From Sampled Outcomes to Capability Distributions: Rethinking Supervision for LLM Routing

# From Sampled Outcomes to Capability Distributions: Rethinking Supervision for LLM Routing
Source: [https://arxiv.org/html/2606.06924](https://arxiv.org/html/2606.06924)
Guannan Lai1,2, Haoran Hu1,2, Long Chen3, Zhenguo Li3,4, Han\-Jia Ye1,2\(🖂\) 1School of Artificial Intelligence, Nanjing University 2National Key Laboratory for Novel Software Technology, Nanjing University 3Hong Kong University of Science and Technology 4Frontier Robotics \{laign,yehj\}@lamda\.nju\.edu\.cn, huhr@smail\.nju\.edu\.cn, longchen@ust\.hk, zhenguol@gmail\.com ![[Uncaptioned image]](https://arxiv.org/html/2606.06924v1/x1.png)[Datasets](https://huggingface.co/datasets/AIGNLAI/DARS)![[Uncaptioned image]](https://arxiv.org/html/2606.06924v1/x2.png)[Source Code](https://github.com/AIGNLAI/DARS)

###### Abstract

Existing LLM routing methods typically treat a model’s single response to a query as its capability label for training routers\. However, because LLM generation is inherently stochastic, such single\-shot supervision provides only a noisy observation of a query\-model pair’s behavior rather than a reliable capability estimate\. We show that this assumption introduces systematic noise into routing supervision, making learned routing policies less reliable\. To address this issue, we proposeDARS\(Distribution\-Aware Routing Supervision\), a framework that constructs routing supervision from a distributional view of model behavior\. Instead of relying on a single generated response, DARS considers uncertainty from both the input side and the output side, capturing how semantically equivalent query formulations and stochastic generations affect model performance\. Based on these distribution\-aware observations, DARS builds more reliable supervision signals for routing\. Experiments across diverse tasks show that single\-shot labels can be misleading for model selection, while distribution\-aware supervision provides more stable labels and improves learned routing behavior\. Our results suggest that reliable LLM routing should move beyond single\-response observations and be grounded in query\-level model capability distributions\.

From Sampled Outcomes to Capability Distributions: Rethinking Supervision for LLM Routing

Guannan Lai1,2, Haoran Hu1,2, Long Chen3, Zhenguo Li3,4, Han\-Jia Ye1,2\(🖂\)1School of Artificial Intelligence, Nanjing University2National Key Laboratory for Novel Software Technology, Nanjing University3Hong Kong University of Science and Technology4Frontier Robotics\{laign,yehj\}@lamda\.nju\.edu\.cn, huhr@smail\.nju\.edu\.cn, longchen@ust\.hk, zhenguol@gmail\.com![[Uncaptioned image]](https://arxiv.org/html/2606.06924v1/x3.png)[Datasets](https://huggingface.co/datasets/AIGNLAI/DARS)![[Uncaptioned image]](https://arxiv.org/html/2606.06924v1/x4.png)[Source Code](https://github.com/AIGNLAI/DARS)

## 1Introduction

![Refer to caption](https://arxiv.org/html/2606.06924v1/x5.png)Figure 1:Illustration of the single\-shot label issue in LLM routing\. The colored regions represent different models that are preferred in different parts of the query\-behavior space\. For the same original query, repeated generations may lead to multiple observed outcomes across these regions, while single\-shot supervision only observes one sampled outcome and may therefore produce an unstable routing label\.Large language models \(LLMs\) are increasingly deployed as heterogeneous model pools rather than isolated single systems\. In practice, different models exhibit distinct strengths: lightweight models can often handle simple instructions or factual queries, while larger or more specialized models may be required for complex reasoning, mathematical problem solving, biomedical question answering, or long\-context understanding\. These models also differ substantially in inference cost, latency, context length, and availability\. As a result,*LLM routing*has become an important mechanism for cost\-effective deployment: given an input query, a router selects an appropriate model from a candidate pool, aiming to balance response quality and serving cost\. Recent routing methods and benchmarks have made substantial progress in learning such query\-dependent model selection policies\(Chenet al\.,[2023](https://arxiv.org/html/2606.06924#bib.bib140); Onget al\.,[2025](https://arxiv.org/html/2606.06924#bib.bib131); Huet al\.,[2024](https://arxiv.org/html/2606.06924#bib.bib148); Huanget al\.,[2025](https://arxiv.org/html/2606.06924#bib.bib168)\)\.

Despite this progress, most routing supervision still represents each query\-model pair with a single generated response and its score\. This point\-estimate view may be adequate for deterministic prediction or closed\-form classification, but it is poorly aligned with free\-form LLM generation, where outputs are inherently stochastic\. Prior work on uncertainty estimation and hallucination detection has shown that LLM generations can be sensitive to sampling randomness and may vary in both surface form and semantic content\(Kuhnet al\.,[2023](https://arxiv.org/html/2606.06924#bib.bib163); Manakulet al\.,[2023](https://arxiv.org/html/2606.06924#bib.bib164); Farquharet al\.,[2024](https://arxiv.org/html/2606.06924#bib.bib167)\)\. As illustrated in Figure[1](https://arxiv.org/html/2606.06924#S1.F1), even for the same original query, different sampled outcomes may suggest different model preferences\. This raises a fundamental question for LLM routing: can a single sampled response reliably serve as the supervision signal for model selection?

This mismatch can propagate through the entire routing pipeline\. Variability in a model’s sampled response may alter its observed score; changes in observed scores may change which model appears preferable for a query; and routers trained on such sample\-dependent labels may learn policies that reflect incidental generation noise rather than stable differences in model capability\. These issues motivate a distribution\-aware view of routing supervision, where labels are constructed from repeated observations of query\-model behavior rather than from isolated sampled outputs\.

In this paper, we proposeDARS\(Distribution\-AwareRoutingSupervision\), a framework for constructing distribution\-aware supervision signals for LLM routing\. DARS considers uncertainty from both the input side and the output side: it uses semantically preserving prompt rewrites to capture sensitivity to query formulation, and repeated decoding to capture stochastic variation in generation\. Based on these observations, DARS constructs more reliable routing labels that reflect query\-level model behavior rather than isolated outputs\. We evaluate DARS on diverse tasks spanning multiple\-choice scientific reasoning, mathematical problem solving, and reading comprehension, using a heterogeneous pool of six LLMs\. Our analysis shows that single\-shot labels can be unstable and misleading for model selection, while a small number of distributional observations already yield substantially more reliable supervision\. Further experiments show that routers trained with DARS supervision achieve more stable and effective routing behavior than those trained from single\-response labels\.

Our contributions are threefold:

- •We identify and systematically analyze the*single\-shot label assumption*in LLM routing, showing that representing query\-model behavior with one generated response can introduce unstable and noisy routing supervision\.
- •We introduceDARS, a distribution\-aware routing supervision framework that accounts for both input\-side and output\-side uncertainty through prompt rewrites and repeated decoding\.
- •We provide empirical evidence across diverse datasets and models that distribution\-aware supervision yields more reliable routing labels and improves learned routing policies compared with single\-response supervision\.

## 2Related Work

#### LLM routing\.

LLM routing aims to select an appropriate model from a heterogeneous model pool for each input query, usually balancing response quality and inference cost\. Early work studies cost\-aware model selection and cascading strategies, showing that using multiple LLMs adaptively can reduce inference cost while maintaining or even improving performance\(Chenet al\.,[2023](https://arxiv.org/html/2606.06924#bib.bib140); Zhanget al\.,[2023](https://arxiv.org/html/2606.06924#bib.bib139); Aggarwalet al\.,[2024](https://arxiv.org/html/2606.06924#bib.bib102); Yueet al\.,[2023](https://arxiv.org/html/2606.06924#bib.bib159); Lai and Ye,[2026](https://arxiv.org/html/2606.06924#bib.bib3)\)\. Recent methods further learn query\-dependent routing policies with different supervision and modeling strategies\. For example, Hybrid LLM and BEST\-Route study adaptive routing under quality\-cost constraints\(Dinget al\.,[2024](https://arxiv.org/html/2606.06924#bib.bib130),[2025](https://arxiv.org/html/2606.06924#bib.bib106)\); RouteLLM learns routing from preference data\(Onget al\.,[2025](https://arxiv.org/html/2606.06924#bib.bib131)\); TensorOpera Router, EmbedLLM, IRT\-Router, GraphRouter, and causal routing develop different learned routers based on regression, representation learning, item response theory, graph modeling, or regret minimization\(Stripeliset al\.,[2024](https://arxiv.org/html/2606.06924#bib.bib132); Zhuanget al\.,[2024](https://arxiv.org/html/2606.06924#bib.bib152); Songet al\.,[2025](https://arxiv.org/html/2606.06924#bib.bib169); Fenget al\.,[2025](https://arxiv.org/html/2606.06924#bib.bib127); Tsiourvaset al\.,[2025](https://arxiv.org/html/2606.06924#bib.bib161); Maet al\.,[2026](https://arxiv.org/html/2606.06924#bib.bib86)\)\. Other studies investigate training\-free, retrieval\-based, controllable, or preference\-aware routing strategies\(Zhaoet al\.,[2024](https://arxiv.org/html/2606.06924#bib.bib138); Li,[2025](https://arxiv.org/html/2606.06924#bib.bib150); Meiet al\.,[2025](https://arxiv.org/html/2606.06924#bib.bib89); Piskalaet al\.,[2025](https://arxiv.org/html/2606.06924#bib.bib104)\)\. Benchmarks such as RouterBench and RouterEval provide systematic evaluation settings for multi\-LLM routing\(Huet al\.,[2024](https://arxiv.org/html/2606.06924#bib.bib148); Huanget al\.,[2025](https://arxiv.org/html/2606.06924#bib.bib168)\)\. Unlike these works, we do not focus on designing a new router architecture; instead, we revisit the supervision used to train or evaluate routers\.

#### Uncertainty\-aware and efficient LLM inference\.

Several lines of work improve LLM inference efficiency by estimating uncertainty, deciding when to defer to stronger models, or allocating computation adaptively\. For example, uncertainty\-based two\-tier selection and related caching or distillation strategies reduce expensive model calls by identifying when a cheaper model is sufficient\(Ramírezet al\.,[2024](https://arxiv.org/html/2606.06924#bib.bib100),[2023](https://arxiv.org/html/2606.06924#bib.bib101)\)\. Other work studies budget\- and quality\-controllable routing, quality\-of\-service\-aware routing, or dynamic routing under deployment constraints\(Meiet al\.,[2025](https://arxiv.org/html/2606.06924#bib.bib89); Yanget al\.,[2025b](https://arxiv.org/html/2606.06924#bib.bib90); Wanget al\.,[2025](https://arxiv.org/html/2606.06924#bib.bib134)\)\. Recent studies also examine routing robustness and vulnerabilities, showing that routing policies can be fragile under distribution shifts, adversarial settings, or lifecycle changes\(Shafranet al\.,[2025](https://arxiv.org/html/2606.06924#bib.bib91); Linet al\.,[2025](https://arxiv.org/html/2606.06924#bib.bib144); Kassemet al\.,[2025](https://arxiv.org/html/2606.06924#bib.bib145)\)\. These works highlight the importance of reliable model selection, but they typically assume that the observed score of a query\-model pair is a suitable supervision signal\. In contrast, DARS argues that such observations are themselves stochastic and should be treated as samples from an underlying capability distribution\.

#### Uncertainty and robustness in language model generation\.

Our work is also related to uncertainty estimation and robustness analysis for LLM generation\. Prior studies show that LLM outputs can vary substantially under sampling randomness, and that inconsistency across generations can reveal uncertainty or hallucination risk\(Kuhnet al\.,[2023](https://arxiv.org/html/2606.06924#bib.bib163); Manakulet al\.,[2023](https://arxiv.org/html/2606.06924#bib.bib164); Farquharet al\.,[2024](https://arxiv.org/html/2606.06924#bib.bib167)\)\. Another line of work studies prompt robustness, showing that semantically similar or adversarially perturbed prompts may lead to different model predictions\(Gan and Mori,[2023](https://arxiv.org/html/2606.06924#bib.bib165); Zhuet al\.,[2023](https://arxiv.org/html/2606.06924#bib.bib166)\)\. These findings suggest that a model’s behavior on a query cannot be fully characterized by a single generated response\. DARS brings this insight to LLM routing: instead of constructing routing labels from isolated sampled outcomes, it uses query rewrites and repeated stochastic decoding to estimate distribution\-aware capability signals for each query\-model pair\.

## 3Preliminaries

### 3\.1Single\-shot Supervision for LLM Routing

LLM routing considers a candidate model poolℳ=\{m1,…,mK\}\\mathcal\{M\}=\\\{m\_\{1\},\\ldots,m\_\{K\}\\\}and aims to select an appropriate model for each input queryxx\. Given a routing functionr​\(⋅\)r\(\\cdot\), the selected model isr​\(x\)∈ℳr\(x\)\\in\\mathcal\{M\}\. The goal is to achieve a favorable trade\-off between response quality and inference cost: the router should assign difficult queries to more capable models when necessary, while avoiding unnecessary use of expensive models for queries that can be handled by cheaper alternatives\.

In a typical routing setup, each modelmmproduces a responseyyfor queryxx\. The response is evaluated along two dimensions\. First, a task\-specific performance functions​\(⋅\)s\(\\cdot\)measures the quality of the response, such as accuracy for multiple\-choice or mathematical tasks, or F1 for reading\-comprehension tasks\. Second, a cost functionκ​\(⋅\)\\kappa\(\\cdot\)measures the inference cost of producing the response, which may depend on the model, the query, the output length, and the decoding process\. Thus, for a query\-model pair\(xi,m\)\(x\_\{i\},m\), a generated response induces both a performance observation and a cost observation\.

Most existing routing datasets construct supervision from a single observed response for each query\-model pair\. Specifically, for each queryxix\_\{i\}and modelmm, one response is generated:

yi,msingle∼P​\(y∣xi,m\),y\_\{i,m\}^\{\\mathrm\{single\}\}\\sim P\(y\\mid x\_\{i\},m\),and the corresponding single\-shot observations are recorded as

qi,msingle=s​\(yi,msingle,xi\),ci,msingle=κ​\(yi,msingle,xi,m\)\.q\_\{i,m\}^\{\\mathrm\{single\}\}=s\(y\_\{i,m\}^\{\\mathrm\{single\}\},x\_\{i\}\),\\quad c\_\{i,m\}^\{\\mathrm\{single\}\}=\\kappa\(y\_\{i,m\}^\{\\mathrm\{single\}\},x\_\{i\},m\)\.Here,qi,msingleq\_\{i,m\}^\{\\mathrm\{single\}\}denotes the observed performance of modelmmon queryxix\_\{i\}, whileci,msinglec\_\{i,m\}^\{\\mathrm\{single\}\}denotes the observed cost of this particular generation\. Importantly, the cost is not necessarily a model\-level constant: even for the same model, it may vary across queries and sampled outputs due to differences in prompt length, completion length, and generation behavior\.

Given a dataset𝒟=\{xi\}i=1N\\mathcal\{D\}=\\\{x\_\{i\}\\\}\_\{i=1\}^\{N\}and a cost budgetCmaxC\_\{\\max\}, the routing objective can be formulated as a constrained optimization problem:

maxr1N∑i=1Nqi,r​\(xi\)singles\.t\.1N∑i=1Nci,r​\(xi\)single≤Cmax\.\\max\_\{r\}\\frac\{1\}\{N\}\\sum\_\{i=1\}^\{N\}q\_\{i,r\(x\_\{i\}\)\}^\{\\mathrm\{single\}\}\\quad\\mathrm\{s\.t\.\}\\quad\\frac\{1\}\{N\}\\sum\_\{i=1\}^\{N\}c\_\{i,r\(x\_\{i\}\)\}^\{\\mathrm\{single\}\}\\leq C\_\{\\max\}\.That is, the router aims to maximize average performance while keeping the average inference cost within a prescribed budget\.

This protocol is simple and scalable, which explains its prevalence in routing benchmarks and learned routing systems\. However, it implicitly treats one sampled generation as a reliable estimate of both the performance and cost behavior of a query\-model pair\. The next section examines why this single\-shot view can be problematic for LLM routing\.

### 3\.2Uncertainty in Query\-Model Behavior

The reliability of single\-shot routing supervision depends on whether a query\-model pair has stable behavior under natural variations\. In this work, we distinguish two sources of uncertainty that are particularly relevant to LLM routing\.

#### Input\-side uncertainty\.

The first source comes from variations in how the same query is expressed\. In real applications, semantically equivalent user requests may differ in wording, structure, level of detail, or prompt format\. Ideally, a model’s capability on a query should be stable under such meaning\-preserving variations\. However, prior work on prompt robustness has shown that LLMs can be sensitive to prompt formulations, and that small changes in prompts may lead to different predictions or performance degradation\(Zhuet al\.,[2023](https://arxiv.org/html/2606.06924#bib.bib166); Gan and Mori,[2023](https://arxiv.org/html/2606.06924#bib.bib165)\)\. For routing, this means that the estimated suitability of a model may depend not only on the underlying task, but also on the particular surface form of the input\. We refer to this source of variability as*input\-side uncertainty*\.

#### Output\-side uncertainty\.

The second source comes from stochastic generation itself\. Even when the input query and model are fixed, decoding can produce different responses, which may vary in factuality, reasoning path, final answer, or semantic content\. Prior work on uncertainty estimation and hallucination detection has shown that such variation is informative: inconsistent generations often indicate higher uncertainty or lower reliability\(Kuhnet al\.,[2023](https://arxiv.org/html/2606.06924#bib.bib163); Manakulet al\.,[2023](https://arxiv.org/html/2606.06924#bib.bib164); Farquharet al\.,[2024](https://arxiv.org/html/2606.06924#bib.bib167)\)\. For routing, this implies that a single generated response may be insufficient to characterize how well a model can handle a query\. We refer to this source of variability as*output\-side uncertainty*\.

![Refer to caption](https://arxiv.org/html/2606.06924v1/x6.png)\(a\)Label instability
![Refer to caption](https://arxiv.org/html/2606.06924v1/x7.png)\(b\)Uncertainty sources
![Refer to caption](https://arxiv.org/html/2606.06924v1/x8.png)\(c\)Router variance

Figure 2:Diagnostic analysis of single\-shot routing supervision\. \(a\) Single\-shot labels are unstable at the outcome, winner, and label levels\. \(b\) Input\-side and output\-side uncertainty contribute differently across datasets\. \(c\) Routers trained from different single\-shot samples exhibit non\-negligible performance variation\.

### 3\.3Limitations of Single\-shot Routing Supervision

To examine whether single\-shot supervision provides reliable routing signals, we conduct a diagnostic analysis across tasks with different output formats and evaluation protocols\. This section focuses only on the limitations of existing supervision protocols, leaving the design of distribution\-aware routing to the next section\.

#### Experimental setup\.

We use three datasets covering complementary task types\.GPQAis a graduate\-level multiple\-choice question answering benchmark covering biology, physics, and chemistry\(Reinet al\.,[2023](https://arxiv.org/html/2606.06924#bib.bib170)\)\.MATH\-500is a 500\-problem subset of the MATH benchmark for mathematical problem solving\(Hendryckset al\.,[2021](https://arxiv.org/html/2606.06924#bib.bib171); Lightmanet al\.,[2024](https://arxiv.org/html/2606.06924#bib.bib172)\)\.DROP\-800is an 800\-problem subset of the reading\-comprehension benchmark requiring discrete reasoning over paragraphs, such as counting, sorting, and arithmetic reasoning\(Duaet al\.,[2019](https://arxiv.org/html/2606.06924#bib.bib173)\)\. For each dataset, we sample 200 training queries for constructing repeated observations\. To capture input\-side uncertainty, we generate five semantically preserving rewrites for each query using GPT\-4o\(Hurstet al\.,[2024](https://arxiv.org/html/2606.06924#bib.bib174)\), while keeping the gold answer and candidate choices unchanged\. To capture output\-side uncertainty, for each rewritten query and each model, we perform five independent decoding runs under the same evaluation protocol\. This results in a5×55\\times 5observation matrix for each query\-model pair, where each entry records both the response quality and the corresponding inference cost\. Our model pool includes Gemma\-3\-12B\-IT\(Teamet al\.,[2025](https://arxiv.org/html/2606.06924#bib.bib175)\), Mistral\-Small\-3\.2\-24B\-Instruct\(Mistral AI,[2025](https://arxiv.org/html/2606.06924#bib.bib176)\), Qwen3\-32B\(Yanget al\.,[2025a](https://arxiv.org/html/2606.06924#bib.bib177)\), Llama\-3\.3\-70B\-Instruct\(Meta AI,[2024](https://arxiv.org/html/2606.06924#bib.bib178)\), Gemini\-2\.5\-Flash\-Lite\(Google,[2025](https://arxiv.org/html/2606.06924#bib.bib179)\), and DeepSeek\-Chat\-V3\.1\(DeepSeek\-AIet al\.,[2024](https://arxiv.org/html/2606.06924#bib.bib180)\)\.

Observation 1: Single\-shot labels are unstable across datasets\.

Figure[2\(a\)](https://arxiv.org/html/2606.06924#S3.F2.sf1)shows that single\-shot supervision is unstable across all three datasets\. We report three diagnostic metrics\.Outcome instabilitymeasures whether repeated observations of the same query\-model pair produce different scores\.Winner flip ratemeasures whether the model selected as the best candidate for a query changes under different single\-shot samples\.Single vs\. DA disagreementmeasures how often a single\-shot routing label disagrees with the distribution\-aware label estimated from repeated observations\.

The results show that instability is not limited to response scores, but propagates to model selection\. GPQA exhibits the strongest instability, with an outcome instability of 0\.715 and a winner flip rate of 0\.970\. DROP\-800 also shows substantial instability, with a winner flip rate of 0\.665 and a single\-vs\.\-DA disagreement rate of 0\.471\. Even on MATH\-500, where the task is more constrained, the winner flip rate remains 0\.645\. These results indicate that single\-shot labels are not merely noisy measurements; they can directly change which model is selected for a query\.

Observation 2: Different tasks exhibit different uncertainty profiles\.

Figure[2\(b\)](https://arxiv.org/html/2606.06924#S3.F2.sf2)decomposes uncertainty into input\-side and output\-side components\. GPQA and MATH\-500 are dominated by output\-side uncertainty: their output\-side variances are0\.0990\.099and0\.0530\.053, respectively, clearly higher than their input\-side variances\. This indicates that repeated decoding alone can substantially change observed model behavior on scientific and mathematical reasoning tasks\. In contrast, DROP\-800 shows comparable input\-side and output\-side uncertainty, with input\-side uncertainty slightly higher \(0\.04280\.0428vs\.0\.03500\.0350\)\. This suggests that reading\-comprehension performance is also sensitive to how the query is formulated\. Therefore, single\-shot supervision can fail in different ways across tasks: some tasks are mainly affected by stochastic generation, while others are also sensitive to meaning\-preserving input variations\.

Observation 3: Single\-shot supervision induces unstable learned routers\.

To test whether label instability affects router learning, we construct100100single\-shot training sets by randomly sampling one observation from the5×55\\times 5matrix for each query\-model pair, and train one router on each sampled training set\. Figure[2\(c\)](https://arxiv.org/html/2606.06924#S3.F2.sf3)reports the resulting test performance distribution\. Since all routers use the same model pool, data split, and architecture, the performance variation mainly comes from the random single\-shot supervision used for training\. The observed spread across the100100routers indicates that single\-shot supervision can lead to non\-negligible policy variance\. Thus, the issue is not limited to label construction: stochastic supervision can propagate into the learned router and make the final model\-selection policy less reliable\.

## 4DARS: Distribution\-Aware Routing Supervision

![Refer to caption](https://arxiv.org/html/2606.06924v1/x9.png)Figure 3:Overview of the proposed DARS framework\. Conventional single\-shot supervision obtains only one stochastic observation for each query\-model pair and constructs routing supervision from the observed performance and cost\. DARS instead collects multiple observations through query rewriting and repeated stochastic decoding, from which it estimates distributional capability signals, including expected performance, expected cost, and performance variability\. These signals are then used to construct risk\-aware supervision for learning a more reliable routing policy\.### 4\.1Overview of DARS

LLM routing aims to select an appropriate model from a candidate poolℳ\\mathcal\{M\}for each input queryxx, typically balancing response quality and inference cost\. Existing routing methods often construct supervision from a single sampled response for each query\-model pair, where the observed quality and cost are used to estimate a single\-shot utilityUsingle​\(x,m\)=q​\(x,m\)−λ​c​\(x,m\)U\_\{\\text\{single\}\}\(x,m\)=q\(x,m\)\-\\lambda c\(x,m\)\. While simple, this point\-estimate protocol is unreliable for stochastic LLM generation: semantically equivalent query formulations and repeated decoding runs may produce different responses, scores, and even model preferences\. As a result, routers trained on single\-shot observations may learn incidental sampling noise rather than stable differences in model capability\.

To address this issue, we proposeDARS, a distribution\-aware supervision framework for LLM routing\. As shown in Figure[3](https://arxiv.org/html/2606.06924#S4.F3), DARS replaces isolated single\-shot observations with repeated observations obtained through query rewriting and stochastic decoding\. For each query\-model pair, these observations are summarized into distributional capability signals, including expected performance, expected cost, and aggregate performance variability\. DARS then constructs a risk\-aware utility:

U​\(x,m\)=μq​\(x,m\)−λ​μc​\(x,m\)−β​σq​\(x,m\),U\(x,m\)=\\mu\_\{q\}\(x,m\)\-\\lambda\\mu\_\{c\}\(x,m\)\-\\beta\\sigma\_\{q\}\(x,m\),whereμq​\(x,m\)\\mu\_\{q\}\(x,m\),μc​\(x,m\)\\mu\_\{c\}\(x,m\), andσq​\(x,m\)\\sigma\_\{q\}\(x,m\)denote the expected quality, expected cost, and overall performance variability, respectively\. By incorporating both cost and risk into supervision, DARS encourages routers to select models that are not only accurate and cost\-effective, but also stable under natural input and output variations\.

### 4\.2DARS Workflow

DARS consists of four main stages: repeated observation construction, distributional capability estimation, risk\-aware supervision construction, and routing policy learning\.

#### Repeated observation construction\.

Given a training queryxx, DARS first uses a strong language model to generate a set of semantically equivalent rewritten queries:

𝒳​\(x\)=\{x\(1\),x\(2\),…,x\(N\)\}\.\\mathcal\{X\}\(x\)=\\\{x^\{\(1\)\},x^\{\(2\)\},\\ldots,x^\{\(N\)\}\\\}\.These rewrites preserve the original task semantics and gold answer, while varying the surface form, wording, or prompt structure\. This step is intended to introduce controlled input\-side perturbations\. By observing how each candidate model behaves across these meaning\-preserving variants, DARS can estimate whether the model’s performance is stable with respect to different formulations of the same underlying query\.

For each candidate modelm∈ℳm\\in\\mathcal\{M\}, DARS then applies stochastic decoding to the rewritten queries multiple times\. This step captures output\-side variability: even when the input formulation is fixed, different decoding runs may lead to different reasoning paths, final answers, or response lengths\. For each generated response, DARS records its task\-specific quality score and inference cost\. Together with the rewritten\-query dimension, these repeated decoding results form a set of observations for each query\-model pair, which will be summarized into distributional capability signals in the next stage\.

#### Distributional capability estimation\.

Let

𝒪x,m=\{\(qm\(n,j\),cm\(n,j\)\)\}\\mathcal\{O\}\_\{x,m\}=\\\{\(q\_\{m\}^\{\(n,j\)\},c\_\{m\}^\{\(n,j\)\}\)\\\}denote the repeated observations collected for modelmmon queryxx\. DARS summarizes this observation set into three capability signals\. The expected quality and expected cost are computed as

μq​\(x,m\)=Mean⁡\(\{qm\(n,j\)\}\(n,j\)\),\\mu\_\{q\}\(x,m\)=\\operatorname\{Mean\}\\left\(\\\{q\_\{m\}^\{\(n,j\)\}\\\}\_\{\(n,j\)\}\\right\),μc​\(x,m\)=Mean⁡\(\{cm\(n,j\)\}\(n,j\)\)\.\\mu\_\{c\}\(x,m\)=\\operatorname\{Mean\}\\left\(\\\{c\_\{m\}^\{\(n,j\)\}\\\}\_\{\(n,j\)\}\\right\)\.The performance risk is estimated by the standard deviation of the observed quality scores:

σq​\(x,m\)=Std⁡\(\{qm\(n,j\)\}\(n,j\)\)\.\\sigma\_\{q\}\(x,m\)=\\operatorname\{Std\}\\left\(\\\{q\_\{m\}^\{\(n,j\)\}\\\}\_\{\(n,j\)\}\\right\)\.This risk term captures the aggregate variability caused by query rewriting and stochastic decoding\. We use this overall variability as a practical risk signal for router training, rather than explicitly decomposing it into input\-side and output\-side components\.

#### Risk\-aware supervision construction\.

DARS combines the estimated capability signals into the following risk\-aware utility:

U​\(x,m\)=μq​\(x,m\)−λ​μc​\(x,m\)−β​σq​\(x,m\)\.U\(x,m\)=\\mu\_\{q\}\(x,m\)\-\\lambda\\mu\_\{c\}\(x,m\)\-\\beta\\sigma\_\{q\}\(x,m\)\.The first term favors models with higher expected performance, the second term penalizes inference cost, and the third term penalizes unstable behavior\. Based on this utility, the preferred model for queryxxis

m∗​\(x\)=arg⁡maxm∈ℳ⁡U​\(x,m\)\.m^\{\*\}\(x\)=\\arg\\max\_\{m\\in\\mathcal\{M\}\}U\(x,m\)\.Compared with single\-shot supervision, this label is estimated from repeated observations and is therefore less sensitive to incidental sampled outcomes\.

#### Routing policy learning\.

Finally, DARS provides distribution\-aware supervision for learning the routing policy\. Since DARS constructs model\-wise capability signals rather than prescribing a specific router architecture, the resulting supervision can be adapted to a variety of learning paradigms\.

For regression\-based routers, DARS provides continuous targets such as expected quality, expected cost, and performance risk\. The router can learn to predict these quantities for each candidate model and derive the final routing score from the predicted capability signals\. For utility\-based routers, the risk\-aware utilityU​\(x,m\)U\(x,m\)can be directly used as the learning target, so that the router learns to approximate the final model\-selection criterion\. For classification\-based routers, the model with the largest DARS utility can be used as a hard routing label, replacing the single\-shot best\-model label with a more stable distribution\-aware label\.

For probabilistic classification routers, DARS utilities can further be converted into a soft categorical target over the candidate model pool:

pDARS​\(m∣x\)=exp⁡\(U​\(x,m\)/τ\)∑m′∈ℳexp⁡\(U​\(x,m′\)/τ\),p\_\{\\text\{DARS\}\}\(m\\mid x\)=\\frac\{\\exp\(U\(x,m\)/\\tau\)\}\{\\sum\_\{m^\{\\prime\}\\in\\mathcal\{M\}\}\\exp\(U\(x,m^\{\\prime\}\)/\\tau\)\},whereτ\\tauis a temperature parameter controlling the sharpness of the target distribution\. This formulation treats model routing as a multi\-class prediction problem, where candidate models are regarded as classes and the router is trained to match the DARS\-induced model preference distribution\. Compared with a hard labelm∗​\(x\)m^\{\*\}\(x\), the soft target preserves relative preferences among models, especially when several candidates have similar risk\-aware utilities\.

DARS can also support ranking\-based routers\. In this case, the utility scores of candidate models are transformed into pairwise or listwise preferences, allowing the router to learn which model should be preferred for a given query\. Similarly, for preference\-based routing methods, DARS can provide more reliable preference labels by comparing models according to their distribution\-aware utilities rather than single sampled outcomes\. For non\-parametric or retrieval\-based routers, the estimated capability signals can be aggregated from similar training queries to infer the suitability of each candidate model for a new query\. For cluster\-based routers, DARS statistics can be summarized at the cluster level, enabling model selection based on the distributional behavior of queries with similar representations\.

At inference time, the router selects the model with the highest predicted routing score, i\.e\.,r​\(x\)=arg⁡maxm∈ℳ⁡U^θ​\(x,m\)r\(x\)=\\arg\\max\_\{m\\in\\mathcal\{M\}\}\\hat\{U\}\_\{\\theta\}\(x,m\)\.

Overall, DARS decouples supervision construction from router design\. By replacing noisy single\-shot labels with distributional capability estimates, it provides a unified supervision source for regression\-based, utility\-based, classification\-based, ranking\-based, preference\-based, retrieval\-based, and cluster\-based routing methods\.

## 5Experiments

### 5\.1Experimental Setup

#### Datasets and model pool\.

We evaluate DARS on the same datasets and model pool used in Section[3\.3](https://arxiv.org/html/2606.06924#S3.SS3)\. The datasets cover different task formats and reasoning abilities, including scientific multiple\-choice reasoning, mathematical problem solving, and reading comprehension\. The candidate model pool contains heterogeneous LLMs with different capability and cost profiles\. This setting allows us to examine whether distribution\-aware supervision improves routing behavior across diverse tasks and model choices\.

#### Test\-time observation protocol\.

For evaluation, we construct two types of test observations for each query\-model pair\. First, we generate three semantically preserving rewrites for each test query to evaluate routing robustness under input\-side variations\. Second, we perform three stochastic decoding runs on the original query to evaluate robustness under output\-side randomness\. Based on these two observation groups, we report two test scores: rewrite\-based utility, denoted asUrewU\_\{\\mathrm\{rew\}\}, and decoding\-based utility, denoted asUdecU\_\{\\mathrm\{dec\}\}\. The former measures whether the selected model remains effective when the query is expressed in different but equivalent forms, while the latter measures whether the selected model remains reliable under repeated stochastic generations\.

#### Compared methods\.

We compare DARS with routing methods from different learning paradigms to evaluate whether distribution\-aware supervision can benefit diverse router architectures\. We consider MLPRouterStripeliset al\.\([2024](https://arxiv.org/html/2606.06924#bib.bib132)\), MIRTSonget al\.\([2025](https://arxiv.org/html/2606.06924#bib.bib169)\), and EmbedLLMZhuanget al\.\([2024](https://arxiv.org/html/2606.06924#bib.bib152)\)as regression\-based routing methods; RM\-SoftmaxTsiourvaset al\.\([2025](https://arxiv.org/html/2606.06924#bib.bib161)\)and GraphRouterFenget al\.\([2025](https://arxiv.org/html/2606.06924#bib.bib127)\)as classification\-based methods; AvengersProZhanget al\.\([2025](https://arxiv.org/html/2606.06924#bib.bib160)\)as a clustering\-based method; and kNNRouterStripeliset al\.\([2024](https://arxiv.org/html/2606.06924#bib.bib132)\)as a non\-parametric retrieval\-based method\. All methods are trained and evaluated under the same data split, model pool, and cost\-quality trade\-off setting for a fair comparison\.

### 5\.2Experimental Results

#### Overall comparison\.

Table[1](https://arxiv.org/html/2606.06924#S5.T1)compares single\-shot routing with DARS\-enhanced routing across different router families, with all utilities computed usingλ=0\.05\\lambda=0\.05\. For each baseline router, the single\-shot result is averaged over 100 independently sampled single\-shot training sets, while DARS keeps the same router architecture and only replaces the supervision with distribution\-aware signals from repeated observations\. This setup isolates the effect of supervision construction\.

Overall, DARS improves almost all routers across the three datasets, showing that its benefit is not tied to a specific router design\. The gains are especially clear on GPQA, where single\-shot supervision is highly unstable, and on MATH\-500, where DARS turns MIRT from a relatively weak baseline into the best\-performing router\. These results indicate that distribution\-aware supervision provides more reliable training signals for stable model selection\.

Table 1:Comparison between single\-shot routing and DARS\-enhanced routing\.![Refer to caption](https://arxiv.org/html/2606.06924v1/x10.png)\(a\)Sample efficiency of DARS\.
![Refer to caption](https://arxiv.org/html/2606.06924v1/x11.png)\(b\)Cost\-quality trade\-off under different cost constraints\.
![Refer to caption](https://arxiv.org/html/2606.06924v1/x12.png)\(c\)Sensitivity to the risk coefficientβ\\beta\.

Figure 4:Further analysis of DARS\. \(a\) Sample efficiency analysis evaluates how the number of repeated observations affects routing performance\. \(b\) Cost\-quality trade\-off analysis compares routing performance under different cost constraints\. \(c\) Sensitivity analysis studies the effect of the risk coefficientβ\\betaon routing utility\.
#### Ablation study\.

Table[2](https://arxiv.org/html/2606.06924#S5.T2)reports the component ablation results of DARS\. We compare several variants to isolate the contribution of different components\.Input\-only DAconstructs supervision using only rewritten queries, whileOutput\-only DAuses only repeated decoding samples from the original query\.DARS w/o Riskremoves the performance\-variability term and uses only expected quality and expected cost\.DARS w/ Rew\. RiskandDARS w/ Dec\. Riskincorporate only rewrite\-induced or decoding\-induced risk, respectively\. The results show that both input\-only and output\-only variants improve over the single\-shot baseline, indicating that query rewriting and repeated decoding each provide useful supervision signals\. However, neither variant matches full DARS, suggesting that input\-side and output\-side observations are complementary\. The risk term is also important: removing it leads to lower average performance, while using either rewrite\-side or decoding\-side risk improves the result\. Full DARS achieves the best average score, confirming that reliable routing supervision benefits from both repeated observations and explicit modeling of performance variability\.

Table 2:Component ablation of DARS\. The average score is computed over all reported rewrite\-based and decoding\-based utilities\.
#### Sample efficiency of DARS\.

Figure[4\(a\)](https://arxiv.org/html/2606.06924#S5.F4.sf1)analyzes the effect of different observation protocols on DARS\. The results show that increasing the number of observations generally improves routing utility, but the gain quickly saturates\. In particular, the3×33\\times 3protocol already achieves performance close to the full5×55\\times 5setting across datasets, suggesting that DARS does not necessarily require a large sampling budget to be effective\. This finding is important for practical deployment: although DARS is built on repeated observations, a moderate number of query rewrites and decoding samples is often sufficient to obtain reliable supervision, making the additional data\-construction cost more manageable than under a fully exhaustive sampling protocol\.

#### Cost\-quality trade\-off under different cost constraints\.

Figure[4\(b\)](https://arxiv.org/html/2606.06924#S5.F4.sf2)compares the cost\-quality trade\-off on GPQA under different routing cost constraints\. DARS consistently improves the Pareto frontier over single\-shot supervision across different router families, indicating that distribution\-aware supervision remains effective under both strict and relaxed cost budgets\. Notably, DARS\-enhanced routers can surpass the best fixed\-model baseline in several cost regions, showing that the learned routing policy can provide a better cost\-quality balance than always using a single strong model\. This result suggests that DARS does not merely improve average routing utility, but also leads to more favorable deployment trade\-offs under practical budget constraints\.

#### Sensitivity to the risk coefficient\.

Figure[4\(c\)](https://arxiv.org/html/2606.06924#S5.F4.sf3)studies the sensitivity of DARS to the risk coefficientβ\\beta\. Overall, the results are stable across a wide range ofβ\\betavalues, with performance differences remaining within a small range\. This indicates that DARS is not highly sensitive to the exact choice of the risk penalty\. Among the tested values,β=0\.2\\beta=0\.2provides a strong and stable setting across datasets, and we therefore useβ=0\.2\\beta=0\.2in our main experiments\.

## 6Conclusion

In this paper, we identify the limitation of single\-shot supervision in LLM routing and propose DARS, a distribution\-aware supervision framework that estimates model capability from repeated observations across query rewrites and stochastic decoding\. Experiments show that DARS provides more reliable routing signals and generally improves routing performance across diverse router families under cost\-quality\-risk trade\-offs\.

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## Appendix

## Appendix ADataset and Model Details

#### Datasets\.

We evaluate DARS on three datasets that cover different task formats and reasoning requirements\.GPQAis a graduate\-level multiple\-choice question answering benchmark covering challenging scientific domains such as biology, physics, and chemistry\(Reinet al\.,[2023](https://arxiv.org/html/2606.06924#bib.bib170)\)\.MATH\-500is a 500\-problem subset of the MATH benchmark, focusing on mathematical problem solving and multi\-step reasoning\(Hendryckset al\.,[2021](https://arxiv.org/html/2606.06924#bib.bib171); Lightmanet al\.,[2024](https://arxiv.org/html/2606.06924#bib.bib172)\)\.DROP\-800is a subset of the DROP reading\-comprehension benchmark, where questions require discrete reasoning over paragraphs, such as counting, sorting, comparison, and arithmetic reasoning\(Duaet al\.,[2019](https://arxiv.org/html/2606.06924#bib.bib173)\)\. These datasets are selected because they differ in output format, reasoning style, and sensitivity to input/output variations, allowing us to evaluate whether distribution\-aware supervision is useful across heterogeneous routing scenarios\.

#### Data split\.

For each dataset, we construct a training split for learning routing policies and a held\-out test split for evaluation\. Following the diagnostic analysis in the main paper, we sample 200 training queries from each dataset to construct repeated observations for supervision\. The remaining held\-out queries are used for test\-time evaluation\. All routers are trained and evaluated using the same split to ensure that performance differences come from the supervision construction method rather than from different data partitions\.

#### Training observation protocol\.

For each training query, we construct distribution\-aware observations for every candidate model\. To capture input\-side variation, we use a strong language model to generate five semantically preserving rewrites of the original query\. The rewriting process preserves the original task semantics, gold answer, and candidate choices when applicable\. To capture output\-side stochasticity, each candidate model is evaluated with five independent decoding runs for each rewritten query\. This gives a5×55\\times 5observation matrix for every query\-model pair, where each entry records the response quality and the corresponding inference cost\. DARS summarizes these repeated observations into expected quality, expected cost, and performance variability, which are then used to construct distribution\-aware supervision signals\.

#### Single\-shot training protocol\.

For the single\-shot baseline, each query\-model pair is represented by one sampled observation\. To reduce the effect of a particular random draw, we construct 100 independently sampled single\-shot training sets\. Each single\-shot training set contains one randomly selected observation for every query\-model pair, and the reported single\-shot result is averaged over the 100 trained routers\. In contrast, the DARS\-enhanced version keeps the router architecture unchanged and replaces only the single\-shot supervision with distribution\-aware supervision constructed from repeated observations\.

#### Test\-time evaluation protocol\.

At test time, the router receives only the original query and selects one model from the candidate pool\. Repeated test observations are used only to evaluate the selected model, not to provide additional information to the router\. For each test query, we construct two evaluation groups\. The first group contains three semantically preserving rewrites of the query, which is used to compute the rewrite\-based utilityUrewU\_\{\\mathrm\{rew\}\}\. The second group contains three independent stochastic decoding runs on the original query, which is used to compute the decoding\-based utilityUdecU\_\{\\mathrm\{dec\}\}\. The former measures whether the routing decision is robust to input\-side variations, while the latter measures whether the selected model remains reliable under output\-side randomness\.

#### Model pool\.

We use a heterogeneous pool of six LLMs with different capability and cost profiles: Gemma\-3\-12B\-IT\(Teamet al\.,[2025](https://arxiv.org/html/2606.06924#bib.bib175)\), Mistral\-Small\-3\.2\-24B\-Instruct\(Mistral AI,[2025](https://arxiv.org/html/2606.06924#bib.bib176)\), Qwen3\-32B\(Yanget al\.,[2025a](https://arxiv.org/html/2606.06924#bib.bib177)\), Llama\-3\.3\-70B\-Instruct\(Meta AI,[2024](https://arxiv.org/html/2606.06924#bib.bib178)\), Gemini\-2\.5\-Flash\-Lite\(Google,[2025](https://arxiv.org/html/2606.06924#bib.bib179)\), and DeepSeek\-Chat\-V3\.1\(DeepSeek\-AIet al\.,[2024](https://arxiv.org/html/2606.06924#bib.bib180)\)\. This model pool contains both open\-source and API\-based models, as well as models with different sizes and deployment costs\. Such heterogeneity is important for evaluating LLM routing, since the router must learn when a cheaper model is sufficient and when a stronger model is worth the additional cost\.

#### Evaluation metric and cost setting\.

For each generated response, we compute a task\-specific quality score and an inference cost\. Quality is measured by the corresponding task metric, such as accuracy for multiple\-choice and mathematical tasks, and F1\-style matching for reading\-comprehension tasks\. Inference cost is normalized within each dataset before computing routing utilities\. Unless otherwise specified, we use the cost coefficientλ=0\.05\\lambda=0\.05and the risk coefficientβ=0\.2\\beta=0\.2in the main experiments\.

## Appendix BQuery Rewriting Prompt and Decoding Protocol

#### Query rewriting\.

For each training query, we generate semantically preserving rewrites using a strong language model\. The purpose of rewriting is to introduce controlled input\-side variation while keeping the underlying task unchanged\. We use dataset\-specific rewriting instructions to avoid changing the gold answer, answer choices, mathematical conditions, or passage\-grounded evidence\. The rewriting constraints are shown below\.

For GPQA, only the question stem is rewritten, while the answer choices are kept unchanged\. For MATH\-500, all mathematical constraints and requested quantities must be preserved\. For DROP\-800, only the question is rewritten and the passage remains fixed\. These constraints ensure that each rewrite remains a valid variant of the original query and can be evaluated using the same gold answer\.

Dataset\-specific rewriting instructionsGPQA\.This is a graduate\-level multiple\-choice science question\. Rewrite only the question stem\. Do not modify, reorder, remove, or paraphrase the answer choices\. The correct option after rewriting must remain exactly the same\.MATH\-500\.This is a free\-form mathematics problem\. Rewrite only the problem statement\. Preserve every mathematical condition, number, variable, equation, unit, constraint, and requested quantity\. The final mathematical answer must remain exactly the same\. Do not simplify the problem, add hints, or reveal the answer\.DROP\-800\.This is a reading\-comprehension question over a fixed passage\. Rewrite only the question\. Do not rewrite the passage\. The rewritten question must be answerable from the same passage and must have the same gold answer\. Do not add information that is not supported by the passage\.

#### Prompt format for model generation\.

After obtaining rewritten queries, we construct dataset\-specific prompts for candidate models\. All prompts require the model to output a clearly markedFinal Answer:line, so that responses can be parsed and evaluated consistently\. For GPQA, the prompt includes the rewritten question and the original answer choices, and asks the model to return one option letter\. For MATH\-500, the prompt asks the model to solve the rewritten mathematics problem and put only the final result afterFinal Answer:\. For DROP\-800, the prompt provides the fixed passage and the rewritten question, and asks the model to answer using only the passage\.

Shared system instruction for generationYou are answering benchmark questions for an LLM routing experiment\. Follow the user instructions exactly\. You may reason internally, but the final response must contain a clearly markedFinal Answer:line\. Do not mention that this is a paraphrase or dataset example\.

#### Training decoding protocol\.

For each training query, DARS uses rewritten prompt variants rather than the original query by default\. Specifically, we useN=5N=5rewritten variants for each query\. For every rewritten variant and every candidate model, we performM=5M=5independent stochastic decoding runs\. This produces5×55\\times 5observations for each query\-model pair\. Each observation records the generated response, task\-specific quality score, token usage, and inference cost\. These observations are then aggregated into expected quality, expected cost, and performance variability for constructing DARS supervision\.

#### Decoding configuration\.

All model generations are collected through the same decoding interface using the same sampling configuration unless otherwise specified\. We use temperature0\.70\.7and top\-p0\.950\.95to induce stochastic generations\. The maximum generation length is capped according to the dataset: 768 tokens for GPQA, 3072 tokens for MATH\-500, and 512 tokens for DROP\-800\. Independent decoding samples are obtained by issuing separate generation calls with the same prompt and decoding configuration\. We use the same candidate model pool as described in Appendix[A](https://arxiv.org/html/2606.06924#A1)\.

#### Test\-time decoding protocol\.

At test time, the router receives only the original query and selects a candidate model\. The repeated test observations are used only for evaluation and are not provided to the router\. For each selected model, we compute two evaluation views\. The rewrite\-based utilityUrewU\_\{\\mathrm\{rew\}\}is evaluated using three semantically preserving rewrites of the test query, while the decoding\-based utilityUdecU\_\{\\mathrm\{dec\}\}is evaluated using three stochastic decoding runs on the original query\. This protocol allows us to separately evaluate robustness to input\-side query variation and output\-side generation randomness\.

## Appendix CSupplementary Experimental Details and Results

### C\.1Baseline Implementation Details

This section provides implementation details for the routing baselines used in our experiments\. All baselines are implemented in the same codebase and share the same data loader, query feature extraction pipeline, candidate model pool, supervision construction protocol, and evaluation procedure unless otherwise specified\.

#### Common setup\.

All methods are evaluated on GPQA, MATH\-500, and DROP\-800 using the same six\-model candidate pool described in Appendix[A](https://arxiv.org/html/2606.06924#A1)\. Each method reads scored generations from the same training and test files\. For query representation, we use one feature vector per query\. By default, the feature text contains both the question and available context fields; context can be removed for ablation\. Neural and retrieval\-based methods use the same query encoder interface\. When available, we use a local sentence\-transformer encoder; otherwise, we fall back to TF\-IDF features\. The default TF\-IDF dimensionality is 20,000\.

#### Single\-shot and DARS supervision\.

Each baseline is evaluated under two supervision modes\. In the single\-shot setting, each query\-model pair is represented by one sampled scored observation\. To reduce the effect of a particular random draw, we repeat this process with 100 independently sampled single\-shot training sets and report the averaged result\. In the DARS setting, the router architecture is kept unchanged, but the supervision signal is replaced by distribution\-aware statistics aggregated from repeated observations\. Specifically, for each query\-model pair, we compute the mean score, mean cost, and score standard deviation\. The default inference utility is based on the risk\-aware objectivescore−λ⋅cost−β⋅risk\\text\{score\}\-\\lambda\\cdot\\text\{cost\}\-\\beta\\cdot\\text\{risk\}, where the cost coefficient is set toλ=0\.05\\lambda=0\.05\.

Table 3:Summary of baseline implementations and how DARS supervision is incorporated\.
#### Test views\.

Each trained router predicts one model for each test query\. The selected model is evaluated under two views\. The rewrite view uses observations generated from semantically preserving query rewrites, while the decoding view uses repeated stochastic decoding samples from the original query\. These test observations are used only for evaluation and are not provided to the router during model selection\.

#### Regression\-based routers\.

MLPtrains separate regressors for model score, model cost, and, under DARS, score standard deviation\. Each regressor predicts a vector over the candidate models for a given query representation, and the router selects the model with the largest predicted risk\-adjusted utility\. We use scikit\-learn MLP regressors with standardized input features\.

MIRTimplements a multi\-dimensional item\-response\-theory router\. It maps each query into a latent ability vector and learns model\-specific discrimination and difficulty parameters\. Separate prediction heads are used for score, cost, and score standard deviation\. Under DARS, MIRT learns all three distributional targets and selects models according to the predicted risk\-aware utility\.

EmbedLLMlearns compact model embeddings and a query projection to predict query\-model performance\. It jointly trains a performance head and a model\-cost head\. Under DARS, the score target is adjusted using the performance\-risk term, while the implementation does not explicitly predict query\-specific uncertainty at inference time; the risk information is incorporated through the training target\.

#### Classification\-based routers\.

RM\-Softmaximplements a regret\-minimization router\. It augments each query embedding with a cost\-weight value and trains a neural classifier over candidate models across a grid of cost coefficients\. Under DARS, the target performance is risk\-adjusted before training, so the classifier learns model preferences induced by distribution\-aware supervision\.

GraphRoutertreats each query\-model pair as an edge between a query node and a model node\. Query features and model descriptions are projected into a shared space, and an edge predictor estimates model\-wise score, cost, and uncertainty\. Under DARS, these targets correspond to the aggregated mean score, mean cost, and score standard deviation for each query\-model pair\.

#### Non\-parametric and cluster\-based routers\.

kNNRouteris a non\-parametric retrieval\-based router\. It stores training query embeddings and their query\-model target matrices\. For a test query, it retrieves the nearest training queries and averages their model\-wise score, cost, and uncertainty statistics using inverse\-distance weights\. Under DARS, the stored targets are distribution\-aware statistics rather than single\-shot observations\.

AvengersProis implemented as a cluster\-based router\. Training queries are clustered in feature space, and each cluster stores the average score, cost, and uncertainty for each candidate model\. For a new query, the router retrieves the nearest clusters and estimates model utility by aggregating their stored statistics\. Under DARS, these cluster\-level statistics are computed from distribution\-aware query\-model observations\.

### C\.2Additional Experimental Results

Table[4](https://arxiv.org/html/2606.06924#A3.T4)provides an expanded version of the main comparison table\. For the single\-shot setting, we report the mean and standard deviation over 100 independently sampled single\-shot training sets when run\-level standard deviations are available\. For DARS, each router is trained once using distribution\-aware supervision constructed from repeated observations\. The results show that single\-shot supervision can exhibit non\-negligible variance across random sampled training sets, especially on GPQA\. In contrast, replacing single\-shot labels with DARS supervision consistently improves the mean performance of most routers, confirming that the gains in the main table come from more reliable supervision construction rather than changes to the router architecture\.

Table 4:Extended comparison between single\-shot routing and DARS\-enhanced routing\. For single\-shot routing, values are averaged over 100 independently sampled training sets, and we report mean±\\pmstandard deviation when run\-level standard deviations are logged\.
### C\.3Ablation Details

This section describes how each ablation variant in Table[2](https://arxiv.org/html/2606.06924#S5.T2)is constructed\. All variants use the same router architecture, training split, candidate model pool, and evaluation protocol\. The only difference is how the supervision signal is constructed from repeated observations\.

#### Single\-shot\.

The single\-shot baseline follows the standard routing supervision protocol\. For each query\-model pair, we randomly sample one scored observation from the available observation set and use it as the training target\. Since the target contains only one observed score and cost, no performance\-variability term is used\. To reduce dependence on a particular random draw, we repeat this process with 100 independently sampled single\-shot training sets and report the averaged result\.

#### Input\-only DA\.

This variant isolates the effect of query rewriting\. For each query\-model pair, we aggregate observations across rewritten queries, but do not use repeated decoding samples for each rewrite\. In practice, this variant estimates model behavior from meaning\-preserving input variations, so the resulting supervision captures sensitivity to query formulation while excluding output\-side stochasticity as much as possible\.

#### Output\-only DA\.

This variant isolates the effect of stochastic decoding\. Instead of aggregating over rewritten queries, it uses repeated decoding samples from the original query\. The supervision signal therefore captures how model performance changes under generation randomness while keeping the input formulation fixed\.

#### DARS w/o Risk\.

This variant uses the same repeated observations as full DARS, but removes the performance\-variability term from the utility\. It constructs supervision only from expected quality and expected cost\. This ablation tests whether the benefit of DARS comes only from averaging repeated observations, or whether explicitly modeling instability provides additional value\.

#### DARS w/ Rew\. Risk\.

This variant includes a risk term estimated only from rewrite\-induced variability\. It measures how much the model’s performance changes across semantically equivalent query formulations, while not explicitly incorporating the variability caused by repeated decoding under the same formulation\.

#### DARS w/ Dec\. Risk\.

This variant includes a risk term estimated only from decoding\-induced variability\. It measures how much the model’s performance changes across stochastic generations of the same input query, while not explicitly incorporating rewrite\-induced variability\.

#### Full DARS\.

Full DARS aggregates observations across both rewritten queries and repeated stochastic decoding runs\. It estimates expected quality, expected cost, and aggregate performance variability from the complete observation set\. This variant therefore captures both input\-side and output\-side uncertainty, and uses the resulting risk\-aware utility to construct the final supervision signal\.

### C\.4Utility Computation and Evaluation Details

This section provides additional details on how routing utilities and evaluation scores are computed\.

#### Per\-observation utility\.

Each generated response is associated with a task\-specific quality score and an inference cost\. Letqi,m\(o\)q\_\{i,m\}^\{\(o\)\}andci,m\(o\)c\_\{i,m\}^\{\(o\)\}denote the quality score and normalized cost of modelmmon queryxix\_\{i\}under observationoo\. The per\-observation utility is computed as

ui,m\(o\)=qi,m\(o\)−λ​ci,m\(o\)\.u\_\{i,m\}^\{\(o\)\}=q\_\{i,m\}^\{\(o\)\}\-\\lambda c\_\{i,m\}^\{\(o\)\}\.In all main experiments, we setλ=0\.05\\lambda=0\.05\.

#### Rewrite\-based utility\.

The rewrite\-based utility evaluates whether the selected model remains effective under semantically equivalent query formulations\. For each test query, the router receives only the original query and selects one modelr​\(xi\)r\(x\_\{i\}\)\. We then evaluate the selected model on the rewrite\-view observations, which contain three rewritten versions of the test query\. The rewrite\-based utilityUrewU\_\{\\mathrm\{rew\}\}is the average per\-observation utility of the selected model over these rewrite\-view observations and all test queries:

Urew​\(r\)=∑xi∈𝒟test∑o∈𝒪irew\(qi,r​\(xi\)\(o\)−λ​ci,r​\(xi\)\(o\)\)\|𝒪irew\|\|𝒟test\|,U\_\{\\mathrm\{rew\}\}\(r\)=\\frac\{\\sum\_\{x\_\{i\}\\in\\mathcal\{D\}\_\{\\mathrm\{test\}\}\}\\frac\{\\sum\_\{o\\in\\mathcal\{O\}^\{\\mathrm\{rew\}\}\_\{i\}\}\\left\(q\_\{i,r\(x\_\{i\}\)\}^\{\(o\)\}\-\\lambda c\_\{i,r\(x\_\{i\}\)\}^\{\(o\)\}\\right\)\}\{\|\\mathcal\{O\}^\{\\mathrm\{rew\}\}\_\{i\}\|\}\}\{\|\\mathcal\{D\}\_\{\\mathrm\{test\}\}\|\},where𝒪irew\\mathcal\{O\}^\{\\mathrm\{rew\}\}\_\{i\}denotes the rewrite\-view observations of queryxix\_\{i\}\.

#### Decoding\-based utility\.

The decoding\-based utility evaluates whether the selected model remains reliable under stochastic generation\. The router again receives only the original query and selects one modelr​\(xi\)r\(x\_\{i\}\)\. We then evaluate this selected model on the decoding\-view observations, which contain three independent stochastic decoding runs on the original query\. The decoding\-based utilityUdecU\_\{\\mathrm\{dec\}\}is computed as

Udec​\(r\)=∑xi∈𝒟test∑o∈𝒪idec\(qi,r​\(xi\)\(o\)−λ​ci,r​\(xi\)\(o\)\)\|𝒪idec\|\|𝒟test\|,U\_\{\\mathrm\{dec\}\}\(r\)=\\frac\{\\sum\_\{x\_\{i\}\\in\\mathcal\{D\}\_\{\\mathrm\{test\}\}\}\\frac\{\\sum\_\{o\\in\\mathcal\{O\}^\{\\mathrm\{dec\}\}\_\{i\}\}\\left\(q\_\{i,r\(x\_\{i\}\)\}^\{\(o\)\}\-\\lambda c\_\{i,r\(x\_\{i\}\)\}^\{\(o\)\}\\right\)\}\{\|\\mathcal\{O\}^\{\\mathrm\{dec\}\}\_\{i\}\|\}\}\{\|\\mathcal\{D\}\_\{\\mathrm\{test\}\}\|\},where𝒪idec\\mathcal\{O\}^\{\\mathrm\{dec\}\}\_\{i\}denotes the decoding\-view observations of queryxix\_\{i\}\.

#### Cost normalization\.

Raw inference costs can have different scales across datasets, models, and generation lengths\. To make the cost term comparable with the quality score, we normalize costs before computing utilities\. Letci,m,orawc^\{\\mathrm\{raw\}\}\_\{i,m,o\}denote the raw inference cost recorded for an observation in datasetdd\. We use a dataset\-level normalization:

ci,m\(o\)=ci,m,orawZd,Zd=maxi,m,o⁡ci,m,oraw\.c\_\{i,m\}^\{\(o\)\}=\\frac\{c^\{\\mathrm\{raw\}\}\_\{i,m,o\}\}\{Z\_\{d\}\},\\quad Z\_\{d\}=\\max\_\{i,m,o\}c^\{\\mathrm\{raw\}\}\_\{i,m,o\}\.After normalization, the cost values are on a comparable scale across models within the same dataset\. All reported utilities use normalized costs\.

#### Choice ofλ\\lambda\.

The cost coefficientλ\\lambdacontrols the trade\-off between response quality and inference cost\. We useλ=0\.05\\lambda=0\.05for all main experiments\. This value gives a moderate cost penalty after normalization: it is large enough to discourage unnecessarily expensive model choices, while avoiding a degenerate setting where the router always prefers the cheapest model\. We keepλ\\lambdafixed across datasets and routers so that differences in performance reflect the routing method and supervision signal rather than per\-method tuning\.

#### Best fixed baseline\.

The best fixed baseline represents the strongest non\-routing strategy\. It selects one single model and uses it for all test queries\. For each dataset and evaluation view, we compute the average utility of every candidate model over all test queries, and report the best\-performing fixed model:

mfixed∗=arg⁡maxm∈ℳ⁡∑xi∈𝒟test∑o∈𝒪i\(qi,m\(o\)−λ​ci,m\(o\)\)\|𝒪i\|\|𝒟test\|\.m\_\{\\mathrm\{fixed\}\}^\{\*\}=\\arg\\max\_\{m\\in\\mathcal\{M\}\}\\frac\{\\sum\_\{x\_\{i\}\\in\\mathcal\{D\}\_\{\\mathrm\{test\}\}\}\\frac\{\\sum\_\{o\\in\\mathcal\{O\}\_\{i\}\}\\left\(q\_\{i,m\}^\{\(o\)\}\-\\lambda c\_\{i,m\}^\{\(o\)\}\\right\)\}\{\|\\mathcal\{O\}\_\{i\}\|\}\}\{\|\\mathcal\{D\}\_\{\\mathrm\{test\}\}\|\}\.
This baseline is useful for evaluating whether a learned router provides a better cost\-quality trade\-off than always using the same strong model\.

#### Use of test observations\.

Repeated test observations are used only for evaluation\. At inference time, the router receives the original query and predicts one model\. The rewrite\-view and decoding\-view observations are not provided to the router and are not used to revise its decision\. This ensures thatUrewU\_\{\\mathrm\{rew\}\}andUdecU\_\{\\mathrm\{dec\}\}evaluate the robustness of the selected model rather than giving the router additional test\-time information\.

### C\.5Detailed Explanation of the Diagnostic Analysis

This section provides additional details on the diagnostic analysis in Section 3\.3\. The goal of this analysis is to examine whether the standard single\-shot supervision protocol provides reliable routing labels\. We focus on three aspects: whether the observed outcome of a query\-model pair is stable, whether the best model selected for a query is stable, and whether single\-shot routing labels agree with labels estimated from repeated observations\.

For each queryxix\_\{i\}and candidate modelm∈ℳm\\in\\mathcal\{M\}, we collect a set of repeated observations through query rewriting and stochastic decoding\. Specifically, let

𝒪i,m=\{\(qi,m\(n,j\),ci,m\(n,j\)\)\}n=1,j=1N,M\\mathcal\{O\}\_\{i,m\}=\\\{\(q^\{\(n,j\)\}\_\{i,m\},c^\{\(n,j\)\}\_\{i,m\}\)\\\}\_\{n=1,j=1\}^\{N,M\}denote the observations of modelmmon queryxix\_\{i\}, wherennindexes rewritten query variants andjjindexes independent decoding runs\. Here,qi,m\(n,j\)q^\{\(n,j\)\}\_\{i,m\}is the task\-specific quality score andci,m\(n,j\)c^\{\(n,j\)\}\_\{i,m\}is the corresponding normalized inference cost\. We also define the per\-observation utility as

ui,m\(n,j\)=qi,m\(n,j\)−λ​ci,m\(n,j\)\.u^\{\(n,j\)\}\_\{i,m\}=q^\{\(n,j\)\}\_\{i,m\}\-\\lambda c^\{\(n,j\)\}\_\{i,m\}\.
#### Outcome instability\.

Outcome instability measures whether repeated observations of the same query\-model pair lead to different observed performance scores\. For each query\-model pair, we check whether the quality scores vary across repeated observations:

Instabout​\(i,m\)=𝕀​\[maxn,j⁡qi,m\(n,j\)−minn,j⁡qi,m\(n,j\)\>ϵ\],\\mathrm\{Instab\}\_\{\\mathrm\{out\}\}\(i,m\)=\\mathbb\{I\}\\left\[\\max\_\{n,j\}q^\{\(n,j\)\}\_\{i,m\}\-\\min\_\{n,j\}q^\{\(n,j\)\}\_\{i,m\}\>\\epsilon\\right\],whereϵ\\epsilonis a small tolerance used to avoid numerical artifacts\. The overall outcome instability is computed by averaging this indicator over all queries and candidate models:

OutcomeInstability=1\|𝒟\|​\|ℳ\|​∑i=1\|𝒟\|∑m∈ℳInstabout​\(i,m\)\.\\mathrm\{OutcomeInstability\}=\\frac\{1\}\{\|\\mathcal\{D\}\|\|\\mathcal\{M\}\|\}\\sum\_\{i=1\}^\{\|\\mathcal\{D\}\|\}\\sum\_\{m\\in\\mathcal\{M\}\}\\mathrm\{Instab\}\_\{\\mathrm\{out\}\}\(i,m\)\.A higher value means that the same model can receive different scores on the same underlying query under natural input or output variations\.

#### Winner flip rate\.

Outcome instability at the query\-model level does not necessarily imply that the routing label changes\. Therefore, we further measure whether the model selected as the best candidate for a query changes across repeated single\-shot observations\. For each observation index\(n,j\)\(n,j\), we construct a single\-shot routing label by selecting the model with the largest observed utility:

wi\(n,j\)=arg⁡maxm∈ℳ⁡ui,m\(n,j\)\.w\_\{i\}^\{\(n,j\)\}=\\arg\\max\_\{m\\in\\mathcal\{M\}\}u^\{\(n,j\)\}\_\{i,m\}\.The winner flip indicator for queryxix\_\{i\}is then defined as

Flip​\(i\)=𝕀​\[\|\{wi\(n,j\):1≤n≤N,1≤j≤M\}\|\>1\]\.\\mathrm\{Flip\}\(i\)=\\mathbb\{I\}\\left\[\\left\|\\\{w\_\{i\}^\{\(n,j\)\}:1\\leq n\\leq N,1\\leq j\\leq M\\\}\\right\|\>1\\right\]\.The winner flip rate is the average of this indicator over all queries:

WinnerFlipRate=1\|𝒟\|​∑i=1\|𝒟\|Flip​\(i\)\.\\mathrm\{WinnerFlipRate\}=\\frac\{1\}\{\|\\mathcal\{D\}\|\}\\sum\_\{i=1\}^\{\|\\mathcal\{D\}\|\}\\mathrm\{Flip\}\(i\)\.This metric captures whether different single\-shot samples can lead to different best\-model labels for the same query\.

#### Single\-shot versus distribution\-aware disagreement\.

We also compare the single\-shot routing label with a distribution\-aware label estimated from all repeated observations\. For each query\-model pair, we first compute the expected quality and expected cost:

μq​\(i,m\)=∑n=1N∑j=1Mqi,m\(n,j\)N​M,\\mu\_\{q\}\(i,m\)=\\frac\{\\sum\_\{n=1\}^\{N\}\\sum\_\{j=1\}^\{M\}q^\{\(n,j\)\}\_\{i,m\}\}\{NM\},μc​\(i,m\)=∑n=1N∑j=1Mci,m\(n,j\)N​M\.\\mu\_\{c\}\(i,m\)=\\frac\{\\sum\_\{n=1\}^\{N\}\\sum\_\{j=1\}^\{M\}c^\{\(n,j\)\}\_\{i,m\}\}\{NM\}\.The distribution\-aware utility used in the diagnostic analysis is

u¯i,m=μq​\(i,m\)−λ​μc​\(i,m\),\\bar\{u\}\_\{i,m\}=\\mu\_\{q\}\(i,m\)\-\\lambda\\mu\_\{c\}\(i,m\),and the corresponding distribution\-aware routing label is

wiDA=arg⁡maxm∈ℳ⁡u¯i,m\.w\_\{i\}^\{\\mathrm\{DA\}\}=\\arg\\max\_\{m\\in\\mathcal\{M\}\}\\bar\{u\}\_\{i,m\}\.For each single\-shot observation\(n,j\)\(n,j\), we compute whether its selected model disagrees with the distribution\-aware label:

Disagree\(n,j\)​\(i\)=𝕀​\[wi\(n,j\)≠wiDA\]\.\\mathrm\{Disagree\}^\{\(n,j\)\}\(i\)=\\mathbb\{I\}\\left\[w\_\{i\}^\{\(n,j\)\}\\neq w\_\{i\}^\{\\mathrm\{DA\}\}\\right\]\.The final disagreement rate is averaged over all queries and single\-shot observations:

∑i=1\|𝒟\|∑n=1N∑j=1MDisagree\(n,j\)\(i\)\.\|𝒟\|​N​M\\frac\{\\sum\_\{i=1\}^\{\|\\mathcal\{D\}\|\}\\sum\_\{n=1\}^\{N\}\\sum\_\{j=1\}^\{M\}\\mathrm\{Disagree\}^\{\(n,j\)\}\(i\)\.\}\{\|\\mathcal\{D\}\|NM\}This metric measures how often a routing label constructed from one sampled observation differs from the label obtained by aggregating repeated observations\.

#### Router variance under single\-shot supervision\.

Finally, we examine whether the instability of single\-shot labels propagates to the learned router\. We construct multiple single\-shot training sets by randomly sampling one observation for each query\-model pair\. Each sampled training set is used to train the same router architecture under the same data split and model pool\. Since the only changing factor is the sampled supervision signal, the variation in test utility reflects the sensitivity of learned routing policies to single\-shot supervision noise\.

## Appendix DValidity Checks

DARS relies on two preprocessing components: query rewriting for constructing input\-side variations, and automatic scoring for evaluating repeated model outputs\. We therefore conduct additional validity checks to examine whether these components introduce obvious artifacts\. These checks are intended as lightweight sanity checks rather than formal guarantees of semantic equivalence or scoring correctness\.

### D\.1Rewrite Validity Check

The purpose of query rewriting is to introduce controlled input\-side variations while preserving the underlying task semantics\. To assess the quality of the generated rewrites, we perform an automatic validity check over all rewritten queries used in our experiments\. The check covers five aspects: \(1\) whether all required rewrites are successfully generated, \(2\) whether rewritten queries duplicate the original query or duplicate each other, \(3\) whether dataset\-specific structural constraints are preserved, \(4\) whether the rewritten query remains close to the original query in sentence\-embedding space, and \(5\) whether numeric tokens are preserved\.

For structural constraints, GPQA rewrites are required to keep the answer choices unchanged; MATH\-500 rewrites are required to preserve the mathematical problem structure; and DROP\-800 rewrites are required to keep the passage unchanged and rewrite only the question\. Sentence similarity is computed using the same sentence\-transformer backend as in our query representation pipeline\. We additionally flag rewrites with low semantic similarity or numeric\-token mismatch for manual inspection\. These flags are conservative: a flagged rewrite is not necessarily invalid, since paraphrasing can legitimately change surface forms, symbols, or numeric formatting\. The results are summarized in Table[5](https://arxiv.org/html/2606.06924#A4.T5)\.

Table 5:Automatic validity check for query rewrites\. Complete denotes the fraction of original queries for which all rewrites are generated\. Original dup\. and Rewrite dup\. denote duplicate rates with the original query and among rewrites, respectively\. Structural pass checks dataset\-specific constraints\. Review flag marks rewrites selected for further inspection due to low similarity or numeric mismatch\.The results show that all datasets achieve full rewrite completion, no duplicate rewrites are detected, and all rewrites pass the dataset\-specific structural checks\. Sentence similarity is also high on average across datasets\. DROP\-800 has the lowest review\-flag rate, while GPQA has a higher numeric\-mismatch flag rate\. This is expected because scientific questions often contain symbols, quantities, abbreviations, and domain\-specific expressions whose surface forms may change under paraphrasing\. We therefore treat the review flag as a conservative diagnostic signal rather than direct evidence of invalid rewrites\.

### D\.2Scoring Validity Check

We also verify the automatic scoring pipeline used to evaluate generated responses\. For each stored generation, we recompute the score using the unified dataset\-specific scoring script and check whether the output can be parsed, whether the score is finite, whether the score lies in the valid range\[0,1\]\[0,1\], whether the expected score type is produced, whether rescoring agrees with the stored score, and whether duplicate generation keys exist\. The results are shown in Table[6](https://arxiv.org/html/2606.06924#A4.T6)\.

Table 6:Validity check for the automatic scoring pipeline\. Parse success measures whether a generated response can be parsed into the expected answer format\. Finite and In\[0,1\]\[0,1\]check numerical validity\. Score type checks whether the dataset\-specific scorer returns the expected score type\. Rescore agreement measures agreement between stored scores and scores recomputed by the unified scorer\.The scoring pipeline achieves near\-perfect parse coverage across all datasets, and all parsed scores are finite and lie within the valid range\. We also find no duplicate generation keys\. The small number of rescoring differences is caused by scorer\-version inconsistencies in intermediate stored files\. To avoid propagating such inconsistencies, all reported experimental results are computed after applying the same unified scoring pipeline to the stored generations\.