Meta-Router: Bridging Gold-standard and Preference-based Evaluations in LLM Routing
Authors: Yichi Zhang
Fangzheng Xie
Shu Yang
Chong Wu
Venue: ICLR 2026 Poster (2026)
Keywords: Causal learning
Meta-learner
Large Language Model
query routing
Primary area: foundation or frontier models, including LLMs

Abstract: In language tasks requiring extensive  human-model interaction, the inference cost of large language models (LLMs) can be substantial. To reduce expenses while preserving the quality of the responses, an LLM router selects among candidate models to balance between the expected response quality and the inference cost. A central challenge in router training is the accuracy and accessibility of reliable supervision. Gold-standard data, obtained from domain experts or benchmark labels, provide accurate quality evaluations of LLM responses but are costly and difficult to scale. In contrast, preference-based data, collected via crowdsourcing or LLM-as-a-judge systems, are cheaper and more scalable, yet often biased in reflecting the true quality of responses. We cast the problem of LLM router training with combined Gold-standard and preference-based data into a causal inference framework by viewing the response evaluation mechanism as the treatment assignment. This perspective further reveals that the bias in preference-based data corresponds to the well-known causal estimand: the conditional average treatment effect (CATE). Based on this new perspective, we develop an integrative causal router training framework that corrects preference-data bias, addresses imbalances between two data sources, and improves routing robustness and efficiency.  Numerical experiments demonstrate that our approach delivers more accurate routing and improves the trade-off between cost and quality. Illustrative code to reproduce our main experiment is available at https://github.com/yichistat/Meta-router.

Published as a conference paper at ICLR 2026
META-ROUTER: BRIDGING GOLD-STANDARD AND PREFERENCE-BASED EVALUATIONS IN LLM ROUTING
Yichi Zhang Fangzheng Xie Shu Yang²
Chong 3u..4.
1 Department of Statistics, Indiana University Bloomington
2 Department of Statistics, North Carolina State University
3 Department of Biostatistics, The University of Texas MD Anderson Cancer Center
4 Institute for Data Science in Oncology. The University of Texas MD Anderson Cancer Center
ABSTRACT
In language tasks that require extensive human-model interaction, deploying a single "best" model for every query can be expensive. To reduce inference cost while preserving the quality of the responses, a large language model (LLM) router selects the most appropriate model from a pool of candidates for each query. A central challenge to training high-quality router is the scarcity of reliable supervision. Gold-standard data (e.g., expert-verified labels or rubric- based scores) provide accurate quality evaluations of LLM responses but are costly and difficult to scale. In contrast, preference-based data, collected via crowdsoureing LLLe-e--a-judge systems, cheaper and more scalable, yet often biased in reflecting the true quality of responses. We cast the problem of LLM router training with combined gold-standard and preference-based data into a causal inference framework by viewing the response evaluation mech- anism the treatment assignment. This perspective further reveals that the bias in preference- based data corresponds to the well-known causal estimand: the conditional average treatment effect (CATE). Based on this new perspec- tive, we develop an integrative causal router training framework that corrects preference-data bias, address imbalances between two data sources, and improve routing robustness and efficiency. Numerical experiments demonstrate that our approach delivers more accurate routing and improves the trade-off between cost and quality. Illustrative code to reproduce our main experiment is available at attps://githiei com
INTRODUCTION
As LLM deployments scale and model size grow, serving every request with the strongest model becomes economically and operationally impractical for commercial success AI applications. LLM routing (Ding et al., 2024; Hu et al., 2024; Ong et al.. 2024) addresses this issue by constructing a decision framework that assigns each incoming query either to larger, more powerful models or to cheaper but potentially weaker ones, thereby balancing cost and performance trade-offs. Traditional cascading routers sequentially process a query through a series of L.Ms,, from light to heavy, until a satisfactory response obtained (Chen al., 2024), but this approach often inefficient and introduces latency from repeated calls. Predictive routers (Ong et al., 2024; Stripelis et al., 2024; Somerstep et al., 2025; Tsiourvas et al., 2025) instead predict the appropriate model in one shot, often by learning a mapping from query feature (such as text embeddings) to target model under cost-quality objective using statistical and machine learning (ML) methods. Another important line of work uses confidence- or eeward-model-basee routing (Chuang et al., 2025; Frick et al., 2025; Wu Lu, 2025), which selects models based on uncertainty estimates or learned reward scores associated with each candidate response.
The effectiveness of predictive routers critically depends on the evaluation metrics available in the training data. Existing works differ in the evaluation mechanisms used. For example, Ong et al. (2024) use the LMArena dataset (Chiang et al., 2024), where model preference are judged by internet
"Corresonndenc:: Yichi Zhang (yiczhan @ @uu.edu) & Chong Wu (ewu 18 (ddanddrsonoorrg

---

Published as a conference paper at ICLR 2026
users, and further combine it with standardized benchmarks such as MMLU (Hendrycks et al., 2020) or with LLM-judge-labblee datasets. In contrast, Tsiourvas et al. (2025); Stripelis et al. (2024) employ accuracy- based benchmarks where queries admit objectively verifiable solutions.
In this work, we consider the LLM routing problem in challenging yet realistic scenarios, where hu- mans and LLMs have complex interactions within expert knowledge domains, such professional healthcare conversations, Al-assisted programming, and exploratory scientific research. In these scenarios, the queries are often open-ended, so accurate evaluation often require domain expertise, multi-criterion rubrics, and careful inspection, making gold labels both costly and labor-intensive to acquire (Chang et al.. 2024). This partially explains why the sample size of benchmark datasets of different professional domains with carefully designed evaluation is small. For example, the sample size of HealthBench (Arora et al., 2025) designed for healthcare dialogue 5000. These challenges hinder the efficient training of routers with sufficient and high-quality samples. Although crowdsoureing or augeees systems may offer scalable alternatives, such evaluations can be systematically biased relative to expert judgments or task-specific rubrics and may not reliably reflect the true quality of responses (Zheng et al., 2023a; Tam et al., 2024).
These limitations highlight the need for principled method that can integrate scarce but accurate gold-standard data with scalable yet potentially biased preference-basee based data efficiently, for debiased LLM router training. We address this challenge from novel angle by casting it into causal infer- ence framework, where the response evaluation mechanism is viewed as the treatment assignment. This perspective links router training and debiasing to the extensive literature on semiparametric causal estimation (Imbens & Rubin, 2015; Cheroozuuov et al., 2018), and further shows that the bias in preference-based data corresponds to the conditional average treatment effect (CATE), which can be efficiently estimated via causal meta-learners (Künzel et al., 2019). Building on this insight, we propose meta-router training framework that corrects preference-dal data bias through R- and DR-learners for CATE estimation (Nie & Wager, 2021; Kennedy, 2023), thereby mitigating sam- ple imbalances across heterogeneous data sources and enabling robust, efficient routing decisions, particularly in human-Al interaction scenarios within high-expertise fields.
2
LLM ROUTING WITH GOLD-STANDARD AND PREFERENCE-BASER DATA
LLM responding process towards a human query can be mathematically represented as a (random) function M : Q - A mapping any query q Q an answer M(4) A. Here, Q and A are the text spaces of queries and answers, respectively. For simplicity, in this work, we consider pairwise LLM routing between two models, namely M, and MG, where M, denotes a premium model with generally higher response quality (e.g., GPT-5 (OpenAI, 2025)), and M,, represents its cost-effective alternative with lower inference cost but potentially lower response quality for certain queries (e.g., GPT-4o mini (OpenAI, 2024)). Given query g. the router learns policy =(q) E {Mp, Ma that maximizes expected utility function involving inference cost and response quality.
2.1 GOLD-STANDARI AND PREFERENCE BASSED DATA
We refer to gold-standard data (GS data) as the high-quality dataset, where response quality is assessed either by domain experts or by "gold labels" (Hendrycks et al., 2020; Arora et al., 2025). Hence, it is generally considered the authoritative ground truth for LLM response evaluation. We consider the GS data in the form of
DG = (()
where 9: denotes the ith query and r3 represents the evaluated quality gain between Mp(g:) and Ma (g:) under the gold standard. Without loss of generality, we assume that >> 0 indicates Mp (q:) outperforms Ma)g;, rs < 0 indicates the opposite, and value near 0 suggests comparable quality. For example, when correctness is objectively defined (e.g., the MMLU dataset), we define Ti 1 if M,(g:) answers correctly and Ma(g:) does not, r; = if both are correct or both are incorrect, and r; -1 when only Ma(g:) is correct. As another example, when r; is evaluated by domain experts, the expert typically rates Mp(a)) and Ma(g:) respectively, based on predefined scoring rubrics, and r, is defined the difference between these ratings.
We consider the standard probabilistic modeling for the generation of Da. In particular, we assume (q, rr).... (gn, Tn) are independent and identically distributed (iid) generated with qi 2 for
2

---

Published as a conference paper at ICLR 2026
some query distribution 2, and
rb(g:) + 6;:
(1)
where the random errors (s)) satisfy E(e; q:) = 0, and m : Q R is the average quality gain of some GS model.
Despite their high accuracy, GS data are typically labor-intensive to obtain and difficult to scale. For open-ended queries, response evaluation often requires expert judgment carefully designed scoring rubrics, particularly in domain-specific professional contexts. Conversely, if only queries with clear standard answers (e.g., the MMLU dataset) are retained, the empirical distribution of (g)) may fail to adequately represent the queries encountered in daily practice.
On the other hand, the preference based evaluation offers more scalable yet typically more subjec- tive alternative for assessing LLM responses. For instance, LMArena (Chiang et al., 2024) evaluates the L.LM responses based on Internet users' preferences, while the L.L--as-a-judee system employs an LLM to directly compare and grade L.M responses (see, e.g., $3.1 in Zheng et al. (2023a)).
Specifically, we denote the preference-basee based data (PB data) by Dp = (4/ y:) where of ~ 2' denotes the ith query from distribution 2', and y/ represents the outcome of comparing the responses from Mp(g)) and Ma(g) through preference-based mechanism. Similar to Da, we assume the samples in Dp are iid and
y: = -((()))))))))))))))))) + c
(2)
where the random errors (4) satisfy E(Ç, | g)) = O, and 17 : :Q ? H R is the average quality gain under a preference-basee evaluation mechanism. Preference-based evaluation mechanisms usually simple and intuitive. For instance, the pairwise comparison in an LLM-as-a-jjdde system or LMArena, returns y: 1 if Mp (q)) is preferred over (1))), 3/ 1 if the opposite holds, and 0 in the case of tic. There are multiple approaches to model the preference data generation and 17(4), e.g., the Pradley-Teryy-aie (Devlin et al., 2019); see 84.2 in Ong et al. (2024).
(BTL) model (Bradley & Terry, 1952) and BERT classifier
Remark Our empirical study suggests that rescaling {r:)mm by a normalization constant c. >> O (e-Ta)- so that the rescaled values are on the same scale as {ye 1n can substantially improve the performance of our proposed router. Some normalization constant could be considered include: (1) e normalizing the magnitude: max rlakvew max (elm)) (2) normalizing the em- pirical variance: Var(e r4) Var(y;); (3) e (approximately) minimizing the distribution distance (e.g.. 2- 2-Wasserstein distance) between the empirical distributions of (c rx)ie(m) and {ys he
2.2 COST FUNCTION
For any LLM M, we define its cost function as CM : Q R50 that quantifies the cost of generating the answer for any input query E Q using LLM model M. Following others (Ong et al., 2024; Ding et al., 2024), in this paper, we assume the cost functions of both models are known a priori, and consider the following normalized cost functions:
CM,, (1),4)=1, 1,
CM,(g)= =
(3)
for any q E Q. Such cost functions treat the call of .M as one unit more expensive than the call of M, for any query. We focus on this normalized cost mainly for the ease of exposition.
Our proposed method can be easily applied more complicated and realistic cost func- tions. Many LLM providers (e.g.. Claude, DeepSeek, Gemini and GPT) adopt a token-based pricing model for developers and enterprises. where the cost of a query is the sum of input tokens times the input rate and output tokens times the output rate (Chen et al., 2023). Formally, for LLM M, Cm(g)= .m.M (T)(q) Cout.M (M)) Cix.M; where TM(()) and (M)) imput and output token counts, Cou..M are known per-token rates, and CRx.M is a fixed cost. Input tokens can be obtained via the tokenizer!, while output tokens can be estimated using gen- eration limits (OpenAl, 2024) or predictive methods (Zheng et al., 2023b). Latency may also be incorporated as an additional cost component.
Remark 2
the
e.g. https::/platteee foratooorooooom www./ooo//ooome zer
3

---

Published as a conference paper at ICLR 2026
2.3 THE ROUTING DECISION RULE
The decision rule of an LLM router is designed to compare the quality gain of choosing Mp over M. with the corresponding answer generation cost in $2.2. To quantitatively measure the quality gain of routing a new query ạ, previous works mainly leverage the average quality gain of different preference data 779) (Ong et al., 2024; Zhang al., 2025). However, as we focus on fields requiring professional knowledge, e.g., healtheare, science, and computer programming. the GS model (g)) arguably more reliable measure of quality gain. Specifically, the proposed utility contrasts the expected quality gain based on the GS with the cost function and strives to balance between the response quality with the cost as follows:
D(g w, m.)
g)) -w (Cx, =b)9)
CM. CM,,
- (C,, CM, (q)) (4)
GS quality gain
cost loss
Here, w 2 0 is a user-specified conversion factor to control the trade-off between the quality gain and the additional cost if the expensive model Mp is preferred over Ma. When ()()) is known and cost function is binary as in (3), the Bayes optimal classifier selects M, over M, in response to the query if and only if the quality gain surpasses the required additional cost based on the decision rule, namely, (44) > w, and selects Ma over MM otherwise.
3
INTEGRATIVE LLM ROUTING THROUGH CAUSAL META-LEARNERS
3.1 ORACLE INTEGRATIVE ROUTER WITH KNOWN SHIFT FUNCTION
To efficiently evaluate the average quality gain function (-) of the GS model, we aim to combine the information from both Dp and Dc. However, due to the uncertainty of human and LLM judge's 's preference ratings, there may exist a potential discrepancy (bias) between the golden-labeled quality gain ())) for DG and the preference-choie model v)()) for Dr (Zheng et al., 2023a; Wataoka et al., 2024; Zhu et al., 2023; Szymanski et al., 2025). This bias can be quantitatively modeled as an unknown shift function for any query
(g)) = (4() - 7(9)-
Consequently, regression approach using the directly combined data De U DD (Ong al., 2024) can suffer from non-negligible estimation bias for (b))) even the sample sizes of both PB data and the GS data are sufficient.
In this section, we estimate (-)) under an oracle scenario that the shift function A(-) is known (a the- oretical scenario for illustration purpose) and leave scenario of unknown (--) to section 3.3, where our new method developed. Under such an ideal condition, one can estimate n7-) by integrating the information in Dp and DG using bias correction process that takes the information of A(-) into account. Specifically, consider the following bias-corrected human preference data:
T(Dp A) = (( r y: + (())))
where r can be roughly interpreted the pseudo-CG -S quality difference as if the human- -refennpreeernce queries are prompted. Then, our newly enriched dataset after bias correction can be described as
D De U T(Dp A) {(g,, r.) U {(g,, r/)
Note that all samples in D are conditionally unbiased for (4))), namely, for any E [r]] and E [m].
v (g:)
E(r: (g), v((o)) E(r)) of)).
Over D+, one can apply any MI. algorithm to estimate (-()) through direct nonparametrie regres- sion. More specifically, v-) solves the following population least-square problem:
(+)) = arg min
Ep+
Σ (r- (r))())) (g+
(5)
h=Q=+R n +
where the expectation is taken with respect to the distribution of D Here h(-) is an arbitrary prediction function mapping a query to scalar estimation of the GS quality gain, and minimizing
4

---

Published as a conference paper at ICLR 2026
(5) over all such h()) identifies the true average GS quality gain ()(.). Then, oracle estimator is obtained by solving the (regularized) empirical counterpart (5):
(- A)
arg min LEHA
(r. h(q))) + Σ y1 +( h(()))
+ A(h),
(6)
m
where Ha is the estimator class specified by the ML algorithm, e.g., Gaussian process regression (Rasmussen & Williams, 2006), deep neural networks (Goodfellow et al., 2016), and random forests (Breiman, 2001a), and A(-) is an optional user-specified regularizer on the complexity of h, e.g. the lg (ridge) regularizer (Tikhonov Arsenin, 1977) and the & (Lasso) regularizer
BriiieeerTiibbaa (Tibshirani, 1996).
By appropriately choosing the ML algorithm (and hereby Hm in (6)), vi( A) serves as a statis- tically principal estimator for 16(-) using all samples in Dz U Dp. For example, if (0()) satisfies certain smoothness condition, then several nonparametric regression estimators can achieve statisti- cal optimality; see e.g., Wasserman (2006); Moutrada al. (2020); Schmidt-Hieeer (2020).
3.2 GS-PB DATA INTEGRATION: CAUSAL INFERENCE PERSPECTIVE
In practice, the shift function A(-) is unknown. Nevertheless, the oracle procedure outlined in $3.1 indicates that, empirically, it crucial to develop a principal statistical estimation framework for the shift function A(-) in order to estimate wh(-) efficiently by combining the information from Da and Dp. the following two sections, we reformulate the data integration problem under the potential outcome framework in causal inference (see e.g., Imbens & Rubin (2015)), and correspondingly, A(-) is the conditional average treatment effect (CATE) under such a new model formulation. One then well-developed CATE estimation approaches in causal inference, e.g., meta-learners (Künzel et al., 2019), to estimate A(-) robustly and efficiently.
To streamline the presentation, unified triple (st, 4, o.) for sample i, where 8; denotes the query of sample i, 4 E (0, 1) is the source indicator (1, if the label is obtained from the gold-standard (GS) mechanism and 4; is obtained from the preference-based (PB) mechanism), evaluated quality gain between Mp (p(6 (s:) and Ma.s:) under the corresponding mechanism. With this notation, the pooled dataset Do U DD can be written as
pool the GS and PB datasets into single collection and use
is the observed outcome, i.e., the
D = {(E,, 44, o0)
where each sample comes from either De or Dp depending on t,. Specifically, when (GS sample), we have o, as in model I; when f4 = 0 (PB sample), we have as in model 2.
Rather than modeling Da and Dp separately, we can alternatively characterize the distribution of the combined dataset D = De U Dp using hierarchical mixture model (Pooled DGP).
Pooled Data Generation Process (Pooled DGP)
For each (s,,t;, 0z) D:
1. Generate t; with Pr(t, = are observed in the joint dataset.
E [0, 1]; here controls how often GS samples
2. Generate 8; with 8; 1, = = 2 and 8; 14 = ~ 2', where 2 and D' are the query distributions of the GS and PB data, respectively;
3. Generate o, T; under model (1) with = if t. 1, and o yi under model
(2) with d. 8; if
Such a joint data generation process naturally leads to the causal potential outcome framework (Rubin, 2005). Specifically,, can view each query a unit, 8, as its covariates, and consider 4 (0, 1} as the binary treatment assignment to indicate whether the evaluation between Mp(s;) and (s.) is carried out by gold standards (t: 1) or is PB (t, 0). For each query s;, the two potential evaluation outcomes follow:
af2h (b(8s) 6:;
10) (788)) 1/b
(8)
5

---

Published as a conference paper at ICLR 2026
(1)
where 0f represents the counterfactual quality assessment of the quality gain shift from Ma(ss) to (s(s;) ) if the evaluation is justified by the gold standards, while (0) represents the quality gain with the same query, but the evaluation is judged through a preference- mechanism. Then, samples in D can be equivalently considered as generated from the following standard causal mechanism.
Lemma Define fa and f as density functions of 2 and 2', respectively. Then the Pooled DGP equivalent to the Causal DGP as follows.
Causal Data Generation Process (Causal DGP)
For each (s,tt,,,) E D:
1. Generate s; ~ R.2 + (1 (1-
x) which the mixture distribution of 2 and 2 with the mixture proportion K;
2. Generate 44 following the propensity score model Pr(ls = 8,) p(
<f2(s))(nf((ss) (s)))
3. Generate following the standard potential outcome model: 100 + (1
,) where o and o are given by (8).
The proof of Lemma 1 in Appendix A.5. Lemma 1 clarifies that the target function A(-) is CATE from the perspective of causal data generation:
(s) (s)) n)s)(() = B(o -0 (0))
The causal identification assumptions such consistency and unconfoundedness could be naturally satisfied under the Causal DGP. In particular, under the data collection procedure considered in this paper (c.f., 1). the no unmeasured confounders is satisfied, whenever there is unobserved random variable, other than the query s, jointly affecting both the treatment assignment mechanism and the outcome. On the other hand, the positivity assumption on the propensity score, i.e., p(s) (e, e)) for some constant >> O, may be violated when the supports of 2 and 2' do not coincide. In particular, violation occurs if there exists a region of q such that fs( > while flr (s) O, or vice versa. In such cases, our proposed method remains valid after data truncation step: we estimate (--) only within the samples in the overlapped region of supports. We defer detailed discussion of this truncation- based extension to future work in 55.
3.3 CAUSAL META-LEARNING FOR A(g) AND
Building on the seminal work of Künzel et al. (2019), many causal meta-learning approaches are de- veloped, aiming to provide principled and flexible frameworks for CATE estimation. Meta-learners can incorporate any off-the-shelf ML algorithm, thereby offering substantial flexibility. Moreover, by leveraging ideas from orthogonal ML and semiparametric statistics (see, e.g., Chernozhukov et al, 2018), 1)), meta-learner such as the R-learner (Nie & Wager, 2021) and the DR learnererr (Kennedy, 2023) enjoy the oracle property. In particular, under mild conditions of nuisance function estimation, CATE meta-learners can be asymptotically equivalent to oracle estimator that has access to the full set of individual treatment effects {o.") -0,0'''
(1) -000) J whereas practice only one of o or (())
is observed for each i. This oracle property implies that R-learner and DR- could achieve the
statistical optimality for the estimation of (-) in our setting (Wu & Yang, 2022; Curth & Van der Schaar, 2021). In this paper, we focus on R- and DR-learners
The implementation details of R- and DR-learners are deferred to $A.3 in the Appendix. Both learners offer robustness against nuisance model misspecification and fit naturally into our estima- tion purpose of (-). In this work, we consider both approaches as benchmark estimators for the shift function A(.), and employ nonparametric ML regressors (e.g., random forests, deep neural networks, and XGBoost) to capture heterogeneous structures of ((-) across the query space.
The samplc-splitting could be further employed into R- and De-learners as discussed in (Nie & Wa- ger, 2021; Kennedy, 2023) to avoid potential biases brought by nuisance function training through MI. algorithms. We omit the details only for simplicity, and note that the sample splitting could be

---

Published as a conference paper at ICLR 2026
straightforwardly incorporated into our method. We refer interested readers to the aforementioned two papers and, e.g., Chemozhukov et al. (2018) for further discussions.
Building the construction of the oracle router in (6), now replace the known shift function (.) with its
based estimator (+)), and thereby formalize our two-step meta-router.
Meta-router
Inputs: D DG Pp; HA. Hm. A(.) specified by selected ML algorithms.
1. Estimate the shift function Á(-) via certain CATE learning approaches, e-g-, the R-learner or DR-learner in (S3) or (S4) with nuisance functions trained over 'D.
2. Meta-router X Á) is obtained by solving (6) wherein (-) is replaced by A())
Although using DR-learner and R-learner as examples, our meta-router is generally framework does not tie on any specific CATE estimation approach. Our meta-router framework could be natu- rally extended to the multiple-LLA scenario, we defer more discussions to $A.2 in the Appendix.
4 NUMERICAL EXPERIMENTS
4.1 HEALTHIBNNCII
HealthBench (Arora et al., 2025) is recently released benchmark designed to evaluate the per- formances of LLMs in open-ended healthcare scenarios. consists 5000 professional user-model dialogues that were selected to span a wide range of healthcare scenarios. In total, 262 physicians across 26 specialties and 60 countries contributed to the creation evaluation rubrics and consensus standards, make the evaluation mechanism precise in reflecting the qualities of LLM responses. The meta-evaluation verifies the trustworthy of these rubrics in faithfully reflecting physician judgment.
In numerical experiments, we Gemini 2.5 Pro the primary model Mp (Comanici al., 2025) and Gemma 12B as the alternative model M,, (Team et al., 2025), and collect their responses to all HealthBench questions. We then employ GPT-5-mini (OpenAI, 2025) for evaluation. For gold-standard evaluations, each score-collecting prompt includes the evaluation rubrics, the original question, and the model response, and GPT-5-mini asked to assign a score strictly following the official rubrics. Notably, generating GS evaluation through LLM could be a limitation of our study, and direct expert validation shall be ideal. The HealthBench study (OpenAI, 2025) reports that GPT-4.1 with rubric achieves marco Fl score of 0.709 against physician annotations on consensus criteria and be able match expert grading (Table in OpenAl (2025)). Thus, we believe that using GPT5-mini with rubric should perform similarly to expert grading for our study. The score difference between Mp and M, for each question is treated as the GS quality differences of two models (see Appendix A.7 for our prompts). For preference- based evaluation, each prompt contains only the question and the two responses, and mini,, asked to act as a medical expert, indicates whether Mp is better (1), comparable (0), worse (-1), and this returedd value is treated as the PB quality gain (see Appendix A.7 for our prompts). We normalize two types of quality gain evaluations align their empirical variance (c.f., Remark (2)). We embedded each query text model. We report in Figure S2 in to a 768-dimensional vector using the gemini-embedding--U0 Appendix, the histogram of the PB-GS quality differences (ri- - 15000 The sample mean of these differences is substantially below zero, as confirmed by a two-sided t-test yielding a p-value smaller than 2.2 x 16, which motivates the training of debiased meta-router.
Experiment Setting For each Monte Carlo (MC) round, we specify the estimator class H. by machine learning algorithm, GS sample size n, and a dimension d such that we further reduce the dimension of query text embedding to d via PCA; for simplicity, we use the same estimator class H. for all nuisance function, CATE function and router training. We then randomly split the data into three parts: testing set Drext of with 500 queries and the corresponding GS evaluation outcomes ri, a GS training set of size n, and a PB training set containing the remaining samples. Each training set only includes its corresponding type of evaluation outcomes. We compare seven types of routers: (1) an oracle benchmark router that has access to the GS evaluation outcomes for all training queries in both GS and PB sets, and trains ()() over H using all these outcomes;
7

---

Published as a conference paper at ICLR 2026
(8) 71 100
(b) 11 500
(c) (2) 1000
Figure 1: The efficiency gains of different routing strategies compared to the random routing base- line, against the primary model usage ratio in the main numerical experiments. Subfigures corre- spond to varying GS sample sizes. Colors indicate different methods: oracle benchmark, meta-router via DR-learner, meta-router via R-learner, predictive router using pooled data, predictive router us- ing GS data only, and predictive router using PS data only.
(a) n 100
(b) n 500
(c) 1000
Figure 2: The efficiency gains of different routing strategies compared with the random routing baseline versus the primary model usage ratio. All regressions are implemented via XGBoost. Other settings are the same as Figure 1.
(2) a predictive router that estimates v5(g) over H on the pooled GS and PB training data, without distinguishing evaluation types; (3) predictive router that estimates (b()) over H using only the
PB training data; (4) predictive router that estimates (())) over H. using only the GS training data; (5) a meta-router based on the R-Iearner trained on GS and PB data, with all involved predictions run by H; (6) meta-router based on the DR-leamer trained on GS and PB data, with all involved predictions run by H: (7) random router that assigns each query to Mp with a fixed assignment probability. The routers based on pooled GS and PB training data, and solely based on PB training data, follow the same framework as Ong et al. (2024), serving as our state-of-art baseline.
Main Experiments We specify H by the learning algorithm of random forest (Breiman, 2001b), set PCA dimension to 50, and test three GS sample sizes n E {100, 500, 1000). For each configuration and each Monte Carlo (MC) round, each router's decision rule follows (4), with v'(q) replaced by the corresponding estimator and binary cost functions as in (3). Given any weight in (4), we compute the total efficiency (TE) of each router TE De Iig: assigned to the primary model} x ri, where F; denotes the realized quality gain. By varying w. or equivalently the assignment probability for the random router, we obtain TE values under different primary model usage ratios (PMUR), defined the proportion of queries assigned to the primary model among all testing samples. We run 200 MC rounds for each configu- ration and report the mean TE across rounds for each router and PMUR level. To quantify relative performance, we further calculate the efficiency gain (EG) of a router as its improvement over the random router, averaging over 500 test samples:
EG of any router Mean TE of any router Mean TE of the random router
500
The EGs of different routers versus PMURs under different sample sizes, reported in Fig- ures 1. The illustrative code to reproduce Figures 1(a) is available at https: /gggttuuu com/ vichi tichisttt//iie -ouue..
8

---

Published as a conference paper at ICLR 2026
(a) 71 100
(b) 1 = 500
(c) m 1000
Figure 3: The efficiency gains of different routing strategies compared with the random routing baseline versus the primary model usage ratio. The setting is same as Figure 1, with an additional curve corresponding to the simple debiased router through linear scaling.
Our simulation results demonstrate the superior efficiency of meta-routers, particularly imbal- anced regimes with very limited GS data. In contrast, the predictive router trained on directly pooled and PB data or only PB data, as considered in e.g. Ong et al. (2024), shows little efficiency im- provement even with relatively large GS sample sizes, highlighting the detrimental effect of bias (g4) in LLM routing. As the GS sample size increases, the efficiency gains of all routers improve, except for the PB-only router, highlighting the value of incorporating GS data for debiasing.
Ablation Studies To investigate the impacts of different meta-router components, we conduct dif- ferent numerical experiments for ablation studies, by changing one key element in our main numer- ical experiments while keeping other settings unchanged, examine the performance changes.
(i) We consider Guestrin, 2016). The EGs are reported in Figure 2. The R-lcarner-based router consistently outperform other routers, which demonstrates its robustness over different regression methods. In general, the EGs in Figure 2 are not as high as the EGs in Figure 1, showing the importance comparing different regression methods when training the meta-router.
be specified by another machine learning algorithm XGBoost (Chen &
(ii)
We consider another router trained using simple debiasing strategy based on linear scaling. We debias the PB data by subtracting the sample mean difference between the PB and GS datasets, and then train router the pooled set consisting of the shifted PB data and the GS data. The resulting EGs are shown in Figure 3. The simple debiasing router has a similar performance with the router trained by directly pooled data, and Meta-Router, consistently outperform it. Such observation indicates that in practice, the bias between PB and GS outcomes are usually heterogeneous across different queries, and more sophisticated CATE estimators such as meta- learners, are essential for an effective debiasing.
(iii) the data preprocess, we do not normalize the GS data following Remark 1(2). The EGs are reported in Figure $4. The meta-routers do not significantly outperform the router trained via GS data only, highlighting the importance of pre-normalization for meta-routers.
(iv) We consider d 100 for the PCA. The EGs are reported in Figure $3. The meta-routers also outperform other routers, further demonstrating the robustness of our approach.
(v) We collect the PB data alternatively from another cheaper L.M judge: Grok 4 Fast, with other settings kept the same the main numerical experiment. The EGs reported in Figure S5. The meta-routers outperform other approaches especially when the sample size of GS data is small, which verifies the adaptivity of our approach with different preferences.
4.2 PRBENCH
PRBench is rubric-based benchmark for high-stakes professional reasoning in the domain of law and finance (Akyrek al., 2025). The dataset comprises 1,100 expert-authored tasks across 114 countries and 47 U.S. jurisdictions, similar to HealthBench, accompanied by 19,356 expert-curated evaluation rubrics. All tasks were contributed by 182 industry professionals, thus offers rich real- world complexity beyond conventional academic benchmarks, enabling deeper analysis of open- ended, economically consequential reasoning. We focus on 676 questions PRBench with one- turn conversation. Similar to $4.1, consider the primary model Gemini 2.5 Pro and altema-

---

Published as a conference paper at ICLR 2026
(a) 12 50
(b) n 100
(c) 7 150
Figure 4: The efficiency gains of different routing strategies trained and tested over PRBench in 54.2. Explanations of subfigures are the same as Figure 1.
tive model Gemma 12 B, and use the same mechanism for the GS-based and PB-based answer evaluations; see Appendix A.7 for our prompts. Due to the limited sample size of PRBench, correspondingly consider small GS sample size n E (50, 100, 150) and small PCA dimension for the query embeddings, namely, d 20. Other numerical experiment settings are the same as the main experiment in $4.1.
The EGs are reported in Figure 4. The DR-leamer-basedd router consistently outperforms the base- lines, demonstrating the effectiveness our approach. Under the limited sample size, the R-learner- based router offers no clear advantage over other methods, highlighting the superior sample effi- ciency of the DR-learner in this setting.
5 FUTURE WORK: TRUNCATION-BASED META-ROUTER UNDER POSITIVITY VIOLATION
Currently, our framework requires that the query distribution of GS data and that of the PB data share the common support, i.e., the positivity of propensity scores shall hold. This requirement can violated in practice when, egg., the GS data focuses on one category where responses can be easily justified, while the PB data are with regard to more subjective queries. One could avoid positivity violation by explicit experiment designs in the data collecting period. On the other hand, positivity violation could also be detected through nigh-dimensional density ratio estimation of the query distributions in GS and PB data, respectively: see e.g., Sugiyama et al. (2012). In particular, the region where the estimated density ratio is well upper- and bower-bounded can be interpreted as the overlap region between the distributions of q in the GS and PB datasets, respectively.
When the propensity scores tend be extreme (i.e., close 0 1), the R-learner and BP-Jearner (van der Laan et al., 2024) may offer more robust debiasing performance. When the positivity as- sumption is totally violated, the distribution supports of query distributions do not fully overlap. A promising direction is to develop truncation-bas based meta-router, which always incorporates all GS data but only retains preference data within the estimated overlap region of the distributions. In particular, the overlap can be identified via efficient density ratio estimation. Then meta-leareer of (--) is trained only through overlapping samples in De Dr, which are considered as belong- ing to this region. If following Remark 1(2), GS data are now only normalized to have the same variance as PB data within these overlapping samples. Finally, when we train our truncation-base meta-router by solving (6) with obtained Á(-) but only incorporating the samples Dp which belong to the detected overlap region. This truncation-basee strategy offers principled way to exploit abundant preference data while avoiding extrapolation bias outside the common support.
Additional discussions other future directions are included in Appendix A.4, including the ap- plications semi-supervised learning and active learning, and the potential extension to out-of- distribution routing.
10

---

Published as a conference paper at ICLR 2026
REFERENCES
Afra Feyza Akyrek, Advait Gosai, Chen Bo Calvin Zhang, Vipul Gupta, Jachwan Jeong, Anisha Gunjal, Tahseen Rabbani, Maria Mazzone, David Randolph, Mohammad Mahmoudi Meymand, Gurshaan Chattha, Paula Rodriguez, Diego Mares, Pavit Singh, Michael Liu, Subodh Chawla, Pete Cline, Lucy Ogaz, Ernesto Hernandez, Zihao Wang. Pavi Bhatter, Marcos Ayestaran, Bing Liu, and Yunzhong He. Prbench: Large-scale expert rubrics for evaluating high-stakes profes- sional reasoning, 2025, URL httpt:tpp//ommm /ttps///axxvvoooo org/ abs/ 2511. 11562.
Rahul K Arora, Jason Wei, Rebecca Soskin Hicks, Preston Bowman, Joaquin Quifonero-Candele Foivos Tsimpourlas, Michael Sharman, Meghan Shah, Andrea Vallone, Alex Beutel, et al. Health- bench: Evaluating large language models towards improved human health. arXiv preprint arXiv:2505.08775. 2025.
Susan Athey, Julie Tibshirani, and Stefan Wager. Generalized random forests. 2019.
Ralph Allan Bradley and Milton E Terry. Rank analysis of incomplete block designs: I. the method of paired comparisons. Biometrika, 39(3/4)3324-345, 1952.
Leo Breiman. Random forests. Machine Learning, 45(1):5-32, 2001a. doi: 10.1023/A:
1010933404324.
Leo Breiman. Random forests. Machine learning. 45(1):5-32, 2001b.
Yupeng Chang, Xu Wang. Jindong Wang, Yuan Wu, Linyi Yang, Kaijie Zhu, Hao Chen, Xiaoyuan Yi, Cunxiang Wang, Yidong Wang, et al. A survey on evaluation of large language models. ACM transactions on intelligent systems and technology. 15(3):1-45, 2024.
Lingjiao Chen, Matei Zaharia, and James Zou. Frugalgpt: How to use large language models while reducing cost and improving performance. arXiv preprint arXiv:2305.00 35565.5.......... 2023.
Lingjiao Chen, Matei Zaharia, and James Zou. FrugalGPT: How to use large language models while reducing cost and improving performance. Transactions on Machine Learning Research, 2024. ISSN 3835-8856. URL https:/ https://openrev//// wwtp//ppwwweeeeeeoe//p net. forum? sediiddmiiii 5mkkc.6.......
Tianqi Chen and Carlos Guestrin. Xgboost: A scalable tree boosting system. In Proceedings of the 22nd acm sigkdd international conference on knowledge discovery and data mining, pp. 785-794, 2016.
David Cheng, Ashwin N Ananthakrishnan, and Tianxi Cai. Robust and efficient semi-supervised estimation of average treatment effects with application to electronic health records data. Biomet- rics, 77(2):413-423, 2021.
Victor Chernozhukov, Denis Chetverikov, Mert Demirer, Esther Duflo, Christian Hansen, Whitney Newey, and James Robins. Double/debiased machine learning for treatment and structural pa- rameters, 2018.
Wei-Lin Chiang, Lianmin Zheng, Ying Sheng, Anastasios Nikolas Angelopoulos, Tianle Li, Dacheng Li, Banghua Zhu, Hao Zhang, Michael Jordan, Joseph E Gonzalez, et al. Chatbot arena: An open platform for evaluating lims by human preference. In Forty-first International Conference on Machine Learning, 2024.
Yu-Neng Chuang, Prathusha Kameswara Sarma, Parikshit Gopalan, John Boccio, Sara Bolouki, Xia Hu, and Helen Zhou. Learning route LLMs with confidence tokens. In Aarti Singh, Maryam Fazel, Daniel Hsu, Simon Lacoste-Julien, Felix Berkenkamp, Tegan Maharaj. Kiri Wagstaff, and Jerry Zhu (eds.), Proceedings of the 42nd International Conference Machine Learning, volume 267 of Proceedings of Machine Learning Research, pp. 10859-10878. PMLR, 13-19 Jul 2025. URL https: httpppppcceengsseee/ee mlr www.eppeessse.. vv 6 https//222//cuau2223333 chw.ng2baabbbb// html.
Gheorghe Comanici, Eric Bieber, Mike Schaekermann, Ice Pasupat, Noveen Sachdeva, Inderjit Dhillon, Marcel Blistein, Ori Ram, Dan Zhang, Evan Rosen, al. Gemini 2.5: Pushing the frontier with advanced reasoning, multimodality, long context, and next generation agentic capa- bilities. arXiv preprint arXiv:2507.0626/ 2025.

---

Published as a conference paper at ICLR 2026
Alicia Curth and Mihacla Van der Schaar. Nonparametric estimation of heterogeneous treatment effects: From theory to learning algorithms. In International Conference on Artificial Intelligence and Statistics, pp. 18 1818.888. PMLR. 2021.
Jacob Devlin, Ming- Wei Chang, Kenton Lee, and Kristina Toutanova. Bert: Pre-training of deep bidirectional transformers for language understanding. In Proceedings of the 2019 conference of the North American chapter of the association for computational linguistics: human language lechnologies, volume (long and shori papers), pp. 4171-4186, 2019.
Dujian Ding, Ankur Mallick, Chi Wang, Robert Sim, Subhabrata Mukherjee, Victor Ruhle, Laks VS Lakshmanan, and Ahmed Hassan Awadallah. Hybrid llm: Cost-efficient and quality- query routing. arXiv preprint arXiv:2404. 14618, 2024.
Evan Frick, Connor Chen, Joseph Tennyson, Tianle Li, Wei-Lin Chiang, Anastasios N Angelopou- los, and Ion Stoica. Prompp-to-leaderbaard. arXiv preprint arXiv:2502 14855, 2025.
Ian Goodfellow, Yoshua Bengio, Aaron Courville, and Yoshua Bengio. Deep learning, volume 1. MIT Press, 2016.
Dan Hendrycks, Collin Burns, Steven Basart, Andy Zou, Mantas Mazeika, Dawn Song, and Jacob Steinhardr. Measuring massive multitask language understanding. arXiv preprint (rXiv:2009.03300 2020.
Jue Hou, Rajarshi Mukherjee, and Tianxi Cai. Efficient and robust semi-supervised estimation of average treatment effect with partially annotated treatment and response. Journal of Machine Learning Research, 26(40):1-77, 2025.
Qitian Jason Hu, Jacob Bieker, Xiuyu Li, Nan Jiang, Benjamin Keigwin, Gaurav Ranganath, Kurt Keutzer, and Shriyash Kaustubh Upadhyay. Routerbench: A benchmark for multi-llm routing system. arXiv preprint arXiv:2403. 12031, 2024.
Guido W Imbens and Donald B Rubin. Causal inference in statistics, social, and biomedical sci- ences. Cambridge university press, 2015.
Edward H Kennedy. Towards optimal doubly robust estimation of heterogeneous causal effects. Electronic Journal of Statistics, 17(2):3088-3049 2023.
Sören R Künzel, Jasjeet S Sekhon, Peter J Bickel, and Bin Yu. Metalearners for estimating heteroge- neous treatment effects using machine learning. Proceedings of the national academy of sciences, 116(10)44556-4165, 2019.
Duncan Luce et al. Individual choice behavior, volume 4. Wiley New York, 1959.
Jaouad Moutrada, Stéphane Gaïffas, and Erwan Scomet. Minim minimax optimal rates for mondrian trees and forests. The Annals of Statistics, 48(4):2253 2020.
Xinkun Nie and Stefan Wager. Quasi-oracle estimation of heterogeneous treatment effects. Biometrika, 108(2):299-319) 2021.
Isaac Ong. Amjad Almahairi, Vincent Wu, Wei-Lin Chiang, Tianhao Wu, Joseph E Gonzalez, M Waleed Kadous, and Ion Stoica. Routellm: Learning route LLMs with preference data. arXiv preprint arXiv:2406. 18665, 2024.
OpenAl. Gpt-4o system card. arXiv preprint arXiv:2410.21276, 2024.
OpenAl. Introducing gpt-5, August 2025. URL https: /top:s:///peaa....// homeenai.commindxx///
introducing-gpt-5// Accessed: August 10, 2025.
Rafael Rafailov, Archit Sharma, Eric Mitchell, Christopher D Manning, Stefano Ermon, and Chelsea Finn. Direct preference optimization: Your language model is secretly a reward model. Advances in neural information processing systems, 36:53728-5533 53741, 2023.
Carl Edward Rasmussen and Christopher K. I. Williams. Gaussian Processes for Machine Learning.
MIT Press, Cambridge, MA, 2006. ISBN 978-0-262-18253-9, doi: 10.755 1 0001. URL httphhtss//ggusssaapppppoooo wwwpc/cceesseeorrooooooooooo org/
12

---

Published as a conference paper at ICLR 2026
Donald B Rubin. Causal inference using potential outcomes: Design, Modeling, Decisions. Journal of the American Statistical Association, 100:322-331, 2005.
Johannes Schmidt-Hieber. Nonparametric regression using deep neural networks with relu activation function. The Annals of Statisties, 48(4): 1875-1887788977, 2020.
Burr Settles. Active learning literature survey. 2009.
Seamus Somerstep, Felipe Maia Polo, Allysson Flavio Melo de Oliveira, Prattyush Mangal, Mírian Silva, Onkar Bhardwaj, Mikhail Yurochkin, and Subha Maity. CARROT: A cost aware rate optimal router. In ICLR 2025 Workshop Foundation Models in the Wild, 2025. URL https: // //pppreeviiww... net fo for um? -
Dimitris Stripelis, Zhaozhuo Xu, Zijian Hu, Alay Dilipbhai Shah, Han Jin, Yuhang Yao, Jipeng Zhang, Tong Zhang. Salman Avestimehr, and Chaoyang He. Tensoropera router: A multi-model router for efficient Ilm inference. In EMNLP (Industry Track), 2024.
Masashi Sugiyama, Taiji Suzuki, and Takafumi Kanamori. Density ratio estimation in machine learning. Cambridge University Press, 2012.
Annalisa Szymanski, Noah Ziems, Heather A Eicher-Miller, Toby Jia-Jun Li, Meng Jiang, and Ronald Metoyer. Limitations of the llm-as-a-judge approach for evaluating Ilm outputs in expert knowledge tasks. In Proceedings of the 30th International Conference Intelligent User Interfaces, pp. 952-966, 2025.
Thomas Yu Chow Tam, Sonish Sivarajkumar, Sumit Kapoor, Alisa Stolyar, Katelyn Polanska, Karleigh R McCarthy, Hunter Osterhoudt, Xizhi Wu, Shyam Visweswaran, Sunyang Fu, al. A framework for human evaluation of large language models healthcare derived from literature review. NPJ digital medicine, 7(1):258, 2024.
Gemma Team, Aishwarya Kamath, Johan Ferret, Shreya Pathak, Nino Vieillard, Ramona Merhej, Sarah Perrin, Tatiana Matejovicova, Alexandre Ramé, Morgane Rivière, et al. Gemma 3 technical report. arXiv preprint arXiv:2503.19786 2025.
Robert Tibshirani. Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology. 58(1):267-288, 1996.
Andrey N. Tikhonov and Vasiliy Y. Arsenin. Solutions of ill-posed problems. V. H. Winston & Sons, Washington, D.C.: John Wiley & Sons, New York, 1977. Translated from the Russian, Preface by translation editor Fritz John, Scripta Series in Mathematics.
Asterios Tsiourvas, Wei Sun, and Georgia Perakis. Causal IIm routing: End-to-end regret minimiza- tion from observational data. arXiv preprint arX1 2025.
Lars der Laan, Marco Carone, and Alex Luedtke. Combining t-learning and dr-learning: A framework for oracle-efficient estimation of causal contrasts. arXiv preprint arXiv:2402.01972. 2024.
Larry Wasserman. All of nonparametric statistics. Springer, 2006.
Koki Wataoka, Tsubasa Takahashi, and Ryokan Ri. Self-preference bias in Ilm-as-a-judge. arXiv preprint arXiv:2410.214 18/9, 2024.
Lili Wu and Shu Yang. Integrative r-learner of heterogeneous treatment effects combining ex- perimental and observational studies. In Conference Causal Learning and Reasoning, pp- 904-926. PMLR, 2022.
Xinle Wu and Yao Lu. Reward model routing in alignment. arXiv preprint arXiv:2510.028550 2025.
Tuo Zhang, Asal Mehradfar, Dimitrios Dimitriadis, and Salman Avestimehr. Leveraging uncertainty estimation for efficient IIm routing. arXiv preprint arXiv:2502. 2025.
Lianmin Zheng, Wei-Lin Chiang, Ying Sheng, Siyuan Zhuang, Zhanghao Wu, Yonghao Zhuang, Zi Lin, Zhuohan Li, Dacheng Li, Eric Xing, et al. Judging llm-as-a-judge with mt- bench and chatbot arena. Advances in neural information processing systems, 36:46595-46621 2023a.
13

---

Published as a conference paper at ICLR 2026
Zangwei Zheng, Xiaozhe Ren, Fuzhao Xue, Yang Luo, Xin Jiang, and Yang You. Response length perception and sequence scheduling: An llm-empowered Ilm inference pipeline. Advances in Neural Information Processing Systems, 36:65517-66553 2023b.
Lianghui Zhu, Xinggang Wang, and Xinlong Wang. Judgelm: Fine-tuned large language models are scalable judges. arXiv preprint Sidd::2310.1763111 2023.
14

---

Published as a conference paper at ICLR 2026
A APPENDIX
A. PRACTICAL DEPLOYMENT AND END-TO-END WORKFLOW
In this section, we discuss design choices and recommendations for deploying meta-router in an end-to-end workflow. To make this section self-contained we briefly recall the key parameters and notations. Following the main paper, we consider pairwise LLM routing between two models: a higher-quality, higher-cost model MM (the "primary") and cheaper alternative Ma. In practice, M, would typically be the strongest model available in the stack (for example, a proprietary frontier LLM), and Ma a smaller open-source model chosen for lower price or latency. A router such as meta-router observes an incoming query 4, E Q and decides whether to assign it to Mp or Ma based the expected gold-standard quality gain (p):)) and the associated cost, following the decision rule in Section 2.3.
The first design choice is the gold-standard quality objective. In practice, the operator must choose the evaluation mechanism that defines the evaluated quality gain ry between My(s;) and M.(g:) and hence the gold-standard quality gain function v-(.). For example, when correctness objectively defined, Ta can be a discrete gain such as 1, 0, or -1 depending on which model answers correctly. When evaluation is rubric-based. domain experts (or a trusted evaluation pipeline) score the two responses separately and r; is defined as the difference between these scores. In other applications, an internal reward model may provide a scalar for each response, and r, can again be taken as the difference between the reward assigned to Mp( and Ma(g). Meta-router does not require access to per-response scores beyond this scalar difference. We recommend that operators define the quality gain using evaluation mechanism that is as objective, stable, and aligned with the target task as possible, since the router is explicitly optimized for this gold-standard objective.
The second design choice is the cost model and the acceptable trade-off between quality and cost, which are encoded through a conversion factor w (see Section 2.2). In deployment, the per-query cost of M, and M, can be measured in monetary units (for example, token-based API pricing), latency, or a weighted combination of the two. Given these costs, w 2 O controls how much gold- standard quality gain is required to justify the additional cost of using Mp instead of M.. larger values of w favor cheaper routing, whereas smaller values favor higher-quality routing. When v(g:) known, the Bayes-optimal policy routes qi to MD if and only if v(g:) exceeds the cost-adjusted threshold implied by w (Section 2.3). In practice, Meta-router learns an estimate mm{s)) and applies the same threshold rule. A practical way select is to evaluate, on a held-out set with gold- standard labels, the average realized cost and average gold-standard quality achieved by the induced routing policy over grid of candidate w values, and then choose the smallest w that satisfies a deployment budget constraint such as maximum fraction of queries routed to Mp or a maximum total cost relative to an "always M," bascline.
The third design choice is how to collect the datasets needed to train the router. Meta-router uses two data sources: a small gold-standard set De and a larger preference-based set Dp. Dr is constructed by sampling queries from the actual traffic in the domain of interest and obtaining r, for each, via expert annotation or a trusted evaluation pipeline as discussed above. The sample size can be adapted to resources; in our experiments, few hundred gold-standard queries already provide measurable gains. In parallel, Dr is obtained for queries drawn from the same traffic by collecting cheaper pairwise judgments (for example, crowdsourced labels or LLM-as-a-judge comparisons) indicating whether Mp is better, similar, or worse than Ma These judgments are coded as E (-1,0, The important practical point is that gold-standard data can be scarce, expensive, and domain-specific, whereas preference data can be plentiful but biased. Meta-router is specifically designed to combine these two data resources and to correct the systematic bias in Dp using the information in Dg-
The fourth design choice is the representation used for training and for incoming queries. In a deployed system, it is natural to reuse an existing embedding service (for example, the same text embedding model already used for retrieval). Each query in DG U Dp is embedded as a numerical vector, once using this service, and the resulting vectors (optionally reduced in dimension by prin- cipal component analysis) serve as features for all downstream components in meta-router. Lever- aging existing infrastructure makes the proposed meta-router highly efficient and flexible, as the additional computational cost at training and inference time is dominated by a single embedding call and lightweight tabular models.
15

---

Published as a conference paper at ICLR 2026
Training (Offline)
Embedding Service
GS-Data {(9. rz)] Ji=1
Embeddings GS
Embedding Service
Causal Meta-Learner
PB-Data (11s U: K=
Embeddings of PB
Estimate GS Quality gain v))))
Estimate Shift (-)
Pseudo-GS Labels (r))
Trained Causal Meta Router (-()
Embedding Service
Embedding of
New Query
No
(gg) Cmр (M) > w?
Yes
Ma
Mp
Inference (Online)
Figure S1: End-to-End workflow of the meta-router. The training stage only involves the GS-data ((g, re) and PB-data {(gh 3m and can be carried out completely offline. The inference stage is based on the trained causal meta router vf-) and runs online with generic incoming new queries.
Given these design choices, meta-router is trained described in Section 3.2: it learns the query- dependent shift between gold-standard and preference-based evaluations using causal meta-learners, uses this estimated shift to transform preference labels into pseudo-gold-stame standard labels, and then fits final regression model ((g) on the union of true and pseudo gold-standard labels. We rec- ommend using simple tabular learners such as gradient-boosted trees or random forests on the fixed embeddings, since these models are already powerful enough for the routing task while keeping computational cost minimal at inference time.
At inference time, the router straightforward to integrate into existing infrastructure, Each incom- ing query from any supported domain is sent to the embedding service, the embedding is passed through the trained router G. and the output is compared to the threshold w. If (gg)) w, the query routed to M,; otherwise it is routed to M,, The incremental latency compared with a system without routing is limited embedding call and single evaluation of small regressor, which
16

---

Published as a conference paper at ICLR 2026
is negligible relative to executing the primary LLM. The entire workflow of the training and infer- ence stages is visualized in Figure S1. Note that the training stage can be done offline with already available GS and PB data, while the trained causal meta-router can be directly applied online with incoming new queries.
Two additional considerations may arise in practice. The first is how to handle multiple evolving domains. When domains are clearly distinct (for example, medical, legal, and coding assistance), the operator may train either separate router for each domain indicator as an additional feature. In the latter case, the causal shift AQq) is allowed to vary by domain, and the router learns to use gold-standard supervision from one domain inform others only to the extent that queries are similar in the shared embedding space. When a completely new domain is introduced, the recommended procedure to start with preference-basee that domain, then gradually collect small amount of gold-standard data and retrain or fine-tune the router, exactly as in the initial deployment. Our experiments show that modest number of domain-specifie gold-standard queries is sufficient obtain benefits from meta-router; without any gold-standard data in a domain, no current method can align routing decisions with that domain's gold-standard objective due to the underlying bias between gold-standard data and preference- -ased data.
single router that takes domain
data in
The second consideration is monitoring costs and benefits after deployment. Because the router's objective is defined in terms of vb(g) and cost, it is natural to monitor, on a rolling basis, (i) the fraction of queries sent Mp. (ii) the realized cost relative to baselines such as always using Mp or always using MM. and (iii) the realized gold-standard quality on a small stream of queries that continue to receive expert evaluation. If the observed cost is too high, the operator can increase w; if the observed quality is lower than desired, the operator can decrease w or collect additional gold-standard labels and retrain. Because router retraining is cheap, these updates can be performed regularly as query distributions or cost constraints change.
A.2 MULTI-MODEL META-
We discuss the natural extension of our meta-routing algorithm to the multi-model routing sce- nario. In particular, we attempt to route each query over N candidate LLMs, indexed by I through N. Following the definition of the pooled dataset for two specific LLMS in (7), we use a quintet (st, k,, C,, 14, oc), to define the ith collected pairwise comparison sample where the interpretations of si, t, o2 are the same as the two-LLM routing scenario, i.c., they are the testing query, GS-PB indicator, and the quality gain measurement, respectively. The new variables ky Ex E [NV] represent that the pair of LLMs being compared are LLM K; and L.LM & in the ith sample; without loss of generality, require k; E, for all E [N]. The overall pairwise comparison dataset, comparing different pairs of LLMs, is
D= ((, kr ly t4, o.))
where / represents the full sample size.
We treat the query s and the LLM pair (k, () together as the covariates, t the treatment assignment and o as the observed outcome, for our causal framework. Then following the potential outcome framework in (8), we define the potential outcomes for the ith sample as
(SI) where the nuisance functions (s) and ms.((s) now depend on both query s, as well as the LLM pair (k, e) being compared. They represent the expected quality difference between the two models (LLM & and LLM E) when assessed through the gold-standard evaluation and the naman--reference evaluation, respectively. Then the bais between e)),((s) and Es,((s) could still be viewed as the CATE function:
of = e, + +
Th.... ($x) + +,
(E.(s) = (s,e(s) es.e(s) = E (0 (0) s (ki 0)
Therefore, by treating (s, k, e) instead of s as the covariates, the meta-learners in 53.3 could still be exploited to obtain an estimator Ás.e(s) of (s.e(s) for any (k, Then similar to (6), the meta-router could be obtained by solving the debiased empirical saust- objective,
$-(- Á) = arg min (.HHHA
Σ (o: (s))) + Σ (os + (s.e(()) her (s:))² -=11
+ A(h), (S2)
17

---

Published as a conference paper at ICLR 2026
where the trained router e Á) now depends on both the query as well as the LLM pair (k, E), and H.S is any user specified estimator class containing functions approximating A.(-) depending on both the LLM pair and query.
When estimating vi. (-),-); (-) and A.(), additional structural assumptions on these functions could be further made. For example, if considering ranking models like Bradley-Tery--Lue Model (Bradley& & Terry, 1952; Luce et al., 1959) for the PB data generation (Rafailov et al., 2023), we have
(x)((x(s))
T exp(Ox(s)) + exp (x)))
where (s) is the preference score function for each LLM k. Such modeling resolves the non- identification issue of es) (s) if (k, E) does not get compared in D, and reduce the sample com- plexity for the estimation of (-). For the practical implementation of meta-router with multiple LLMs in the above procedure, it would be important to investigate reasonable functional assump- tions in order to improve the estimation flexibility and efficiency, which we leave for future work.
A.3 R-LEARNER AND DR-LEARNER
R-learner Let (s) E(o s) denote the marginal regression of the evaluation outcome the covariates (query) 8, and let p(s) = Pr(t = 8) denote the propensity score of receiving GS evaluation. R-learner (Nie & Wager, 2021) constructs the orthogonalized residuals:
ã 0. (s:). & = p(s)),
where y and ş are any sensible comple-based estimators for y and p. The R-learner then estimates (--) by solving the generalized least squares problem
(n(-)
arg min H
Σ (ō, -a-E,A (.h(s))))) A(h),
(S3)
N +
where Ha is pre-specified hypothesis space (e.g., linear functions, random forests, or neural net- works), and A(h) regularizer to control complexity. This formulation is quasi-oracle efficient under mild conditions on nuisance estimators. Specifically, causal forests (Athey et al., 2019) is associated with the tree-based function class H. that can flexibly capture heterogeneous structures of (-) across different
DR-learner An alternative is the doubly robust (DR) learner of Kennedy (2023). It constructs pseudo-outcome for each sample by combining outcome regression and propensity adjustment, thereby guarantecing consistency if either component is correctly specified. Specifically, DR-learner considers (e(s) (o f), denoting the conditional regression under treatment status & {0, 1). With no unmeasured confounders, we further have (-(-) = +v(-), (-(-) = n(-). Then, the DR pseudo-outcome
6: (
t, ((s:) ((s)) ((s;)) (o: (si)) + (s. (Ao(s:).
The DR-learner estimates A(.) by regressing o; 5;
ÅDR()) arg min
( (h.-1(s.))² A(h).
(S4)
HEHA n2 +
The doubly robust property ensures that ÂDR(-) is consistent if either (e(-) or p(-) is estimated consistently. Such a feature is particularly appealing in our setting, because the distributional dis- crepancy between DG and Dp may induce misspecification in one nuisance model.
Remark (Computational Cost) In practice, the computational cost of -eeta-eeareers modest. The overall complexity is essentially the same order training the underlying machine-learning models used within the learner. More concretely, the computation consists of: (i) fitting the nui- sance models (propensity score and outcome regressions) and the final CATE regression, and (ü)
18

---

Published as a conference paper at ICLR 2026
for certain meta-learners, constructing pseudo-outcome Step (i) has the same computational der as training the chosen ML algorithm for nuisance and CATE function approximations. Step (ii) requires only single pass through the data (e.g.. computing R-learner DR-learner pseudo- outcomes), which is linear in the sample size. Therefore, the additional overhead introduced by meta-learning mild relative the MI. models used.
A.4 FUTURE DIRECTIONS
Semi-supervised learning From a causal perspective, fully semi-supervised CATE estimation is technically challenging because the target a high-dimensional function of the covariates; most existing semi-supervised learning based work (Cheng et al., 2021; Hou et al., 2025) focused on average treatment effects (ATEs) rather than CATEs. That said, we believe unlabeled data can still be very useful in our setting by helping to learn better query representations. One natural extension to augment the current methods with learnable representation that trained on both labeled and unlabeled queries. Unlabeled queries from real traffic can regularize that it reflects the true deployment distribution (e.g.. via smoothness/consiseency or clustering objectives), while GS+PB queries drive the CATE loss in this learned space. We believe that such representation can reduce the distribution difference between GS, PB, and incoming queries, thereby improving and downstream routing quality.
Active learning Active learning offers complementary and appealing extension (Settles, 2009). particular, rather than treating the GS pool as fixed, one could use an initial Meta-Router to adap- tively select which queries receive expensive GS evaluation to maximize routing accuracy within fixed GS budget. For example, one can view the evaluation mechanism (GS vs PB) as treatment and design acquisition rules that prioritize queries where the current router is most uncertain or most decision critical, such as queries near the routing decision boundary.
Handling -uitt--distribution routing Our current work focuses on the in-distribution setting, where deployment queries are drawn from the same population as the GS and PB data used to train Meta-Router. For truly -uitt--distributionn (OOD) queries, a practical platform may collect responses from both models and obtain PB or GS evaluations for these new queries. This naturally forms an online-lcarning process in which the system gradually expands the coverage of the in-distribution domain. Integrating such OOD-aware data collection into Meta-Router is interesting direction for future work, and we have noted this in the revised manuscript.
A.5 PROOF OF LEMMA 1
The density function of (s,t, in GS-PB DGP could be written as
f(8,4,0) = x² =(=
8)
where /.(- () and [.(. 8) represent the conditional probability density function of r; and y; given
4a 8, following (1) and (2), respectively. This could be further written
x² ((())))=((())ff (s)
( (1-
fo (oos)ff- s), (S5)
(1
faæ1 (1
(ols)
recalling the notation in Causal DGP, and thereby show the distributional equivalence of two pro- cesses.
19

---

Published as a conference paper at ICLR 2026
Figure S2: The histogram of all queries' PB-based and GS-based evaluation differences, i.e., {r² 55000 of HeacthBench evaluations. The red vertical line represents the sample mean around -0.47.
Y: Ti=1
A.6 ADDITIONAL. NUMERICAL. RESULTS FOR 54.
Additional results for 84.1 are reported in Figures S2-S5.
Winey Madiiuaag
Piautry ROT
Hinory Made urage R0E
100
(b) 500
(c) 1000
Figure S3: The efficiency gains of different routing strategies compared with the random routing baseline versus the primary model usage ratio. The query embedding dimension reduced 100. Other explanations are the same as Figure 1.
PRowcy
06
14
Hoail
03
1.4 Hodd
85
(a)
(b) 71
(c) n1 1000
Figure $4: The efficiency gains of different routing strategies compared with the random routing bascline versus the primary model usage ratio. GS data are not normalized to align the variance with PB data. Other settings are the same Figure 1.
20

---

Published as a conference paper at ICLR 2026
33
15
42
24 15
C6
06
02
03
34 Mode
C6
(a)
(b) F
(c) 71
1000
Figure S5: The efficiency gains of different routing strategies compared with the random routing baseline versus the primary model usage ratio. PB data are collected by another LLM judger: Grok 4 Fast. Other settings the same as Figure 1.
21

---

Published as a conference paper at ICLR 2026
A.7 EVALUATION- PROMPTS IN NUMERICAL EXPERIMENTS
In this section, we present the four prompts we use to collect the GS and PB evaluation data for our numerical experiments in $4.1 and $4.2 follows.
Prompt collecting the gold-standen standard evaluation from GPT-5 mini for HealthBench
You medical educator evaluator. Evalate this single medical according provided enttria.
-OOiginal Questio: [groupt
""Evaluzion Chiteri: frubrice lorrice)) los)
**ideal Completion (Reference):* [-daal wwwpplooooppaa iona.da:a]
"Rpposee to Evaliate:** {r [respee
-eellution Rules:**
Read each critecion evactly as writien
Award points the criterion completely satisfied
Award points for puntial matches or close appoximatios
details
3. Be precise with medicaltechmical decails
7. For negative criteia (penaliees), apply the full pesaity if viclated
OOnguut Rommor:
*total score": Meriterion Perstarion_ spoints exordes/dedustedh. cersterioo_2* spoints nordee//eeduceed resstarion_3"; spoints exordes/dcdurteds
criterion points>,
Prompt collecting the preference- preeeeence-bused evaluation from GPT-5 mini for HealthBench
You expert evaluating responses questions. significantly better from a medical perspective.
compare two responses
*PEeslution Cittria;**
Medical socuracy safety
Completeness of advice
Approprinteness the yutients 's situntion
Professional
Adherence to medical best practices
Insttructionss"*
1. Read the patient's question carefully
Evaluate both Respouse Response
Return ONLY a single number:
Respome significanlyy
Response significantly hetter
Of both responses roughly equivalens in quality
**Patient Question:* (prompt)
*RRssponse a
[xeaponee
"Respouse [re (response 1t)
"Yuour evaluation (return only -1, digre
22

---

Published as a conference paper at ICLR 2026
Prompt collecting the gold-standard evaluation from GPT-5 mini for PRBench
You are am expent legal and finunce edocator and respunse evalatorr Evaluase this single legal finance respunse secording to the provided criteria.
-Original Quoston: {prampt)
"Follution [rubricn Lox)
""Rpposse to Evallaate {response
Rules:*4
1. Read each critetion exactly writeen
points only
complotely
Awand 0 points for pertial motches or close approcimaiios
spacifies exeet (sumbers, procedures), raquie oxact
Be with medicallecnncal decails
When in doube, award points rather than partial credit
For negaiive criteta (penaliies), apply the full penalty violated
Ouupuut Formar:44
"coraa_ocare": <oun criterion pointss,
Nor tor I
Mesitarion spoints axarded/deductest
resstarion_2" spoints wworecs/dddcteeds "esiterion spointa awarchd/deducted
Prompt collecting the preference- based evaluation from GPT-5 mini for PRBench
financial/legal" expent evalusting responses fimaneolegal questions. is ficnntly better from financaallggll perspective.
compure responees
"Evlllutionn Criteria** (if the query is related to fatance)
(if the quecy is relased logal)
Financial Accuney
Logal Accuracy
Provess Transparncy & Audinbility
Application of Law to the Facts
Hundling Uncertainty
Procedural
Practical thility
Handling Unvertainty
Disclosure
Uality
Supplemental Insight
Risk & Ethical Disclesure
Instraction Following
Supplemental Insight
Instruction Following
*Insttuctions
Read the clien's carefiilly
Evalunte both Response and Response
Return ONLY single number:
Response A is significunlly better
Respunse significantly beiler
responses roughly equivalent in quality
PTPiticnt Question (pxampt)
Resspons [response_s]
*RRspoose Bitx fresponse.b)
=YYor ovaliation
0y>*
"Uinn the woeds "faancia"" "leggl" degends on the type of the conesponding query. For the A/B format helow, the same logie applics.
23

---

Published as a conference paper at ICLR 2026
A.8 THE USE OF LARGE LANGUAGE MODELS (LLM)
For this project, LLMs were used to polish the writing of the main paper and to assist with coding for the numerical experiments.
24


Reviews:
[{"review_id": "526mivGp1G", "reviewer": "Reviewer_K5JD", "summary": "This paper tackles a key challenge in LLM routing: balancing response quality and inference cost amid the constraints of two data types—scarce but accurate gold-standard (GS) data (from experts/benchmark labels) and scalable but biased preference-based (PB) data (from crowdsourcing/LLM-as-a-judge).\nTo address this, the work frames LLM router training within a\ncausal inference framework\n: it treats GS and PB evaluations as two \"treatment assignments\" for query quality assessment, identifying PB data biasas the\nConditional Average Treatment Effect (CATE)\n—the expected quality gain difference between GS and PB evaluations for a given query.\n\nExperiments on the\nHealthBench benchmark\n(5,000 healthcare dialogues) use M_p (Gemini 2.5 Pro, high-cost/high-quality) and (Gemma 3 12B, low-cost/low-quality), with GPT-5-mini generating GS (rubric-based) and PB (preference) evaluations. Results show Meta-Router outperforms baselines (pooled GS- PB, GS-only, random routing) across GS sample sizes (n=100, 500, 1000), achieving higher efficiency gain (EG) by cutting M_p usage while preserving quality. R-learner-based Meta-Router is most robust, especially with scarce GS data.\n\nThe framework provides a principled way to integrate biased scalable data with accurate scarce data, advancing reliable LLM routing in high-expertise domains (e.g., healthcare). Future work will explore truncation-based extensions for non-overlapping GS-PB query distributions.", "questions": "1. Is \"LMM judge\" in Section 3.1 a typo for \"LLM judge,\" and why was no alternative notation used to avoid symbol \"m\" overloading?\n\n2. Why is the function h(q) in Equation (5) not explicitly defined, and how does this affect reproducibility?\n\n3. Can the \"GS–PB joint Data Generation Process\" be simplified to reduce comprehension costs?\n\n4. Does the paper provide any end-to-end workflow or hyperparameter guidance for Meta-Router’s real-world deployment?\n\n5. Were empirical checks conducted for the propensity score positivity assumption, and how is performance affected by its violation?\n\n6. Why are distinct PB bias sources not disentangled, and is there experimental proof that blurring them is acceptable?\n\n7. Has Meta-Router been tested on non-healthcare domains or multi-model routing scenarios to verify generalization?\n\n8. Why is there no comparison to SOTA routing methods or a \"PB-only predictive router\" baseline in experiments?\n\n9. What ablation experiments are missing to validate the necessity of Meta-Router’s core components (e.g., CATE estimation)?", "limitations": "", "rating": 2, "confidence": 3, "soundness": 1, "presentation": 3, "contribution": 2, "strengths": "Novel Causal Framing for a Critical Practical Gap:\nThe paper addresses a longstanding tension in LLM routing—scarce but accurate gold-standard (GS) data vs. scalable but biased preference-based (PB) data—by innovatively framing the problem as causal inference. By treating GS/PB evaluations as \"treatment assignments\" and PB bias as the Conditional Average Treatment Effect (CATE), it links router training to rigorous semiparametric causal estimation (Imbens & Rubin, 2015), providing a principled solution to debias PB data rather than relying on ad-hoc fusion.\n\nRobust and Flexible Meta-Router Framework:\nThe proposed Meta-Router is both theoretically grounded and practically adaptable. Its two-step design—CATE estimation via R/DR-learners (Nie & Wager, 2021; Kennedy, 2023) and debiased data fusion—enjoys the oracle property and supports off-the-shelf ML models (random forests, XGBoost) for nonparametric regression , . This flexibility allows it to handle heterogeneous query spaces (e.g., healthcare dialogues) and avoids over-reliance on specific model architectures.\n\nTargeted Validation on High-Stakes, Expert-Driven Benchmark:\nExperiments are conducted on HealthBench (Arora et al., 2025)—a 5,000-sample healthcare benchmark with expert-designed rubrics and physician consensus—addressing a critical limitation of prior routing work (e.g., using MMLU with objective answers). By testing on open-ended, professional tasks (healthcare dialogues) and using realistic model pairs (Gemini 2.5 Pro as M_p, Gemma 3 12B as M_p), the study validates Meta-Router’s relevance to high-stakes domains where quality-cost trade-offs matter most .", "weaknesses": "Lack of Practical Implementation Guidance:\nThe Meta-Router framework is presented up to its theoretical formulation but provides no actionable guidance for real-world deployment. For instance, it does not outline end-to-end workflows. This gap undermines its utility, as the paper addresses a problem with clear real-world relevance but fails to bridge the gap between theory and application.\n\n\nInadequate Experimental Scope:\nThe experimental design is overly limited. The main figures (Figure 1, Figure 2) only compare variants of the proposed Meta-Router against basic baselines (pooled GS- PB data, GS-only data) , with no comparison to state-of-the-art (SOTA) routing methods. This omission prevents readers from assessing Meta-Router’s competitive advantage. Furthermore, the absence of a \"PB-only predictive router\" baseline (a logical counterpoint to GS-only routing) leaves unaddressed how much value GS data adds to debiased PB data.\n\nMissing Ablation Studies:\nThe paper lacks ablation experiments to isolate the impact of critical Meta-Router components. For example, it does not test: (1) whether CATE estimation (via R/DR-learners) is indispensable (e.g., comparing to simpler bias correction methods like linear scaling); (2) the necessity of data normalization (per Remark 1) by omitting this step and measuring performance degradation; or (3) how different nonparametric regressors (random forests vs. XGBoost) influence accuracy. Without these ablations, the contribution of individual components to Meta-Router’s performance remains unsubstantiated.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "Wwexckcgmu", "reviewer": "Reviewer_Ejhf", "summary": "This paper proposes Meta-Router, an LLM router training method that leverages both expensive, scarce gold data (GS) with cheap, scalable preference-based (PB) supervision. Recognizing that PB can be biased relative to GS, the authors frame the discrepancy between GS and PB as a causal shift, and use meta-learners like R-learner or DR learner to estimate it. The estimated shift is then incorporated into router training to leverage both data sources in a consistent manner. Experiments on HealthBench, using random forest and XGBoost, show higher efficiency gains than the baselines trained on pooled GS+PB or GS-only data across primary model usage ratios.", "questions": "Please see weaknesses.\n\n1. How would this method extend to more than two LLMs?", "limitations": "", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "1. The paper explicitly identifies the bias in preference-based (PB) supervision, and provides clear motivation of framing the discrepancy between PB and GS data as a causal shift. Then it leverages meta-learning to estimate the bias, thereby aligning PB with GS evaluation.\n2. The framework is derived through clear, step-by-step equations. The equations are coherent and easy to follow.\n3. The experiments include multiple ablated variants, which effectively demonstrates the strengths of Meta-Router.", "weaknesses": "1. Both GS and PB scores are obtained from GPT5-mini. This would potentially reduce the real discrepancy between them and add confounding factors. Using a less capable LLM for PB may better reflect the bias . The paper also lacks validation of GPT5-mini GS scores against human judgements, especially in an expert-dependent domain.\n2. PB scores are normalized to have same variance as GS, and training is restricted to overlapping supports. Normalization may be unstable when overlap is small. The paper mentions GS sample size, but the overlap size is unclear.\n3. Experiments are done on a single domain, Health. Mixed-domain tests might be needed to assess whether the meta-learner generalizes well across domains.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "KlgfM5eMMn", "reviewer": "Reviewer_ujoT", "summary": "This paper presents a novel framework for LLM routing, specifically addressing the challenge of cost-effective routing between LLMs while ensuring optimal response quality. The core idea is to balance the expected quality of responses against the computational cost of using different LLMs. The paper introduces an integrative causal router training framework that combines gold-standard data with preference-based data, with the goal of reducing bias and improving routing efficiency and robustness. The framework is empirically validated using a healthcare benchmark, HealthBench, demonstrating improved efficiency in routing decisions.", "questions": "1. How does the method handle situations where the preference-based data distribution does not match the gold-standard data distribution, particularly in more complex or subjective queries?\n2. The use of meta-learners such as R-learner and DR-learner is central to the paper. While the paper highlights their benefits, how does the method perform when faced with data heterogeneity or imbalances across large-scale datasets? Could the integration of more flexible or adaptive meta-learners improve performance further in such cases?", "limitations": "", "rating": 6, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 2, "strengths": "1. The proposed method models the challenge as a causal inference problem, framing the task as correcting biases in preference-based data using causal meta-learners.\n\n2. These meta-learners, specifically the R-learner and DR-learner, are used to estimate the Conditional Average Treatment Effect, which quantifies the discrepancy between gold-standard and preference-based evaluations.", "weaknesses": "The oracle integrative router in Section 3.1 assumes that the shift function (∆(q)) between the gold-standard and preference-based evaluations is known a priori. This assumption, while ideal in theoretical settings, is unrealistic in practical scenarios where the true shift function is unknown and must be estimated from data. Exploring semi-supervised learning or incorporating active learning strategies to iteratively estimate the shift function could make the approach more applicable.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "CsULiMnpE2", "reviewer": "Reviewer_uL2P", "summary": "This paper addresses the challenge of training LLM routers to balance response quality and inference cost. It proposes a novel causal inference framework, viewing gold-standard and preference-based evaluation data as a treatment assignment problem. The key insight is that bias in preference-based data corresponds to the conditional average treatment effect (CATE). The proposed Meta-Router framework corrects this bias via meta-learners (S-learner, T-learner, X-learner), incorporates propensity score weighting to address data source imbalances, and applies distributionally robust optimization (DRO) for routing robustness. Experiments on benchmarks (AlpacaEval, MT-Bench, MMLU, GSM8K) show improved cost-quality trade-offs compared to standard baselines.", "questions": "1. What is the computational overhead compared to baselines (training and inference)? Can you quantify the trade-off?\n\n2. Can you empirically measure and report the actual bias in preference data vs. gold-standard benchmarks used?\n\n3. How does performance scale with model pool size—do you have results for routing among 8–10 models?\n\n4. Could you compare to modern LLM routing baselines, such as confidence-based or learned reward model routing?\n\n5. Have you performed error analysis outlining types of queries or inputs where Meta-Router fails?\n\n6. Why does T-learner outperform X-learner in Table 2 occasionally—are simpler meta-learners sometimes preferable?\n\n7. How is out-of-distribution or universally poor query detection handled by the method?", "limitations": "", "rating": 6, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "**Originality**: The framing of LLM router training as a causal inference problem is novel and insightful; the connection between preference evaluation bias and CATE is a principled advancement. Creative application of meta-learners to this setting broadens the impact of causal inference in LLM routing.\n\n**Quality**: The methodology is sound, with solid theoretical framing and systematic derivations. Experiments are thorough, covering benchmarks, model combinations, and ablations.\n\n**Clarity**: The paper is well-written, with clear motivation, notation, and visualizations.\n\n**Significance**: Addresses a highly relevant and practical problem. The framework is generalizable to different model pools and routing architectures.", "weaknesses": "**Limited Baseline Scope**: Compares mostly to classical methods (IPW, DR); recent LLM routing methods and stronger supervised baselines are needed.\n\n**Scalability**: Only small model pools tested, unclear if methods scale to larger pools or what challenges emerge.\n\n**Missing Error Analysis**: No detailed analysis of failure cases, e.g., when Meta-Router misroutes queries.\n\n**Computational Overhead**: No empirical or theoretical analysis of the increased training cost for meta-learners and propensity score modeling, particularly for the X-learner.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Decision: Accept (Poster)