\\Omega\}=F\_\{\\phi\}\(E\_\{u\}\),\\qquad W\_\{u,\\ell\}=W\_\{\\theta,\\ell\}\+\\Delta W\_\{u,\\ell\}\.\(2\)The updatesΔ​θu=\(Δ​Wu,ℓ\)ℓ∈Ω\\Delta\\theta\_\{u\}=\(\\Delta W\_\{u,\\ell\}\)\_\{\\ell\\in\\Omega\}produce the adapted policyπθ\+Δ​θu\\pi\_\{\\theta\+\\Delta\\theta\_\{u\}\}, with all other parameters unchanged\. The objective is to maximize task success onuuwithout target demonstrations or parameter optimization on the target interface\.FϕF\_\{\\phi\}has three parts, described below: a frozen bank of adapters, a small router with the trainable parametersϕ\\phi, and a frozen scout that maps the recordEvE\_\{v\}of an interfacevvto a representationhv∈ℝdch\_\{v\}\\in\\mathbb\{R\}^\{d\_\{c\}\}\.

##### The adapter bank\.

The bank holds one LoRA adapter per source interface\. For eachvkv\_\{k\}, the bank contains factors\{\(Aℓ\(k\),Bℓ\(k\)\)\}ℓ∈Ω\\\{\(A^\{\(k\)\}\_\{\\ell\},B^\{\(k\)\}\_\{\\ell\}\)\\\}\_\{\\ell\\in\\Omega\}of rankrrand scaleα\\alpha, with updatesΔ​Wℓ\(k\)=\(α/r\)​Bℓ\(k\)​Aℓ\(k\)\\Delta W^\{\(k\)\}\_\{\\ell\}=\(\\alpha/r\)\\,B^\{\(k\)\}\_\{\\ell\}A^\{\(k\)\}\_\{\\ell\}\. We writeΔ​θ\(k\)=\(Δ​Wℓ\(k\)\)ℓ∈Ω\\Delta\\theta^\{\(k\)\}=\(\\Delta W^\{\(k\)\}\_\{\\ell\}\)\_\{\\ell\\in\\Omega\}for adapterkk, andπ\(k\)=πθ\+Δ​θ\(k\)\\pi^\{\(k\)\}=\\pi\_\{\\theta\+\\Delta\\theta^\{\(k\)\}\}for the policy using it\. The bank\{Δ​θ\(1\),…,Δ​θ\(K\)\}\\\{\\Delta\\theta^\{\(1\)\},\\dots,\\Delta\\theta^\{\(K\)\}\\\}is the source knowledge thatWebMixtransfers to an unseen interface\. All bank adapters share the base model, the index setΩ\\Omega, the rankrr, and the scaleα\\alpha, so they can be combined matrix\-wise\. The bank is trained once \(Stage 1, §[3\.3\.1](https://arxiv.org/html/2610.22539#S3.SS3.SSS1)\) and stays frozen afterwards\.

##### What the hypernetwork predicts\.

A direct hypernetwork, such as Text\-to\-LoRA\([Charakorn et al\., 2025](https://arxiv.org/html/2610.22539#bib.bib35)\), predicts every entry of every LoRA factor\. The router ofWebMixpredicts onlyKKmixing coefficients, one per bank adapter\. It is a two\-layer perceptron with parametersϕ=\(U1,b1,U2,b2\)\\phi=\(U\_\{1\},b\_\{1\},U\_\{2\},b\_\{2\}\)that mapshvh\_\{v\}to a vector of mixture weights over theKKbank adapters,

cv=softmax⁡\(U2​SiLU⁡\(U1​hv\+b1\)\+b2\)∈ℝK,c\_\{v\}=\\operatorname\{softmax\}\\\!\\left\(U\_\{2\}\\,\\operatorname\{SiLU\}\(U\_\{1\}h\_\{v\}\+b\_\{1\}\)\+b\_\{2\}\\right\)\\in\\mathbb\{R\}^\{K\},\(3\)whereU1∈ℝdh×dcU\_\{1\}\\in\\mathbb\{R\}^\{d\_\{h\}\\times d\_\{c\}\},b1∈ℝdhb\_\{1\}\\in\\mathbb\{R\}^\{d\_\{h\}\},U2∈ℝK×dhU\_\{2\}\\in\\mathbb\{R\}^\{K\\times d\_\{h\}\},b2∈ℝKb\_\{2\}\\in\\mathbb\{R\}^\{K\}, anddhd\_\{h\}is the hidden width \(dh=256d\_\{h\}=256, Appendix[A\.3](https://arxiv.org/html/2610.22539#A1.SS3)\)\. The coefficientcv,kc\_\{v,k\}is the weight of bank adapterkkin the adapter forvv\. A single coefficient vector is shared across all adapted matrices\. We initializeU2U\_\{2\}andb2b\_\{2\}to zero, so the untrained router weights all bank adapters equally\.

##### How the bank is combined\.

The adapter for interfacevvis the weighted sum of the bank’s updates,

Δ​Wv,ℓ=∑k=1Kcv,k​Δ​Wℓ\(k\)\.\\Delta W\_\{v,\\ell\}=\\sum\_\{k=1\}^\{K\}c\_\{v,k\}\\,\\Delta W^\{\(k\)\}\_\{\\ell\}\.\(4\)We compute this sum by concatenating the bank factors along their rank dimension,

Av,ℓ=\[cv,1​Aℓ\(1\)cv,K​Aℓ\(K\)\],Bv,ℓ=\[Bℓ\(1\)⋯Bℓ\(K\)\]\.A\_\{v,\\ell\}=\\begin\{bmatrix\}c\_\{v,1\}A^\{\(1\)\}\_\{\\ell\}\\\\ \\vdots\\\\ c\_\{v,K\}A^\{\(K\)\}\_\{\\ell\}\\end\{bmatrix\},\\qquad B\_\{v,\\ell\}=\\begin\{bmatrix\}B^\{\(1\)\}\_\{\\ell\}&\\cdots&B^\{\(K\)\}\_\{\\ell\}\\end\{bmatrix\}\.\(5\)The product satisfiesBv,ℓ​Av,ℓ=∑kcv,k​Bℓ\(k\)​Aℓ\(k\)B\_\{v,\\ell\}A\_\{v,\\ell\}=\\sum\_\{k\}c\_\{v,k\}B^\{\(k\)\}\_\{\\ell\}A^\{\(k\)\}\_\{\\ell\}\. Averaging the two factors separately would instead weight each adapter’s own term bycv,k2c\_\{v,k\}^\{2\}and add cross termsBℓ\(j\)​Aℓ\(k\)B^\{\(j\)\}\_\{\\ell\}A^\{\(k\)\}\_\{\\ell\}withj≠kj\\neq k, which by Eq\.[9](https://arxiv.org/html/2610.22539#A1.E9)\(Appendix[A\.1](https://arxiv.org/html/2610.22539#A1.SS1)\) depends on how each adapter happens to be factored\. The composed factors have inner dimensionrout=K​rr\_\{\\mathrm\{out\}\}=Kr, and we setαout=K​α\\alpha\_\{\\mathrm\{out\}\}=K\\alpha, so thatαout/rout=α/r\\alpha\_\{\\mathrm\{out\}\}/r\_\{\\mathrm\{out\}\}=\\alpha/r\. The result is a standard LoRA adapter of rank at mostK​rKrand scaleαout\\alpha\_\{\\mathrm\{out\}\}, and installing it through Eq\.[2](https://arxiv.org/html/2610.22539#S3.E2)adds the same update as Eq\.[4](https://arxiv.org/html/2610.22539#S3.E4)to the base weights\. It is served by an unmodified LoRA inference stack\.

##### The scout\.

The scout turns a bounded visit to an interfacevvinto the representationhvh\_\{v\}that the router reads\. It has three parts: an explorer collects an exploration recordEvE\_\{v\}, a writer summarizesEvE\_\{v\}into a text*interface manual*mvm\_\{v\}, and a frozen sentence encoderψ\\psiembedsmvm\_\{v\}\. None of the three is trained at any stage\. Appendix[A\.5](https://arxiv.org/html/2610.22539#A1.SS5)describes each part\. The scout returns two objects for an interfacevv\. The first is the manualmvm\_\{v\}, a readable text summary of the interface\. The second is the conditioning vector, the normalized encoding of the manual,

hv=ψ⁡\(mv\)‖ψ⁡\(mv\)‖2∈ℝdc\.h\_\{v\}=\\frac\{\\psi\(m\_\{v\}\)\}\{\\\|\\psi\(m\_\{v\}\)\\\|\_\{2\}\}\\in\\mathbb\{R\}^\{d\_\{c\}\}\.\(6\)Onlyhvh\_\{v\}enters the hypernetwork, where the router maps it to the mixing coefficientscvc\_\{v\}of Eq\.[3](https://arxiv.org/html/2610.22539#S3.E3)\.

### 3\.3Training and Adaptation

WebMixassembles the parameters from low\-rank adapters\. It trains one adapter per source interface, keeps the adapters in a frozen bank, and learns a small network, the router, that decides how much of each adapter to use on a given interface\. Learning and adaptation proceed in three stages:

1. 1\.Stage 1: Adapter Bank Training \(§[3\.3\.1](https://arxiv.org/html/2610.22539#S3.SS3.SSS1)\)\.For each source interface, we fine\-tune one adapter on its demonstrations and freeze it in the bank\.
2. 2\.Stage 2: Router Training \(§[3\.3\.2](https://arxiv.org/html/2610.22539#S3.SS3.SSS2)\)\.The router learns to map the scout’s encoded manual of each source interface to mixing weights over the bank, by imitating its demonstrations through the frozen base model and bank\.
3. 3\.Stage 3: Adaptation \(§[3\.3\.3](https://arxiv.org/html/2610.22539#S3.SS3.SSS3)\)\.The scout writes a manual of the unseen interface, and the router mixes the bank into the agent’s adapter for it in one forward pass\.

Both training stages use only source\-interface data, so the scouting visit is the only information from the unseen interface thatWebMixuses\. Adding a source interface later adds one adapter to the bank and retrains the router, leaving the existing adapters unchanged\.

#### 3\.3\.1Stage 1: Adapter Bank Training

Each adapterkkis trained with supervised fine\-tuning \(SFT\) to reproduce the expert on its own interface\. An anchoring term also limits how far the adapter moves the base policy on the other source interfaces\. We call𝒟k\\mathcal\{D\}\_\{k\}the adapter’s*own corpus*, and𝒟∖k=⋃j≠k𝒟j\\mathcal\{D\}\_\{\\setminus k\}=\\bigcup\_\{j\\neq k\}\\mathcal\{D\}\_\{j\}its*anchor corpus*, which pools the demonstrations of the otherK−1K\-1source interfaces\. Neither corpus contains data from an unseen interface\. We minimize the objective

ℒbank\(k\)=𝔼\(x,y\)∼𝒟k\[−1\|y\|∑i=1\|y\|logπ\(k\)\(yi∣x,y<i\)\]⏟SFT on training interface\+β​𝔼\(x,y\)∼𝒟∖k​\[1\|y\|​∑i=1\|y\|κi\(k\)\]⏟anchor on other source interfaces,\\mathcal\{L\}\_\{\\mathrm\{bank\}\}^\{\(k\)\}=\\underbrace\{\\mathbb\{E\}\_\{\(x,y\)\\sim\\mathcal\{D\}\_\{k\}\}\\left\[\-\\frac\{1\}\{\|y\|\}\\sum\_\{i=1\}^\{\|y\|\}\\log\\pi^\{\(k\)\}\(y\_\{i\}\\mid x,y\_\{<i\}\)\\right\]\}\_\{\\text\{SFT on training interface\}\}\+\\beta\\,\\underbrace\{\\mathbb\{E\}\_\{\(x,y\)\\sim\\mathcal\{D\}\_\{\\setminus k\}\}\\left\[\\frac\{1\}\{\|y\|\}\\sum\_\{i=1\}^\{\|y\|\}\\kappa^\{\(k\)\}\_\{i\}\\right\]\}\_\{\\text\{anchor on other source interfaces\}\},\(7\)whereβ≥0\\beta\\geq 0controls the anchoring strength andβ=0\\beta=0recovers plain SFT\. The first term is the SFT loss on the adapter’s own interface\. The second term penalizes changes to the base policy’s likelihood of the demonstrated tokens of the other source interfaces, whereκi\(k\)\\kappa^\{\(k\)\}\_\{i\}is a clipped estimate of the KL divergence \(Appendix[A\.2](https://arxiv.org/html/2610.22539#A1.SS2)\)\.

#### 3\.3\.2Stage 2: Training the Router

Stage 2 trains the router with the bank frozen\. Before training, the scout visits each source interfacevkv\_\{k\}once and returns its representationhvkh\_\{v\_\{k\}\}\. Fromhvkh\_\{v\_\{k\}\}, Eqs\.[3](https://arxiv.org/html/2610.22539#S3.E3)and[4](https://arxiv.org/html/2610.22539#S3.E4)construct the composed updateΔ​θϕ​\(hvk\)=\(Δ​Wvk,ℓ\)ℓ∈Ω\\Delta\\theta\_\{\\phi\}\(h\_\{v\_\{k\}\}\)=\(\\Delta W\_\{v\_\{k\},\\ell\}\)\_\{\\ell\\in\\Omega\}\. Letπϕ,k=πθ\+Δ​θϕ​\(hvk\)\\pi\_\{\\phi,k\}=\\pi\_\{\\theta\+\\Delta\\theta\_\{\\phi\}\(h\_\{v\_\{k\}\}\)\}denote the resulting policy\. We train the router with the behavior\-cloning \(BC\) loss averaged over all response tokens of the pooled source corpus,

ℒrouter\(ϕ\)=−1N∑k=1K∑\(x,y\)∈𝒟k∑i=1\|y\|logπϕ,k\(yi∣x,y<i\),N=∑k=1K∑\(x,y\)∈𝒟k\|y\|\.\\mathcal\{L\}\_\{\\mathrm\{router\}\}\(\\phi\)=\-\\frac\{1\}\{N\}\\sum\_\{k=1\}^\{K\}\\sum\_\{\(x,y\)\\in\\mathcal\{D\}\_\{k\}\}\\sum\_\{i=1\}^\{\|y\|\}\\log\\pi\_\{\\phi,k\}\(y\_\{i\}\\mid x,y\_\{<i\}\),\\qquad N=\\sum\_\{k=1\}^\{K\}\\sum\_\{\(x,y\)\\in\\mathcal\{D\}\_\{k\}\}\|y\|\.\(8\)Prompt tokens are masked, so only the expert’s response tokens contribute\. Gradients reach the router through the composed updates and the frozen base model, so the router learns which mixtures reproduce the expert on each source interface\. Onlyϕ\\phiis trained\. The base model, the bank, and the scout stay frozen, and the representationshvkh\_\{v\_\{k\}\}are fixed inputs\.

#### 3\.3\.3Stage 3: Adaptation

In Stage 3, the scout visits the unseen interfaceuuonce, with the same procedure and budget as for the source interfaces, and producesmum\_\{u\}andhuh\_\{u\}\. The router mapshuh\_\{u\}tocuc\_\{u\}through Eq\.[3](https://arxiv.org/html/2610.22539#S3.E3), Eq\.[5](https://arxiv.org/html/2610.22539#S3.E5)composes the factors, and Eq\.[2](https://arxiv.org/html/2610.22539#S3.E2)adds the resulting updateΔ​Wu,ℓ=∑kcu,k​Δ​Wℓ\(k\)\\Delta W\_\{u,\\ell\}=\\sum\_\{k\}c\_\{u,k\}\\,\\Delta W^\{\(k\)\}\_\{\\ell\}to the frozen base weights\. For version transfer, the visit performs no tasks\. For domain transfer, the base policy attemptsnntasks from the training split ofuu, and the writer reads their intents and answer\-stripped reference procedures\. Adaptation therefore costs one scouting visit and one router forward pass\. It uses no demonstrations fromuuand no gradient updates\.

## 4Experiments

We answer four research questions\.RQ1 \(transfer\)\.DoesWebMixtransfer to unseen domains and versions better than in\-context learning and LoRA?RQ2 \(retention\)\.Does it keep its performance on the seen interfaces?RQ3 \(design\)\.Which design choices matter: composing a bank instead of generating weights, the scout, the scouting budget, and the KL anchor?RQ4 \(analysis\)\.What do the bank adapters and the router’s mixtures look like?

### 4\.1Experimental Setup

##### Benchmarks and evaluation\.

We evaluate on \(i\) TimeWarp \(TW\)\([Ishmam and Marino, 2026](https://arxiv.org/html/2610.22539#bib.bib32)\), a containerized web benchmark with three domains \(Wiki, News, Shop\), each rendered in six versions, and \(ii\) WebArena\-Lite\-v2 \(WAL\)\([Liu et al\., 2026](https://arxiv.org/html/2610.22539#bib.bib20)\), an updated version of WebArena\-Lite\([Liu et al\., 2024](https://arxiv.org/html/2610.22539#bib.bib3)\), itself a curated subset of WebArena\([Zhou et al\., 2024](https://arxiv.org/html/2610.22539#bib.bib38)\), with five domains \(Reddit, GitLab, CMS, Map, OSS\)\. We evaluate*domain transfer*, in which the unseen interface is a new domain, on both benchmarks, and*version transfer*, in which it is a new version, on TW, holding out v1 and v5 in turn\. We conduct all the experiments with the BrowserUse harness\([Müller and Žunič, 2024](https://arxiv.org/html/2610.22539#bib.bib31)\), with non\-rationale tokens and memory enabled\.

##### Baselines\.

We compareWebMixwith three baselines on two open\-weight models, Qwen\-3\.5 4B\([Qwen Team, 2026](https://arxiv.org/html/2610.22539#bib.bib28)\)and Llama\-3\.1 8B\([Grattafiori et al\., 2024](https://arxiv.org/html/2610.22539#bib.bib27)\): the base model zero\-shot, in\-context learning \(ICL\)\([Brown et al\., 2020](https://arxiv.org/html/2610.22539#bib.bib21)\), and LoRA\([Hu et al\., 2022](https://arxiv.org/html/2610.22539#bib.bib34)\)\. We evaluate each method in two settings\. In theUnseen setting, the evaluated domain or version is left out of the training data and the in\-context examples\. In theSeen setting, it is included\. In the seen setting, we report LoRA trained only on the evaluated interface \(in\), which is a target\-trained specialist, and LoRA trained on all interfaces \(all\)\. Each number is the average of three runs\. §[A\.4](https://arxiv.org/html/2610.22539#A1.SS4)gives the training details\.

Table 1:TimeWarp results \(RQ1, RQ2\)\.Success rate \(%\) on held\-out domains and held\-out versions\. U/S: unseen/seen setting\. LoRA \(in\) is trained on the evaluated interface only, and LoRA \(all\) on all interfaces\. Avg\. is pooled over all tasks\. Best result per model and setting inbold\.Table 2:WebArena\-Lite\-v2 results \(RQ1, RQ2\)\.Success rate \(%\) on held\-out domains\. U/S: unseen/seen setting\. Overall is pooled over all tasks\. Best result per model and setting inbold\.

### 4\.2Results and Analysis

Table 3:Hypernetwork design \(RQ3\)\.Success rate \(%\) on unseen TimeWarp domains with Qwen\-3\.5 4B\.T2Lrows generate every adapter weight,Mixturecomposes the frozen bank, and \+Scoutconditions the router on the scout’s manual built fromnnscouted tasks\. Avg is pooled over all tasks\. Best result per column inbold, second bestunderlined\.
101020203030404050506060Score \(%\)WikiNewsShopAvg \(weighted\)Best per seriesBest on Avg
Figure 3:KL anchor strength \(RQ3\)\.Success rate \(%\) on unseen TimeWarp domains with Qwen\-3\.5 4B as the anchor weightβ\\betaof Eq\.[7](https://arxiv.org/html/2610.22539#S3.E7)varies\. The shaded column marks the best average,β=1\.0\\beta=1\.0\.
##### RQ1: Transfer to unseen interfaces\.

Tables[1](https://arxiv.org/html/2610.22539#S4.T1)and[2](https://arxiv.org/html/2610.22539#S4.T2)report transfer toUnseen domains and versions on TW and WAL, respectively\.WebMixoutperforms every baseline on both models and transfer types, often matching the performance of baselines trained on those interfaces\. For example, it matches LoRA \(in\) on TW domains with both models\. Gains are largest for domain transfer, at 7\.5–8\.3 percentage points \(pp\) over the best baseline on TW and 3\.3–4\.6pp on WAL\. Version transfer gains are smaller \(1\.8–3\.2pp\), and LoRA is a strong baseline there\. Transfer is strongest for domains that are similar to the training domains\. With Qwen\-3\.5, News gains 26pp over zero\-shot, compared to∼\\sim5pp on Wiki and Shop\. On WAL, OSS gains the most for both models \(14–15pp\)\. The one failure case is Map with Llama\-3\.1, where every unseen method scores zero\. Composing adapters trained on other interfaces thus recovers most of the benefit of training on the target itself\.

##### RQ2: Retention on seen interfaces\.

WebMix’s transfer gains do not come at the cost ofSeen interface performance\. In Tables[1](https://arxiv.org/html/2610.22539#S4.T1)and[2](https://arxiv.org/html/2610.22539#S4.T2),WebMixis the best or tied\-best method in every column\. It improves over the strongest seen baseline, LoRA \(all\), by 4\.6–6\.3pp on TW domains, 1\.8–2\.4pp on TW versions, and 2\.6–4\.3pp on WAL\. Training LoRA on all domains adds only 0\.2–1\.3pp over in\-domain training, suggesting thatWebMix’s gains do not solely come from more training data\.

##### RQ3: Design choices\.

Hypernetwork design\.Table[3](https://arxiv.org/html/2610.22539#S4.T3)isolates the contribution of each component\.T2L Recons\(Appendix[B\.2](https://arxiv.org/html/2610.22539#A2.SS2)\), which regresses the stored adapter updates, is the weakest variant\.T2L SFT\(Appendix[B\.3](https://arxiv.org/html/2610.22539#A2.SS3)\), which trains the same generator through the policy, improves on it by 9\.17pp\. Replacing weight generation with the mixture hypernetwork \(§[3\.2](https://arxiv.org/html/2610.22539#S3.SS2)\) adds a further 3\.75pp, mostly from News \(\+9pp\)\. Conditioning the router on the scout then adds another 4\.58pp, for a total gain of 8\.33pp over the strongest T2L variant\. Composing a frozen bank is easier to learn from a handful of interfaces than generating weights\. Appendix[D\.2](https://arxiv.org/html/2610.22539#A4.SS2)analyzes an idealization ofT2L Recons\.Scouting budget\.ScoutoutperformsMixtureon average for every number of scouted tasksnn, butn=1n\{=\}1is best\. Largernngives a marginal gain on News \(\+1\.5pp atn=3n\{=\}3\) but costs 6–8 points on Wiki, so we usen=1n\{=\}1throughout\.KL regularization\.Figure[3](https://arxiv.org/html/2610.22539#S4.F3)varies the anchor weightβ\\beta\.β=1\.0\\beta\{=\}1\.0is the best of the tested values on every domain and on average\. Removing the KL term \(β=0\\beta\{=\}0\) lowers the average by over 7pp\. A stronger penalty \(β=3\.0\\beta\{=\}3\.0\) also hurts\.WebMixtherefore benefits from moderate regularization\.

##### RQ4: Analysis\.

The domain adapters of the bank are nearly orthogonal \(cosine below 0\.16, Figure[4\(a\)](https://arxiv.org/html/2610.22539#A3.F4.sf1)\), and without the KL anchor they are about twice as similar \(Figure[5](https://arxiv.org/html/2610.22539#A3.F5)\)\. The router builds a held\-out version from its nearest versions, e\.g\., v1 from v2 and v3 \(Figure[4\(b\)](https://arxiv.org/html/2610.22539#A3.F4.sf2)\)\. With v6 in the bank, it puts 96% of the weight on v6, so the mixture weights track interface similarity\.

## 5Conclusion

Experience with one website makes the next one easier to use, andWebMixgives small, open\-weight web agents the same advantage\. A scout writes a short manual of the new interface, and a router, trained only on source interfaces, maps the encoded manual to mixing weights over a frozen adapter bank\. The result is a standard LoRA adapter, produced without gradient updates on the target\.WebMixtransfers to unseen domains and versions better than in\-context learning and LoRA, often matches target\-trained specialists\. When the web changes, a web agent does not need retraining\. It can simplyremixwhat it already knows\.

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- Zhouet al\.\(2024\)S\. Zhou, F\. F\. Xu, H\. Zhu, X\. Zhou, R\. Lo, A\. Sridhar, X\. Cheng, Y\. Bisk, D\. Fried, U\. Alon,et al\.WebArena: a realistic web environment for building autonomous agents\.InInternational Conference on Learning Representations \(ICLR\),Cited by:[§2](https://arxiv.org/html/2610.22539#S2.SS0.SSS0.Px1.p1.1),[§3\.1](https://arxiv.org/html/2610.22539#S3.SS1.SSS0.Px1.p1.1),[§4\.1](https://arxiv.org/html/2610.22539#S4.SS1.SSS0.Px1.p1.1)\.

## Appendix AMethod Details

### A\.1Adapter Factorization

The factors of an adapter are not unique\. For any invertibleQ∈ℝr×rQ\\in\\mathbb\{R\}^\{r\\times r\},

\(Q​Aℓ,Bℓ​Q−1\)induces\(Bℓ​Q−1\)​\(Q​Aℓ\)=Bℓ​Aℓ,\(QA\_\{\\ell\},\\;B\_\{\\ell\}Q^\{\-1\}\)\\quad\\text\{induces\}\\quad\(B\_\{\\ell\}Q^\{\-1\}\)\(QA\_\{\\ell\}\)=B\_\{\\ell\}A\_\{\\ell\},\(9\)so different factor pairs give the same update and the same policy\. Two adapters trained on similar data can therefore lie far apart in factor space and still behave alike\.WebMixcombines adapters through their updatesΔ​Wℓ\\Delta W\_\{\\ell\}, which Eq\.[9](https://arxiv.org/html/2610.22539#A1.E9)leaves unchanged\. Every comparison we report is either functional \(task success\) or computed on the updatesΔ​Wℓ\\Delta W\_\{\\ell\}\(Figures[4\(a\)](https://arxiv.org/html/2610.22539#A3.F4.sf1)and[5](https://arxiv.org/html/2610.22539#A3.F5)\), never on the factors\.

### A\.2KL Anchor

d~i\(k\)\\displaystyle\\tilde\{d\}^\{\(k\)\}\_\{i\}:=clip⁡\(log⁡πθ​\(yi∣x,y<i\)−log⁡π\(k\)​\(yi∣x,y<i\),τmin,τmax\),\\displaystyle:=\\operatorname\{clip\}\\\!\\left\(\\log\\pi\_\{\\theta\}\(y\_\{i\}\\mid x,y\_\{<i\}\)\-\\log\\pi^\{\(k\)\}\(y\_\{i\}\\mid x,y\_\{<i\}\),\\ \\tau\_\{\\min\},\\ \\tau\_\{\\max\}\\right\),\(10\)κi\(k\)\\displaystyle\\kappa^\{\(k\)\}\_\{i\}:=exp⁡\(d~i\(k\)\)−d~i\(k\)−1,\\displaystyle:=\\exp\\big\(\\tilde\{d\}^\{\(k\)\}\_\{i\}\\big\)\-\\tilde\{d\}^\{\(k\)\}\_\{i\}\-1,whereτmin<0<τmax\\tau\_\{\\min\}<0<\\tau\_\{\\max\}are clipping bounds, set toτmin=−20\\tau\_\{\\min\}=\-20andτmax=8\\tau\_\{\\max\}=8\. This penalty is nonnegative and vanishes when the two policies assign the same probability to the demonstrated token\. It has the form of thek3k\_\{3\}estimator of the KL divergence\([Schulman, 2020](https://arxiv.org/html/2610.22539#bib.bib48)\), but it is evaluated on demonstrated tokens instead of samples fromπ\(k\)\\pi^\{\(k\)\}\. It therefore anchors the probabilities of those tokens without constraining the full output distribution\. Own\-corpus examples contribute only the SFT term, and anchor\-corpus examples contribute only the anchoring term\. We sample the anchor examples from𝒟∖k\\mathcal\{D\}\_\{\\setminus k\}to match the size of𝒟k\\mathcal\{D\}\_\{k\}\. The objective encourages each adapter to specialize to its own interface with less interference when the adapters are mixed\.

### A\.3Router Details

The router is the two\-layer perceptron of Eq\.[3](https://arxiv.org/html/2610.22539#S3.E3)with hidden widthdh=256d\_\{h\}=256and a SiLU nonlinearity\. It hasdc​dh\+dh\+\(dh\+1\)​Kd\_\{c\}d\_\{h\}\+d\_\{h\}\+\(d\_\{h\}\+1\)Kparameters, about0\.260\.26M fordc=1024d\_\{c\}=1024and anyK≤5K\\leq 5used here\. Its output layer starts at zero, so the initial mixture is uniform and the initial adapter is the mean of the bank\. One coefficient vector is shared across all adapted matrices\.

### A\.4Training Hyperparameters

Bank adapters use rankr=16r=16and scaleα=32\\alpha=32on the attention and MLP projections, and the KL anchor usesβ=1\.0\\beta=1\.0\(Figure[3](https://arxiv.org/html/2610.22539#S4.F3)\)\. The router has hidden widthdh=256d\_\{h\}=256and trains for 500 steps with AdamW\. The scout’s encoder is BGE\-large\-en\-v1\.5\([Xiao et al\., 2024](https://arxiv.org/html/2610.22539#bib.bib49)\)withdc=1024d\_\{c\}=1024\. For domain transfer, the scout attemptsn=1n=1task \(§[4\.2](https://arxiv.org/html/2610.22539#S4.SS2), RQ3\)\. Table[4](https://arxiv.org/html/2610.22539#A1.T4)lists the hyperparameters of every component\.

ComponentSettingValueBank adaptersrankrr/ scaleα\\alpha/ dropout16 / 32 / 0adapted matricesΩ\\Omegaattention q, k, v, o and MLP gate, up, downepochs6learning rate and schedule10−410^\{\-4\}, cosine, 10% warmupeffective batch size4 \(1 per device×\\times4 accumulation\)precision and hardwarebf16, DeepSpeed ZeRO\-3, one 80 GB GPUKL anchorweightβ\\beta1\.0clipping\[τmin,τmax\]\[\\tau\_\{\\min\},\\tau\_\{\\max\}\]\[−20,8\]\[\-20,\\,8\]anchor examplesas many as own\-corpus examplesRouterhidden widthdhd\_\{h\}256optimizerAdamW, learning rate10−410^\{\-4\}, no weight decayschedule20 linear warmup steps, then cosinesteps / batch size / gradient clip500 / 16 / 1\.0maximum sequence length8192 tokensScoutexplorer budget25 pages, link depth 3scouted tasksnn\(domain transfer\)1writerQwen\-3\.5 9B, temperature 0encoderBGE\-large\-en\-v1\.5,dc=1024d\_\{c\}=1024, 512 tokensAgent decodingtemperature0\.01maximum new tokens per step2048Table 4:Training and evaluation hyperparameters\.
### A\.5Scout Details

##### Explorer\.

For version transfer, the explorer runs a scripted, task\-free, breadth\-first crawl of same\-origin links from each site’s landing page\. It visits up to2525pages per site at a maximum link depth of33\. For every page it records the title, headings, link labels and targets, buttons, and form fields\. It also performs one search probe, which submits the label of the first suitable link on the landing page to the first text or search field the crawl finds\. For domain transfer, the explorer is task\-grounded instead\. The frozen base policy, without an adapter, attemptsnntasks drawn from the training split of the site, and the explorer replays the pages that the policy visited\.

##### Writer\.

A frozen language model writes the manualmvm\_\{v\}from the exploration record\. The manual follows a fixed schema of seven sections, which cover purpose and top\-level navigation, search, page anatomy, lists and pagination, forms and multi\-step flows, interaction pitfalls, and visual style\. The writer is instructed to use only labels and paths observed during exploration, and to omit comparisons with other interfaces and any assessment of task performance\. For version transfer, an interface spans several sites, and its manual concatenates the per\-site manuals\. For domain transfer, the writer also receives each attempted task’s intent and its reference procedure with the final answer removed\. The manual then adds a deterministic eighth section that lists, for each attempted task, its intent and the sequence of actions with the visible label of each target\.

##### Encoder\.

A frozen sentence encoderψ\\psi\(BGE\-large\-en\-v1\.5\([Xiao et al\., 2024](https://arxiv.org/html/2610.22539#bib.bib49)\)\) reads the tokens of the manual and returns their mean\-pooled representationψ⁡\(mv\)∈ℝdc\\psi\(m\_\{v\}\)\\in\\mathbb\{R\}^\{d\_\{c\}\}, withdc=1024d\_\{c\}=1024\.

## Appendix BTheT2LBaseline in Detail

T2L\([Charakorn et al\., 2025](https://arxiv.org/html/2610.22539#bib.bib35)\)is a direct hypernetwork\. It maps the same conditioninghvh\_\{v\}thatWebMixuses to the two LoRA factors of every adapted matrix\. We index the adapted matrices bys∈Ωs\\in\\Omegain this appendix and the next\.

### B\.1Architecture

A query addresses one factor of one adapted matrix through three learnable embedding tables, indexed by the matrix’s layer indexlsl\_\{s\}, its module typemsm\_\{s\}, and the*role*ρ\\rhoof the requested factor,

zs,ρ=Tϕ​\(\[hv;Elyr​\(ls\);Emod​\(ms\);Erole​\(ρ\)\]\),ρ∈\{A,B\},z\_\{s,\\rho\}\\;=\\;T\_\{\\phi\}\\\!\\Big\(\\big\[\\,h\_\{v\};\\;E\_\{\\mathrm\{lyr\}\}\(l\_\{s\}\);\\;E\_\{\\mathrm\{mod\}\}\(m\_\{s\}\);\\;E\_\{\\mathrm\{role\}\}\(\\rho\)\\,\\big\]\\Big\),\\qquad\\rho\\in\\\{\\mathrm\{A\},\\mathrm\{B\}\\\},\(11\)where each embedding has width3232and the trunkTϕT\_\{\\phi\}is a linear projection to widthdT=128d\_\{T\}=128followed by a SiLU nonlinearity and two residual blocks\. Output heads are shared at the coarsest granularity the target shapes allow\. There is one linear headPκP\_\{\\kappa\}per distinct weight shape\(din,dout\)\(d^\{\\mathrm\{in\}\},d^\{\\mathrm\{out\}\}\)realized in the target set, shared across layers and across both roles\. The head emitsmax⁡\(r​din,r​dout\)\\max\(r\\,d^\{\\mathrm\{in\}\},\\,r\\,d^\{\\mathrm\{out\}\}\)values, and the leading block is reshaped into the requested factor,

As=matr×dsin​\(Pκ⁡\(s\)​zs,A\),Bs=matdsout×r​\(Pκ⁡\(s\)​zs,B\)\.A\_\{s\}\\;=\\;\\mathrm\{mat\}\_\{r\\times d^\{\\mathrm\{in\}\}\_\{s\}\}\\\!\\big\(P\_\{\\kappa\(s\)\}\\,z\_\{s,\\mathrm\{A\}\}\\big\),\\qquad B\_\{s\}\\;=\\;\\mathrm\{mat\}\_\{d^\{\\mathrm\{out\}\}\_\{s\}\\times r\}\\\!\\big\(P\_\{\\kappa\(s\)\}\\,z\_\{s,\\mathrm\{B\}\}\\big\)\.\(12\)Because layer, module, and role enter as embeddings, all2​\|Ω\|2\|\\Omega\|queries are batched and the full adapter is emitted in a single forward pass\.

### B\.2Reconstructor

T2L Reconsfollows the original training recipe ofT2L\([Charakorn et al\., 2025](https://arxiv.org/html/2610.22539#bib.bib35)\)\. It regresses the stored bank adapters: for each source interfacevkv\_\{k\}, it minimizes theℓ1\\ell\_\{1\}distance between the generated updateBs​AsB\_\{s\}A\_\{s\}and the stored updateΔ​Ws\(k\)\\Delta W^\{\(k\)\}\_\{s\}, scaled by10410^\{4\}, over 16 randomly drawn matrices per step\. It trains for 2000 steps with AdamW \(learning rate10−410^\{\-4\}, 100 warmup steps, cosine decay, gradient clipping at 1\.0\) from the initialization described below\.

### B\.3SFT

T2L SFTtrains the same generator end to end through the frozen policy with the router objective of Eq\.[8](https://arxiv.org/html/2610.22539#S3.E8), so it receives exactly the training signal ofWebMix\. It uses the router’s optimizer settings \(Appendix[A\.4](https://arxiv.org/html/2610.22539#A1.SS4)\)\. Its output heads start at zero forBBand at a LoRA\-style random bias forAA, so training starts at the base policy\.

### B\.4Initialization at the oracle’s scale

A generator that writes weights must write them at the right order of magnitude from step00\. Our trained per\-version adapters have mean update magnitude\|Δ​W\|≈5×10−5\|\\Delta W\|\\approx 5\\times 10^\{\-5\}\. With every emitted entry i\.i\.d\.𝒩⁡\(0,σ02\)\\mathcal\{N\}\(0,\\sigma\_\{0\}^\{2\}\), an entry ofBs​AsB\_\{s\}A\_\{s\}sumsrrproducts of independent pairs, so its typical scale isr​σ02\\sqrt\{r\}\\,\\sigma\_\{0\}^\{2\}\. This is*quadratic*inσ0\\sigma\_\{0\}, which is why an innocuous\-looking initialization can emit updates orders of magnitude too large and swamp the frozen computation\. We therefore setσ0=\(ε/r\)1/2\\sigma\_\{0\}=\(\\varepsilon/\\sqrt\{r\}\)^\{1/2\}withε=10−6\\varepsilon=10^\{\-6\}, which places the generated update well below the oracle scale at initialization\. It is near a no\-op, but without the exactly\-zero factor that would stall the other factor’s gradient through the product\. WhereWebMixstarts at the mean trained adapter,T2Lstarts from almost nothing\.

## Appendix CAnalysis of the Bank and the Router

This appendix answers RQ4 of §[4](https://arxiv.org/html/2610.22539#S4): what the bank adapters and the router’s mixtures look like\.

##### The bank\.

Figure[4\(a\)](https://arxiv.org/html/2610.22539#A3.F4.sf1)shows that the domain adapters of the WAL bank are close to orthogonal\. Every pairwise cosine similarity of their updates is below 0\.16, so each adapter contributes a distinct direction to the mixture\. Figure[5](https://arxiv.org/html/2610.22539#A3.F5)measures the same similarity per layer\. Adapters trained on the same domain become more similar with depth, reaching a cosine of about 0\.5 \(Qwen\) and 0\.6 \(Llama\) in the last layers\. Adapters of different domains stay below 0\.14 with the KL anchor, and are about twice as similar without it\. The anchor therefore keeps the bank’s adapters distinct, which is consistent with its gain in Figure[3](https://arxiv.org/html/2610.22539#S4.F3)\.

##### The router\.

Figure[4\(b\)](https://arxiv.org/html/2610.22539#A3.F4.sf2)shows the weights the router assigns on TW versions\. With v1 held out, the router builds its adapter mostly from v2 \(81%\) and v3 \(17%\), so the weights fall with distance from v1 in time\. With v6 in the bank, it puts 96% of the weight on the v6 adapter\. The coefficients can thus be read as the model’s judgment of which interfaces are similar\. Appendix[D\.3](https://arxiv.org/html/2610.22539#A4.SS3)shows that, for a linearly independent bank, they are determined by the served adapter\.

Qwen\-3\.5 4B–0\.100\.080\.090\.090\.10–0\.100\.100\.120\.080\.10–0\.110\.120\.090\.100\.11–0\.150\.090\.120\.120\.15–RedditGitLabCMSMapOSSLlama\-3\.1 8B–0\.080\.080\.080\.060\.08–0\.130\.100\.120\.080\.13–0\.110\.140\.080\.100\.11–0\.100\.060\.120\.140\.10–RedditGitLabCMSMapOSSRedditGitLabCMSMapOSS0\.060\.080\.100\.120\.140\.16cosine ofΔ​W\\Delta W

\(a\)Pairwise cosine ofΔ​W\\Delta W
v1 \(unseen\)leave\-v1\-outv281%v317%v42%v6all 6v41%v53%v696%

\(b\)Router mixture weights

Figure 4:The bank’s domain adapters are close to orthogonal, and the router mixes nearby versions \(RQ4\)\.\(a\)Pairwise cosine similarity of the weight updatesΔ​W=αr​B​A\\Delta W=\\tfrac\{\\alpha\}\{r\}BAbetween the five domain adapters of the KL\-anchored WebArena\-Lite\-v2 bank, computed over the whole model\. All pairs stay below0\.160\.16\. Reddit is the least aligned with the other domains in both models, while OSS overlaps most with Map \(Qwen\) and CMS \(Llama\)\.\(b\)Mixture weights that the router assigns on TimeWarp versions\. With v1 held out, it composes v1 mostly from the adjacent version v2 and then v3\. With all six versions in the bank, it puts almost all weight on the matching v6 adapter\.005510101515202025253030000\.10\.10\.20\.20\.30\.30\.40\.40\.50\.50\.60\.60\.70\.7LayerPairwise Cosine ofΔ​W\\Delta WQwen\-3\.5 4B005510101515202025253030LayerLlama\-3\.1 8BSame\-domainCross\-domain, with KLCross\-domain, w/o KL

Figure 5:The KL anchor keeps adapters of different domains apart \(RQ4\)\.Cosine similarity of the weight updatesΔ​W\\Delta Wper layer, averaged over pairs of adapters trained on the same domain, on different domains with the KL anchor, and on different domains without it\.

## Appendix DTheoretical Analysis

We analyze the one design choice that separatesWebMixfrom direct generation\. The emitted adapter is confined to the convex hull of a frozen bank, and the interface choosesKKcoefficients rather than every weight\. We ask what this confinement costs, what it buys, and how direct generation behaves in a tractable idealization\. Full proofs are in Appendix[E](https://arxiv.org/html/2610.22539#A5)\.

##### Setting\.

In this appendix,s∈Ωs\\in\\Omegaindexes the adapted matrices \(writtenℓ\\ellin §[3](https://arxiv.org/html/2610.22539#S3), whereℓ\\ellis kept free for the logits in Appendix[E](https://arxiv.org/html/2610.22539#A5)\), andpθ,Δ​Wp\_\{\\theta,\\Delta W\}is the policyπθ\+Δ​θ\\pi\_\{\\theta\+\\Delta\\theta\}written as a function of the update\. We identify an adapter with its induced updateΔ​W=\(Δ​Ws\)s∈Ω\\Delta W=\(\\Delta W\_\{s\}\)\_\{s\\in\\Omega\}in the product space𝒲:=∏s∈Ωℝdsout×dsin\\mathcal\{W\}:=\\prod\_\{s\\in\\Omega\}\\mathbb\{R\}^\{d^\{\\mathrm\{out\}\}\_\{s\}\\times d^\{\\mathrm\{in\}\}\_\{s\}\}, with norm‖Δ​W‖:=\(∑s‖Δ​Ws‖F2\)1/2\\\|\\Delta W\\\|:=\(\\sum\_\{s\}\\\|\\Delta W\_\{s\}\\\|\_\{F\}^\{2\}\)^\{1/2\}and distancedist⁡\(Δ​W,ℋ\):=minW′∈ℋ⁡‖Δ​W−W′‖\\mathrm\{dist\}\(\\Delta W,\\mathcal\{H\}\):=\\min\_\{W^\{\\prime\}\\in\\mathcal\{H\}\}\\\|\\Delta W\-W^\{\\prime\}\\\|to a compact setℋ\\mathcal\{H\}\. The bank is\{Δ​W\(k\)\}k=1K⊂𝒲\\\{\\Delta W^\{\(k\)\}\\\}\_\{k=1\}^\{K\}\\subset\\mathcal\{W\},K≥1K\\geq 1, withΔ​Ws\(k\)=αr​Bs\(k\)​As\(k\)\\Delta W^\{\(k\)\}\_\{s\}=\\tfrac\{\\alpha\}\{r\}B^\{\(k\)\}\_\{s\}A^\{\(k\)\}\_\{s\}\. Forc∈ΔK−1c\\in\\Delta^\{K\-1\}, the probability simplex\{c∈ℝK:ck≥0,∑kck=1\}\\\{c\\in\\mathbb\{R\}^\{K\}:c\_\{k\}\\geq 0,\\ \\sum\_\{k\}c\_\{k\}=1\\\}with verticese1,…,eKe\_\{1\},\\dots,e\_\{K\}, the mixture and its reachable set are

M⁡\(c\):=∑k=1Kck​Δ​W\(k\),ℋK:=M⁡\(ΔK−1\)=conv⁡\{Δ​W\(1\),…,Δ​W\(K\)\},M\(c\):=\\sum\_\{k=1\}^\{K\}c\_\{k\}\\,\\Delta W^\{\(k\)\},\\qquad\\mathcal\{H\}\_\{K\}:=M\(\\Delta^\{K\-1\}\)=\\mathrm\{conv\}\\\{\\Delta W^\{\(1\)\},\\dots,\\Delta W^\{\(K\)\}\\\},\(13\)and by Eq\.[4](https://arxiv.org/html/2610.22539#S3.E4),M⁡\(c\)M\(c\)is exactly the updateWebMixserves under coefficientscc\. For an interfacevvletpθ,Δ​Wp\_\{\\theta,\\Delta W\}denote the likelihood of the frozen base with updateΔ​W\\Delta Winstalled, and let

ℛv​\(Δ​W\):=𝔼\(x,y\)∼𝒟v​\[−log⁡pθ,Δ​W​\(y∣x\)\]\\mathcal\{R\}\_\{v\}\(\\Delta W\):=\\mathbb\{E\}\_\{\(x,y\)\\sim\\mathcal\{D\}\_\{v\}\}\\big\[\-\\log p\_\{\\theta,\\Delta W\}\(y\\mid x\)\\big\]\(14\)be the behavior\-cloning risk of that interface\. On𝒟v\\mathcal\{D\}\_\{v\}, the token\-averaged objective of Eq\.[8](https://arxiv.org/html/2610.22539#S3.E8)equalsℛv\\mathcal\{R\}\_\{v\}times the positive constant\|𝒟v\|/Nv\|\\mathcal\{D\}\_\{v\}\|/N\_\{v\}, whereNvN\_\{v\}is the number of response tokens in𝒟v\\mathcal\{D\}\_\{v\}, so the constant changes no minimizer and no result below except the value of the Lipschitz constantΛu\\Lambda\_\{u\}of Assumption[1](https://arxiv.org/html/2610.22539#Thmassumption1)\.ℛv\\mathcal\{R\}\_\{v\}is a finite average of smooth functions of the weights, so it is continuous on𝒲\\mathcal\{W\}\. The held\-out interface isuu, and its oracleΔ​W\(u\)\\Delta W^\{\(u\)\}is an adapter trained on the demonstrations ofuuitself, the LoRA \(in\) specialist of §[4](https://arxiv.org/html/2610.22539#S4)\. We writec^:=cu\\hat\{c\}:=c\_\{u\}for the router output of Eq\.[3](https://arxiv.org/html/2610.22539#S3.E3)on the held\-out embedding\. The demonstrations𝒟u\\mathcal\{D\}\_\{u\}ofuuare never used byWebMix; the analysis uses them only to defineℛu\\mathcal\{R\}\_\{u\}and the oracle\.

### D\.1What the hull can reach

###### Assumption 1\(Lipschitz risk\)\.

Let𝒦u:=conv⁡\(ℋK∪\{Δ​W\(u\)\}\)\\mathcal\{K\}\_\{u\}:=\\mathrm\{conv\}\\big\(\\mathcal\{H\}\_\{K\}\\cup\\\{\\Delta W^\{\(u\)\}\\\}\\big\)\. There is a constantΛu<∞\\Lambda\_\{u\}<\\inftysuch that\|ℛu​\(Δ​W\)−ℛu​\(Δ​W′\)\|≤Λu​‖Δ​W−Δ​W′‖\|\\mathcal\{R\}\_\{u\}\(\\Delta W\)\-\\mathcal\{R\}\_\{u\}\(\\Delta W^\{\\prime\}\)\|\\leq\\Lambda\_\{u\}\\\|\\Delta W\-\\Delta W^\{\\prime\}\\\|for allΔ​W,Δ​W′∈𝒦u\\Delta W,\\Delta W^\{\\prime\}\\in\\mathcal\{K\}\_\{u\}\.

The set𝒦u\\mathcal\{K\}\_\{u\}is convex and compact, and the base network is continuously differentiable in its weights, so by the mean value inequality the assumption holds withΛu\\Lambda\_\{u\}equal to the largest gradient norm ofℛu\\mathcal\{R\}\_\{u\}on𝒦u\\mathcal\{K\}\_\{u\}\. The constant can be large\. We use the bound for its direction, not its value\.

###### Theorem 1\(Hull decomposition of the held\-out risk\)\.

Under Assumption[1](https://arxiv.org/html/2610.22539#Thmassumption1), a minimizerc⋆∈arg⁡minc∈ΔK−1​ℛu​\(M⁡\(c\)\)c^\{\\star\}\\in\\arg\\min\_\{c\\in\\Delta^\{K\-1\}\}\\mathcal\{R\}\_\{u\}\(M\(c\)\)exists, and

ℛu​\(M⁡\(c^\)\)−ℛu​\(Δ​W\(u\)\)=ℛu​\(M⁡\(c^\)\)−ℛu​\(M⁡\(c⋆\)\)⏟routing≥0\+ℛu​\(M⁡\(c⋆\)\)−ℛu​\(Δ​W\(u\)\)⏟approximation,\\mathcal\{R\}\_\{u\}\(M\(\\hat\{c\}\)\)\-\\mathcal\{R\}\_\{u\}\(\\Delta W^\{\(u\)\}\)\\;=\\;\\underbrace\{\\mathcal\{R\}\_\{u\}\(M\(\\hat\{c\}\)\)\-\\mathcal\{R\}\_\{u\}\(M\(c^\{\\star\}\)\)\}\_\{\\text\{routing\}\\;\\geq\\,0\}\\;\+\\;\\underbrace\{\\mathcal\{R\}\_\{u\}\(M\(c^\{\\star\}\)\)\-\\mathcal\{R\}\_\{u\}\(\\Delta W^\{\(u\)\}\)\}\_\{\\text\{approximation\}\},\(15\)where the approximation term and the best mixture satisfy

ℛu​\(M⁡\(c⋆\)\)−ℛu​\(Δ​W\(u\)\)\\displaystyle\\mathcal\{R\}\_\{u\}\(M\(c^\{\\star\}\)\)\-\\mathcal\{R\}\_\{u\}\(\\Delta W^\{\(u\)\}\)≤Λu​dist​\(Δ​W\(u\),ℋK\),\\displaystyle\\leq\\;\\Lambda\_\{u\}\\,\\mathrm\{dist\}\\big\(\\Delta W^\{\(u\)\},\\mathcal\{H\}\_\{K\}\\big\),\(16\)ℛu​\(M⁡\(c⋆\)\)\\displaystyle\\mathcal\{R\}\_\{u\}\(M\(c^\{\\star\}\)\)≤mink≤K⁡ℛu​\(Δ​W\(k\)\)\.\\displaystyle\\leq\\;\\min\_\{k\\leq K\}\\mathcal\{R\}\_\{u\}\(\\Delta W^\{\(k\)\}\)\.

*Takeaway: the excess held\-out risk ofWebMixsplits into a term set by the bank, the target and the risk and bounded by the geometry of the bank and the target, which the head cannot change, and a term set by the head and the conditioning\. The best mixture is never worse than the best single bank adapter transferred to the held\-out interface\.*

###### Proof sketch\.

The decomposition is an identity\. For the first bound, take a pointΠ\\Piof the compact setℋK\\mathcal\{H\}\_\{K\}nearest toΔ​W\(u\)\\Delta W^\{\(u\)\}\. It is someM⁡\(cΠ\)M\(c\_\{\\Pi\}\), and Lipschitz continuity on𝒦u\\mathcal\{K\}\_\{u\}givesℛu​\(M⁡\(c⋆\)\)≤ℛu​\(M⁡\(cΠ\)\)≤ℛu​\(Δ​W\(u\)\)\+Λu​dist​\(Δ​W\(u\),ℋK\)\\mathcal\{R\}\_\{u\}\(M\(c^\{\\star\}\)\)\\leq\\mathcal\{R\}\_\{u\}\(M\(c\_\{\\Pi\}\)\)\\leq\\mathcal\{R\}\_\{u\}\(\\Delta W^\{\(u\)\}\)\+\\Lambda\_\{u\}\\,\\mathrm\{dist\}\(\\Delta W^\{\(u\)\},\\mathcal\{H\}\_\{K\}\)\. For the second, every vertexeke\_\{k\}lies inΔK−1\\Delta^\{K\-1\}andM⁡\(ek\)=Δ​W\(k\)M\(e\_\{k\}\)=\\Delta W^\{\(k\)\}\. ∎

###### Corollary 2\(Growing the bank\)\.

Let a bank ofK\+1K\+1adapters extend the bank ofKKby one adapterΔ​W\(K\+1\)\\Delta W^\{\(K\+1\)\}, and writeMK,ℋKM\_\{K\},\\mathcal\{H\}\_\{K\}andMK\+1,ℋK\+1M\_\{K\+1\},\\mathcal\{H\}\_\{K\+1\}for the two mixtures and reachable sets\. ThenMK\+1​\(\(c,0\)\)=MK​\(c\)M\_\{K\+1\}\(\(c,0\)\)=M\_\{K\}\(c\)for everyc∈ΔK−1c\\in\\Delta^\{K\-1\}, henceℋK⊆ℋK\+1\\mathcal\{H\}\_\{K\}\\subseteq\\mathcal\{H\}\_\{K\+1\}, and for every interfacevvand everyΔ​W∈𝒲\\Delta W\\in\\mathcal\{W\},

dist⁡\(Δ​W,ℋK\+1\)\\displaystyle\\mathrm\{dist\}\(\\Delta W,\\mathcal\{H\}\_\{K\+1\}\)≤dist⁡\(Δ​W,ℋK\),\\displaystyle\\leq\\mathrm\{dist\}\(\\Delta W,\\mathcal\{H\}\_\{K\}\),\(17\)ℛv⋆​\(K\+1\)\\displaystyle\\mathcal\{R\}^\{\\star\}\_\{v\}\(K\+1\)≤ℛv⋆​\(K\),whereℛv⋆​\(K\):=minc∈ΔK−1⁡ℛv​\(MK​\(c\)\)\.\\displaystyle\\leq\\mathcal\{R\}^\{\\star\}\_\{v\}\(K\),\\qquad\\text\{where\}\\quad\\mathcal\{R\}^\{\\star\}\_\{v\}\(K\):=\\min\_\{c\\in\\Delta^\{K\-1\}\}\\mathcal\{R\}\_\{v\}\(M\_\{K\}\(c\)\)\.

*Takeaway: appending an adapter enlarges what the mixture can reach and can only lower the best achievable risk on every interface, old or new\. Any degradation on an old interface after the bank grows is routing error, not lost capacity\.*

###### Corollary 3\(Convex twin\)\.

Ifℛu\\mathcal\{R\}\_\{u\}is convex on𝒦u\\mathcal\{K\}\_\{u\}, then for everyc∈ΔK−1c\\in\\Delta^\{K\-1\},

ℛu​\(M⁡\(c\)\)≤∑k=1Kck​ℛu​\(Δ​W\(k\)\),in particularℛu​\(M⁡\(𝟏/K\)\)≤1K​∑k=1Kℛu​\(Δ​W\(k\)\)\.\\mathcal\{R\}\_\{u\}\(M\(c\)\)\\;\\leq\\;\\sum\_\{k=1\}^\{K\}c\_\{k\}\\,\\mathcal\{R\}\_\{u\}\(\\Delta W^\{\(k\)\}\),\\quad\\text\{in particular\}\\quad\\mathcal\{R\}\_\{u\}\(M\(\\mathbf\{1\}/K\)\)\\;\\leq\\;\\frac\{1\}\{K\}\\sum\_\{k=1\}^\{K\}\\mathcal\{R\}\_\{u\}\(\\Delta W^\{\(k\)\}\)\.\(18\)

*Takeaway: in the convex twin, the held\-out risk of any mixture is at most the corresponding average of the held\-out risks of the transferred bank adapters, and the risk of the uniform mixture is at most their mean\.*

### D\.2Direct generation in a frozen\-trunk twin

TheT2L Reconsbaseline trains a trunk and per\-shape heads and regresses the stored bank updates\. We analyze an idealization that keeps the regression and the heads and drops the rest\.

###### Assumption 2\(Frozen\-trunk twin\)\.

A direct generator consists of a fixed feature mapξ:ℝdc→ℝdξ\\xi:\\mathbb\{R\}^\{d\_\{c\}\}\\to\\mathbb\{R\}^\{d\_\{\\xi\}\}and a linear headΓ∈ℝDΩ×dξ\\Gamma\\in\\mathbb\{R\}^\{D\_\{\\Omega\}\\times d\_\{\\xi\}\}that emits the vectorized update directly,vec​Δ​WΓ​\(h\)=Γ​ξ​\(h\)\\mathrm\{vec\}\\,\\Delta W\_\{\\Gamma\}\(h\)=\\Gamma\\,\\xi\(h\), whereDΩ:=∑s∈Ωdsout​dsinD\_\{\\Omega\}:=\\sum\_\{s\\in\\Omega\}d^\{\\mathrm\{out\}\}\_\{s\}d^\{\\mathrm\{in\}\}\_\{s\}\. Writeξk:=ξ⁡\(hvk\)\\xi\_\{k\}:=\\xi\(h\_\{v\_\{k\}\}\),Ξ:=\[ξ1,…,ξK\]∈ℝdξ×K\\Xi:=\[\\xi\_\{1\},\\dots,\\xi\_\{K\}\]\\in\\mathbb\{R\}^\{d\_\{\\xi\}\\times K\}, andℬ:=\[vec​Δ​W\(1\),…,vec​Δ​W\(K\)\]∈ℝDΩ×K\\mathcal\{B\}:=\[\\mathrm\{vec\}\\,\\Delta W^\{\(1\)\},\\dots,\\mathrm\{vec\}\\,\\Delta W^\{\(K\)\}\]\\in\\mathbb\{R\}^\{D\_\{\\Omega\}\\times K\}\. The features of the source interfaces are linearly independent,rank​Ξ=K\\mathrm\{rank\}\\,\\Xi=K, and the head is trained by gradient descent fromΓ=0\\Gamma=0on the squared reconstruction loss∑k‖Γ​ξk−vec​Δ​W\(k\)‖2\\sum\_\{k\}\\\|\\Gamma\\xi\_\{k\}\-\\mathrm\{vec\}\\,\\Delta W^\{\(k\)\}\\\|^\{2\}\.

###### Proposition 4\(Direct generation is unnormalized routing\)\.

Under Assumption[2](https://arxiv.org/html/2610.22539#Thmassumption2), gradient descent with a sufficiently small step converges toΓ^=ℬ​Ξ\+\\hat\{\\Gamma\}=\\mathcal\{B\}\\,\\Xi^\{\+\}withΞ\+=\(Ξ⊤​Ξ\)−1​Ξ⊤\\Xi^\{\+\}=\(\\Xi^\{\\top\}\\Xi\)^\{\-1\}\\Xi^\{\\top\}, the unique minimum\-norm interpolant of the bank updates\. For every conditioninghh,

vec​Δ​WΓ^​\(h\)=ℬ​η​\(h\)=∑k=1Kηk​\(h\)​vec​Δ​W\(k\),η⁡\(h\):=Ξ\+​ξ​\(h\)∈ℝK\.\\mathrm\{vec\}\\,\\Delta W\_\{\\hat\{\\Gamma\}\}\(h\)\\;=\\;\\mathcal\{B\}\\,\\eta\(h\)\\;=\\;\\sum\_\{k=1\}^\{K\}\\eta\_\{k\}\(h\)\\,\\mathrm\{vec\}\\,\\Delta W^\{\(k\)\},\\qquad\\eta\(h\):=\\Xi^\{\+\}\\xi\(h\)\\in\\mathbb\{R\}^\{K\}\.\(19\)Hence \(a\) the output at any conditioning lies in the linear span of the bank, \(b\) the coefficientsη⁡\(h\)\\eta\(h\)depend on the feature geometry\(Ξ,ξ⁡\(h\)\)\(\\Xi,\\xi\(h\)\)alone and not on the bank or on the policy, and \(c\)η⁡\(h\)\\eta\(h\)need not lie inΔK−1\\Delta^\{K\-1\}\. For any two training featuresξ1,ξ2\\xi\_\{1\},\\xi\_\{2\}, a conditioning withξ⁡\(h\)=ξ1\+ξ2\\xi\(h\)=\\xi\_\{1\}\+\\xi\_\{2\}receivesΔ​W\(1\)\+Δ​W\(2\)\\Delta W^\{\(1\)\}\+\\Delta W^\{\(2\)\}, twice the scale of the mean of the two, and one withξ⁡\(h\)=ξ1−ξ2\\xi\(h\)=\\xi\_\{1\}\-\\xi\_\{2\}receivesΔ​W\(1\)−Δ​W\(2\)\\Delta W^\{\(1\)\}\-\\Delta W^\{\(2\)\}\.

*Takeaway: in the twin, direct generation at a new interface is routing over the same bank, with coefficients that are neither normalized nor chosen through the policy\. Its outputs lie in the span of the bank, so it reaches beyond the hull ofWebMixonly through coefficients outside the simplex, whose scale nothing in its objective controls \(Corollary[5](https://arxiv.org/html/2610.22539#Thmtheorem5)\)\.*

###### Corollary 5\(Scale control\)\.

For everyc∈ΔK−1c\\in\\Delta^\{K\-1\}and every adapted matrixss,

‖M⁡\(c\)‖≤maxk≤K⁡‖Δ​W\(k\)‖and‖M​\(c\)s‖F≤maxk≤K⁡‖Δ​Ws\(k\)‖F\.\\\|M\(c\)\\\|\\leq\\max\_\{k\\leq K\}\\\|\\Delta W^\{\(k\)\}\\\|\\qquad\\text\{and\}\\qquad\\\|M\(c\)\_\{s\}\\\|\_\{F\}\\leq\\max\_\{k\\leq K\}\\\|\\Delta W^\{\(k\)\}\_\{s\}\\\|\_\{F\}\.\(20\)No such bound holds in the twin over feature vectors, for a nonzero bank\. For anyt\>0t\>0and anykk, a conditioning with feature vectorξ⁡\(h\)=t​ξk\\xi\(h\)=t\\,\\xi\_\{k\}yields the outputt​Δ​W\(k\)t\\,\\Delta W^\{\(k\)\}, so the scale of the output follows the scale of the features\.

*Takeaway: the mixture never emits an update larger than the largest adapter in the bank, at any conditioning and with no tuning, whereas the scale of a directly generated update is set by the feature geometry\.*

### D\.3Well\-posedness of the mixture

###### Proposition 6\(Gauge invariance and coefficient identifiability\)\.

\(a\) The objectiveℒrouter\\mathcal\{L\}\_\{\\mathrm\{router\}\}of Eq\.[8](https://arxiv.org/html/2610.22539#S3.E8)depends onFϕ​\(Ev\)F\_\{\\phi\}\(\{E\_\{v\}\}\)only through the induced updates\{Bs​As\}s∈Ω\\\{B\_\{s\}A\_\{s\}\\\}\_\{s\\in\\Omega\}, because the policy uses the weightsWθ,s\+αr​Bs​AsW\_\{\\theta,s\}\+\\tfrac\{\\alpha\}\{r\}B\_\{s\}A\_\{s\}\. In particular, replacing bank memberkkby\(Qk​As\(k\),Bs\(k\)​Qk−1\)\(Q\_\{k\}A^\{\(k\)\}\_\{s\},\\,B^\{\(k\)\}\_\{s\}Q\_\{k\}^\{\-1\}\)for any invertibleQkQ\_\{k\}leavesM⁡\(c\)M\(c\)unchanged for everyccandℒrouter\\mathcal\{L\}\_\{\\mathrm\{router\}\}unchanged for everyϕ\\phi\. \(b\) IfΔ​W\(1\),…,Δ​W\(K\)\\Delta W^\{\(1\)\},\\dots,\\Delta W^\{\(K\)\}are linearly independent in𝒲\\mathcal\{W\}, the mapc↦M⁡\(c\)c\\mapsto M\(c\)is injective onΔK−1\\Delta^\{K\-1\}\. \(c\) Theℓ1\\ell\_\{1\}loss on factors‖A−A∗‖1\+‖B−B∗‖1\\\|A\-A^\{\\ast\}\\\|\_\{1\}\+\\\|B\-B^\{\\ast\}\\\|\_\{1\}against a stored target\(A∗,B∗\)\(A^\{\\ast\},B^\{\\ast\}\)withA∗≠0A^\{\\ast\}\\neq 0, is not constant over the targets\(Q​A∗,B∗​Q−1\)\(QA^\{\\ast\},B^\{\\ast\}Q^\{\-1\}\)asQQvaries, although all of them are the same model\.

Part \(c\) is a general remark about objectives on factors\. OurT2L Reconsbaseline does not have this problem, because it regresses the productBs​AsB\_\{s\}A\_\{s\}\(Appendix[B\.2](https://arxiv.org/html/2610.22539#A2.SS2)\), which is gauge\-invariant by Eq\.[9](https://arxiv.org/html/2610.22539#A1.E9)\.

*Takeaway: the training objective sees the model and not its coordinates, and for a linearly independent bank the coefficientsWebMixemits are a well\-defined function of the adapter it serves\. A factor\-space objective such as theℓ1\\ell\_\{1\}loss of \(c\) is not a function of the target model\.*

## Appendix EProofs

### E\.1Proof of Theorem[1](https://arxiv.org/html/2610.22539#Thmtheorem1)

###### Proof\.

*Existence\.*ΔK−1\\Delta^\{K\-1\}is compact and nonempty \(K≥1K\\geq 1\),MMis linear and hence continuous, andℛu\\mathcal\{R\}\_\{u\}is continuous onℋK⊆𝒦u\\mathcal\{H\}\_\{K\}\\subseteq\\mathcal\{K\}\_\{u\}by Assumption[1](https://arxiv.org/html/2610.22539#Thmassumption1), soc↦ℛu​\(M​\(c\)\)c\\mapsto\\mathcal\{R\}\_\{u\}\(M\(c\)\)is continuous on a compact set and attains its minimum at somec⋆c^\{\\star\}\.

*Decomposition\.*Eq\.[15](https://arxiv.org/html/2610.22539#A4.E15)follows from adding and subtractingℛu​\(M⁡\(c⋆\)\)\\mathcal\{R\}\_\{u\}\(M\(c^\{\\star\}\)\)\. The routing term is non\-negative becausec^∈ΔK−1\\hat\{c\}\\in\\Delta^\{K\-1\}andc⋆c^\{\\star\}minimizes overΔK−1\\Delta^\{K\-1\}\.

*First bound\.*ℋK=M⁡\(ΔK−1\)\\mathcal\{H\}\_\{K\}=M\(\\Delta^\{K\-1\}\)is the image of a compact convex set under a linear map, so it is compact and convex\. The functionW′↦‖Δ​W\(u\)−W′‖W^\{\\prime\}\\mapsto\\\|\\Delta W^\{\(u\)\}\-W^\{\\prime\}\\\|is continuous, so it attains its minimum onℋK\\mathcal\{H\}\_\{K\}at some pointΠ\\Pi, and by definition‖Δ​W\(u\)−Π‖=dist⁡\(Δ​W\(u\),ℋK\)\\\|\\Delta W^\{\(u\)\}\-\\Pi\\\|=\\mathrm\{dist\}\(\\Delta W^\{\(u\)\},\\mathcal\{H\}\_\{K\}\)\. SinceΠ∈ℋK\\Pi\\in\\mathcal\{H\}\_\{K\}there iscΠ∈ΔK−1c\_\{\\Pi\}\\in\\Delta^\{K\-1\}withΠ=M⁡\(cΠ\)\\Pi=M\(c\_\{\\Pi\}\)\. BothΠ\\PiandΔ​W\(u\)\\Delta W^\{\(u\)\}lie in𝒦u\\mathcal\{K\}\_\{u\}, so Assumption[1](https://arxiv.org/html/2610.22539#Thmassumption1)applies and

ℛu​\(M⁡\(c⋆\)\)\\displaystyle\\mathcal\{R\}\_\{u\}\(M\(c^\{\\star\}\)\)≤ℛu​\(M⁡\(cΠ\)\)=ℛu​\(Π\)\\displaystyle\\;\\leq\\;\\mathcal\{R\}\_\{u\}\(M\(c\_\{\\Pi\}\)\)\\;=\\;\\mathcal\{R\}\_\{u\}\(\\Pi\)≤ℛu​\(Δ​W\(u\)\)\+Λu​‖Π−Δ​W\(u\)‖\\displaystyle\\;\\leq\\;\\mathcal\{R\}\_\{u\}\(\\Delta W^\{\(u\)\}\)\+\\Lambda\_\{u\}\\\|\\Pi\-\\Delta W^\{\(u\)\}\\\|=ℛu​\(Δ​W\(u\)\)\+Λu​dist​\(Δ​W\(u\),ℋK\)\.\\displaystyle\\;=\\;\\mathcal\{R\}\_\{u\}\(\\Delta W^\{\(u\)\}\)\+\\Lambda\_\{u\}\\,\\mathrm\{dist\}\(\\Delta W^\{\(u\)\},\\mathcal\{H\}\_\{K\}\)\.
*Second bound\.*For eachkk, the vertexeke\_\{k\}lies inΔK−1\\Delta^\{K\-1\}andM⁡\(ek\)=Δ​W\(k\)M\(e\_\{k\}\)=\\Delta W^\{\(k\)\}, soℛu​\(M⁡\(c⋆\)\)≤ℛu​\(M⁡\(ek\)\)=ℛu​\(Δ​W\(k\)\)\\mathcal\{R\}\_\{u\}\(M\(c^\{\\star\}\)\)\\leq\\mathcal\{R\}\_\{u\}\(M\(e\_\{k\}\)\)=\\mathcal\{R\}\_\{u\}\(\\Delta W^\{\(k\)\}\)\. Taking the minimum overkkgives Eq\.[16](https://arxiv.org/html/2610.22539#A4.E16)\. ∎

### E\.2Proof of Corollary[2](https://arxiv.org/html/2610.22539#Thmtheorem2)

###### Proof\.

Forc∈ΔK−1c\\in\\Delta^\{K\-1\}, the vector\(c,0\)\(c,0\)lies inΔK\\Delta^\{K\}andMK\+1​\(\(c,0\)\)=∑k≤Kck​Δ​W\(k\)\+0⋅Δ​W\(K\+1\)=MK​\(c\)M\_\{K\+1\}\(\(c,0\)\)=\\sum\_\{k\\leq K\}c\_\{k\}\\Delta W^\{\(k\)\}\+0\\cdot\\Delta W^\{\(K\+1\)\}=M\_\{K\}\(c\)\. HenceℋK=MK​\(ΔK−1\)⊆MK\+1​\(ΔK\)=ℋK\+1\\mathcal\{H\}\_\{K\}=M\_\{K\}\(\\Delta^\{K\-1\}\)\\subseteq M\_\{K\+1\}\(\\Delta^\{K\}\)=\\mathcal\{H\}\_\{K\+1\}\. A minimum over a superset is no larger than the minimum over the subset\. Applied toW′↦‖Δ​W−W′‖W^\{\\prime\}\\mapsto\\\|\\Delta W\-W^\{\\prime\}\\\|this gives the distance inequality, and applied toℛv∘MK\+1\\mathcal\{R\}\_\{v\}\\circ M\_\{K\+1\}over\{\(c,0\):c∈ΔK−1\}⊆ΔK\\\{\(c,0\):c\\in\\Delta^\{K\-1\}\\\}\\subseteq\\Delta^\{K\}it givesℛv⋆​\(K\+1\)≤ℛv⋆​\(K\)\\mathcal\{R\}^\{\\star\}\_\{v\}\(K\+1\)\\leq\\mathcal\{R\}^\{\\star\}\_\{v\}\(K\)\. Both minima exist becauseℛv\\mathcal\{R\}\_\{v\}is continuous and both simplices are compact and nonempty\. ∎

### E\.3Proof of Corollary[3](https://arxiv.org/html/2610.22539#Thmtheorem3)

###### Proof\.

M⁡\(c\)=∑kck​Δ​W\(k\)M\(c\)=\\sum\_\{k\}c\_\{k\}\\Delta W^\{\(k\)\}is a convex combination of points of𝒦u\\mathcal\{K\}\_\{u\}, andℛu\\mathcal\{R\}\_\{u\}is convex on𝒦u\\mathcal\{K\}\_\{u\}, so Jensen’s inequality givesℛu​\(M⁡\(c\)\)≤∑kck​ℛu​\(Δ​W\(k\)\)\\mathcal\{R\}\_\{u\}\(M\(c\)\)\\leq\\sum\_\{k\}c\_\{k\}\\mathcal\{R\}\_\{u\}\(\\Delta W^\{\(k\)\}\)\. The second statement is the casec=𝟏/Kc=\\mathbf\{1\}/K\.

For the scope stated in Remark[3](https://arxiv.org/html/2610.22539#Thmremark3): if the output logits are an affine function ofΔ​W\\Delta W, the logits for a fixed input and prefix areℓ⁡\(Δ​W\)=ℓ0\+𝒜⁡\(Δ​W\)\\ell\(\\Delta W\)=\\ell\_\{0\}\+\\mathcal\{A\}\(\\Delta W\)for a fixed vectorℓ0\\ell\_\{0\}and a fixed linear map𝒜\\mathcal\{A\}\. The per\-token negative log\-likelihoodℓ↦log∑jexp\(ℓj\)−ℓy\\ell\\mapsto\\log\\sum\_\{j\}\\exp\(\\ell\_\{j\}\)\-\\ell\_\{y\}is convex inℓ\\ell\(by Hölder’s inequality\), the composition of a convex function with an affine map is convex, and a nonnegative combination of convex functions is convex \(ℛu\\mathcal\{R\}\_\{u\}weights every response token by1/\|𝒟u\|1/\|\\mathcal\{D\}\_\{u\}\|\)\. Henceℛu\\mathcal\{R\}\_\{u\}is convex on all of𝒲\\mathcal\{W\}in that configuration\. ∎

### E\.4Proof of Proposition[4](https://arxiv.org/html/2610.22539#Thmtheorem4)

###### Proof\.

WriteL⁡\(Γ\):=‖Γ​Ξ−ℬ‖F2L\(\\Gamma\):=\\\|\\Gamma\\Xi\-\\mathcal\{B\}\\\|\_\{F\}^\{2\}for the training loss\. Sincerank​Ξ=K≤dξ\\mathrm\{rank\}\\,\\Xi=K\\leq d\_\{\\xi\}, the matrixΞ⊤​Ξ∈ℝK×K\\Xi^\{\\top\}\\Xi\\in\\mathbb\{R\}^\{K\\times K\}is invertible andΞ\+:=\(Ξ⊤​Ξ\)−1​Ξ⊤\\Xi^\{\+\}:=\(\\Xi^\{\\top\}\\Xi\)^\{\-1\}\\Xi^\{\\top\}satisfiesΞ\+​Ξ=IK\\Xi^\{\+\}\\Xi=I\_\{K\}\. HenceΓ^:=ℬ​Ξ\+\\hat\{\\Gamma\}:=\\mathcal\{B\}\\Xi^\{\+\}obeysΓ^​Ξ=ℬ\\hat\{\\Gamma\}\\Xi=\\mathcal\{B\}andL⁡\(Γ^\)=0L\(\\hat\{\\Gamma\}\)=0, soΓ^\\hat\{\\Gamma\}is a minimizer and the minimizer set is the affine space\{Γ:Γ​Ξ=ℬ\}=Γ^\+\{N:N​Ξ=0\}\\\{\\Gamma:\\Gamma\\Xi=\\mathcal\{B\}\\\}=\\hat\{\\Gamma\}\+\\\{N:N\\Xi=0\\\}\.

*Minimum norm\.*Every row ofΓ^=ℬ​Ξ\+\\hat\{\\Gamma\}=\\mathcal\{B\}\\Xi^\{\+\}is a linear combination of the rows ofΞ⊤\\Xi^\{\\top\}, that is, of the vectorsξ1⊤,…,ξK⊤\\xi\_\{1\}^\{\\top\},\\dots,\\xi\_\{K\}^\{\\top\}\. Every row of anNNwithN​Ξ=0N\\Xi=0is orthogonal to eachξk\\xi\_\{k\}\. Therefore⟨Γ^,N⟩F=0\\langle\\hat\{\\Gamma\},N\\rangle\_\{F\}=0and‖Γ^\+N‖F2=‖Γ^‖F2\+‖N‖F2≥‖Γ^‖F2\\\|\\hat\{\\Gamma\}\+N\\\|\_\{F\}^\{2\}=\\\|\\hat\{\\Gamma\}\\\|\_\{F\}^\{2\}\+\\\|N\\\|\_\{F\}^\{2\}\\geq\\\|\\hat\{\\Gamma\}\\\|\_\{F\}^\{2\}, with equality only atN=0N=0\. SoΓ^\\hat\{\\Gamma\}is the unique minimum\-norm minimizer\.

*Gradient descent\.*∇L​\(Γ\)=2​\(Γ​Ξ−ℬ\)​Ξ⊤\\nabla L\(\\Gamma\)=2\(\\Gamma\\Xi\-\\mathcal\{B\}\)\\Xi^\{\\top\}\. Let the step beγ\\gammawith0<γ<1/λmax0<\\gamma<1/\\lambda\_\{\\max\}, whereλmax:=λmax​\(Ξ​Ξ⊤\)\\lambda\_\{\\max\}:=\\lambda\_\{\\max\}\(\\Xi\\Xi^\{\\top\}\), letΓt\+1=Γt−γ∇L\(Γt\)\\Gamma\_\{t\+1\}=\\Gamma\_\{t\}\-\\gamma\\nabla L\(\\Gamma\_\{t\}\)withΓ0=0\\Gamma\_\{0\}=0, and letXt:=Γt−Γ^X\_\{t\}:=\\Gamma\_\{t\}\-\\hat\{\\Gamma\}\. SinceΓ^​Ξ=ℬ\\hat\{\\Gamma\}\\Xi=\\mathcal\{B\}, the error obeysXt\+1=Xt​\(I−2​γ​Ξ​Ξ⊤\)X\_\{t\+1\}=X\_\{t\}\(I\-2\\gamma\\,\\Xi\\Xi^\{\\top\}\)\. \(i\) The rows ofXtX\_\{t\}stay inspan⁡\{ξ1,…,ξK\}\\mathrm\{span\}\\\{\\xi\_\{1\},\\dots,\\xi\_\{K\}\\\}, that is,Xt​P=XtX\_\{t\}P=X\_\{t\}for the projectorP:=Ξ​Ξ\+P:=\\Xi\\Xi^\{\+\}\. This holds forX0=−Γ^X\_\{0\}=\-\\hat\{\\Gamma\}becauseΓ^​P=Γ^\\hat\{\\Gamma\}P=\\hat\{\\Gamma\}, and it is preserved becauseΞ​Ξ⊤​P=Ξ​Ξ⊤\\Xi\\Xi^\{\\top\}P=\\Xi\\Xi^\{\\top\}\. \(ii\) Each step decreases the error\. Expanding the square, and using⟨X,X​Ξ​Ξ⊤⟩F=‖X​Ξ‖F2\\langle X,X\\,\\Xi\\Xi^\{\\top\}\\rangle\_\{F\}=\\\|X\\Xi\\\|\_\{F\}^\{2\}and‖Z​Ξ⊤‖F2≤λmax​‖Z‖F2\\\|Z\\Xi^\{\\top\}\\\|\_\{F\}^\{2\}\\leq\\lambda\_\{\\max\}\\\|Z\\\|\_\{F\}^\{2\}\(the matricesΞ⊤​Ξ\\Xi^\{\\top\}\\XiandΞ​Ξ⊤\\Xi\\Xi^\{\\top\}have the same nonzero eigenvalues\),

‖Xt\+1‖F2=‖Xt‖F2−4​γ​‖Xt​Ξ‖F2\+4​γ2​‖Xt​Ξ​Ξ⊤‖F2≤‖Xt‖F2−4​γ​\(1−γ​λmax\)​‖Xt​Ξ‖F2\.\\\|X\_\{t\+1\}\\\|\_\{F\}^\{2\}=\\\|X\_\{t\}\\\|\_\{F\}^\{2\}\-4\\gamma\\\|X\_\{t\}\\Xi\\\|\_\{F\}^\{2\}\+4\\gamma^\{2\}\\\|X\_\{t\}\\Xi\\Xi^\{\\top\}\\\|\_\{F\}^\{2\}\\leq\\\|X\_\{t\}\\\|\_\{F\}^\{2\}\-4\\gamma\\,\(1\-\\gamma\\lambda\_\{\\max\}\)\\,\\\|X\_\{t\}\\Xi\\\|\_\{F\}^\{2\}\.\(iii\) Summing \(ii\) overttgives∑t‖Xt​Ξ‖F2≤‖X0‖F2/\(4​γ​\(1−γ​λmax\)\)<∞\\sum\_\{t\}\\\|X\_\{t\}\\Xi\\\|\_\{F\}^\{2\}\\leq\\\|X\_\{0\}\\\|\_\{F\}^\{2\}/\\big\(4\\gamma\(1\-\\gamma\\lambda\_\{\\max\}\)\\big\)<\\infty, soXt​Ξ→0X\_\{t\}\\Xi\\to 0\. \(iv\) By \(i\),Xt=Xt​P=\(Xt​Ξ\)​Ξ\+→0X\_\{t\}=X\_\{t\}P=\(X\_\{t\}\\Xi\)\\,\\Xi^\{\+\}\\to 0, soΓt→Γ^\\Gamma\_\{t\}\\to\\hat\{\\Gamma\}\.

*Output\.*For anyhh,vec​Δ​WΓ^​\(h\)=Γ^​ξ​\(h\)=ℬ​Ξ\+​ξ​\(h\)=ℬ​η​\(h\)\\mathrm\{vec\}\\,\\Delta W\_\{\\hat\{\\Gamma\}\}\(h\)=\\hat\{\\Gamma\}\\xi\(h\)=\\mathcal\{B\}\\,\\Xi^\{\+\}\\xi\(h\)=\\mathcal\{B\}\\,\\eta\(h\), which is Eq\.[19](https://arxiv.org/html/2610.22539#A4.E19)\. Statements \(a\) and \(b\) read off this expression\. For \(c\),Ξ\+​Ξ=IK\\Xi^\{\+\}\\Xi=I\_\{K\}givesη⁡\(ξk\)=Ξ\+​Ξ​ek=ek\\eta\(\\xi\_\{k\}\)=\\Xi^\{\+\}\\Xi e\_\{k\}=e\_\{k\}for everykk, so by linearityη⁡\(ξ1±ξ2\)=e1±e2\\eta\(\\xi\_\{1\}\\pm\\xi\_\{2\}\)=e\_\{1\}\\pm e\_\{2\}, and the outputs areℬ⁡\(e1±e2\)=vec​Δ​W\(1\)±vec​Δ​W\(2\)\\mathcal\{B\}\(e\_\{1\}\\pm e\_\{2\}\)=\\mathrm\{vec\}\\,\\Delta W^\{\(1\)\}\\pm\\mathrm\{vec\}\\,\\Delta W^\{\(2\)\}\. The coordinates ofe1\+e2e\_\{1\}\+e\_\{2\}sum to22ande1−e2e\_\{1\}\-e\_\{2\}has a negative coordinate, so neither lies inΔK−1\\Delta^\{K\-1\}; orthonormality is not needed\. ∎

### E\.5Proof of Corollary[5](https://arxiv.org/html/2610.22539#Thmtheorem5)

###### Proof\.

Forc∈ΔK−1c\\in\\Delta^\{K\-1\}, the triangle inequality andck≥0c\_\{k\}\\geq 0give‖M⁡\(c\)‖≤∑kck​‖Δ​W\(k\)‖≤\(∑kck\)​maxk​‖Δ​W\(k\)‖=maxk⁡‖Δ​W\(k\)‖\\\|M\(c\)\\\|\\leq\\sum\_\{k\}c\_\{k\}\\\|\\Delta W^\{\(k\)\}\\\|\\leq\\big\(\\sum\_\{k\}c\_\{k\}\\big\)\\max\_\{k\}\\\|\\Delta W^\{\(k\)\}\\\|=\\max\_\{k\}\\\|\\Delta W^\{\(k\)\}\\\|\. The per\-matrix statement is the same argument applied toM​\(c\)s=∑kck​Δ​Ws\(k\)M\(c\)\_\{s\}=\\sum\_\{k\}c\_\{k\}\\Delta W^\{\(k\)\}\_\{s\}in the Frobenius norm\. For the twin,ξ⁡\(h\)=t​ξk\\xi\(h\)=t\\,\\xi\_\{k\}givesη⁡\(h\)=Ξ\+​\(t​ξk\)=t​Ξ\+​Ξ​ek=t​ek\\eta\(h\)=\\Xi^\{\+\}\(t\\,\\xi\_\{k\}\)=t\\,\\Xi^\{\+\}\\Xi e\_\{k\}=t\\,e\_\{k\}and hencevec​Δ​WΓ^​\(h\)=t​vec​Δ​W\(k\)\\mathrm\{vec\}\\,\\Delta W\_\{\\hat\{\\Gamma\}\}\(h\)=t\\,\\mathrm\{vec\}\\,\\Delta W^\{\(k\)\}, whose norm ist​‖Δ​W\(k\)‖t\\\|\\Delta W^\{\(k\)\}\\\|and is unbounded inttwhenΔ​W\(k\)≠0\\Delta W^\{\(k\)\}\\neq 0\. ∎

### E\.6Proof of Proposition[6](https://arxiv.org/html/2610.22539#Thmtheorem6)

###### Proof\.

\(a\) In Eq\.[8](https://arxiv.org/html/2610.22539#S3.E8),pθ,Fϕ​\(Ev\)p\_\{\\theta,F\_\{\\phi\}\(\{E\_\{v\}\}\)\}is the likelihood of the network whose weight at matrixssisWθ,s\+αr​Bs​AsW\_\{\{\\theta,\}s\}\+\\tfrac\{\\alpha\}\{r\}B\_\{s\}A\_\{s\}\. Replacing\(As,Bs\)\(A\_\{s\},B\_\{s\}\)by\(Qs​As,Bs​Qs−1\)\(Q\_\{s\}A\_\{s\},B\_\{s\}Q\_\{s\}^\{\-1\}\)leaves this weight, hence the likelihood and the loss, unchanged\. ForWebMix, re\-gauging bank memberkkchanges the concatenated factors of Eq\.[5](https://arxiv.org/html/2610.22539#S3.E5)but not the product∑kck​Bs\(k\)​Qk−1​Qk​As\(k\)=∑kck​Bs\(k\)​As\(k\)\\sum\_\{k\}c\_\{k\}B^\{\(k\)\}\_\{s\}Q\_\{k\}^\{\-1\}Q\_\{k\}A^\{\(k\)\}\_\{s\}=\\sum\_\{k\}c\_\{k\}B^\{\(k\)\}\_\{s\}A^\{\(k\)\}\_\{s\}of Eq\.[4](https://arxiv.org/html/2610.22539#S3.E4), soM⁡\(c\)M\(c\)is unchanged for everyccand so isℒrouter\\mathcal\{L\}\_\{\\mathrm\{router\}\}\.

\(b\) IfM⁡\(c\)=M⁡\(c′\)M\(c\)=M\(c^\{\\prime\}\)then∑k\(ck−ck′\)​Δ​W\(k\)=0\\sum\_\{k\}\(c\_\{k\}\-c^\{\\prime\}\_\{k\}\)\\Delta W^\{\(k\)\}=0, and linear independence forcesck=ck′c\_\{k\}=c^\{\\prime\}\_\{k\}for everykk\.

\(c\) Fix a generator output\(A,B\)\(A,B\)and takeQ=t​IrQ=tI\_\{r\}witht\>0t\>0, so the re\-gauged target is\(t​A∗,B∗/t\)\(tA^\{\\ast\},B^\{\\ast\}/t\)\. The loss‖A−t​A∗‖1\+‖B−B∗/t‖1\\\|A\-tA^\{\\ast\}\\\|\_\{1\}\+\\\|B\-B^\{\\ast\}/t\\\|\_\{1\}is finite att=1t=1and tends to infinity ast→∞t\\to\\inftybecauseA∗≠0A^\{\\ast\}\\neq 0\. It is therefore not constant intt, while every\(t​A∗,B∗/t\)\(tA^\{\\ast\},B^\{\\ast\}/t\)induces the same updateB∗​A∗B^\{\\ast\}A^\{\\ast\}\. ∎

## Declaration of LLM Usage

The authors acknowledge the use of LLMs to polish the draft, to brainstorm ideas and research directions, to orchestrate experiments, to verify every proof in Lean, and to review code\. We take full responsibility for the paper’s content and correctness\.

## Reproducibility Statement

We have taken several steps to ensure our results are reproducible\. The complete source code is included in the supplementary material and will be released publicly upon acceptance\. It comes with a README describing environment setup, dependencies, and each component\. All proofs are formalized in Lean 4, and the formalization is included in the supplementary material, so every theorem can be machine\-checked\. The source code maps every numbered result of Appendix[D](https://arxiv.org/html/2610.22539#A4)to the Lean declaration that proves it\. All datasets are publicly available and have been cited\.