Chance-Constrained Personalized LLM Routing with Online Preference Learning

Ameya Anjarlekar; Rasoul Etesami; R Srikant

cs.LGOct 15, 2026

Chance-Constrained Personalized LLM Routing with Online Preference Learning

Authors: Ameya Anjarlekar, Rasoul Etesami, R Srikant

Abstract

We study personalized LLM routing, where a router selects among heterogeneous models with different costs and capabilities. The router must minimize cumulative inference cost while ensuring reliable realized response quality, even though user preferences are unknown and must be learned from feedback during deployment. We formulate this problem as a constrained contextual bandit and develop an algorithm that learns preferences while balancing cost and quality. Under a logistic preference model and a strict-feasibility condition, we establish sublinear expected cost regret relative to an optimal stationary randomized contextual oracle and a high-probability guarantee on realized cumulative satisfaction. The framework extends beyond LLM routing to broader online model-selection problems with reliability requirements.

Figures & tables

Method	Online	Personalized	Safety
FrugalGPT	$\times$	$\times$	$\times$
RouteLLM	$\times$	$\times$	$\times$
PersonalizedRouter	$\times$	$\checkmark$	$\times$
MetaRouter	$\checkmark$	$\checkmark$	$\times$
BaRP	$\checkmark$	$\checkmark$	$\times$
Conformal Routing	$\times$	$\times$	$\checkmark$

Table 1: Comparison of LLM routing approaches.

a_{t}\leftarrow\arg\min_{a\in\mathcal{A}}\left[c(a)-\lambda_{t}\operatorname{UCB}_{t}(a)\right].

Algorithm 1 Primal-Dual GLM-UCB Router

Panel of Figure 1 — Figure 1: Comparison of the proposed routing algorithm with alternative routing strategies as the satisfaction tolerance $\epsilon$ varies. In the third panel, the green and orange curves overlap.

Panel of Figure 2 — Figure 2: Overall performance and ablation results. Error bars denote $95\%$ confidence intervals. The dotted lines indicate the required satisfaction level $1-\epsilon=0.8$ and constraint probability $1-\delta=0.95$ , respectively.

Panel of Figure 3 — Figure 3: Real-LLM routing POC: (a) average routing cost as a function of $\epsilon$ and (b) model-selection composition as a function of $\epsilon$ .

Figure 4: Personalized and global routing costs for Users 1–3.

Figure 5: Per-round and cumulative cost gap relative to the oracle across horizons.

Appendix figures & tables3 assets

Supplementary material from the paper’s appendix.

Appendix

Figure 6: Sensitivity to the satisfaction tolerance $\epsilon$ . Dashed lines indicate the corresponding satisfaction and constraint thresholds.

Figure 8: Empirical probability of satisfying the chance constraint over 5000 independent trials.

LLM Routing

Large language model routing selects models, tools, experts, or inference strategies to balance answer quality, cost, latency, and deployment constraints.

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Large language models (LLMs) present a trade-off between performance and cost, where more powerful models incur greater expense. LLM routing aims to mitigate expenses while maintaining performance by sending queries to the most suitable model. However, existing methods cannot perform well for different user cost-performance preferences. To address this gap, we introduce a novel perceptive LLM routing paradigm for personalized and user-centric cost-performance optimization, which efficiently learns users' implicit preferences through little interaction. To handle the challenge of heterogeneous user needs, we formulate preference profiles as a set of distinct tasks in contextual bandit and propose MetaRouter, a meta-learning framework designed for preference-aware LLM routing. Experimental results show that MetaRouter outperforms strong baselines on both in-distribution and out-of-distribution tasks. Furthermore, it exhibits high efficiency in learning user preferences, robustness to changes in the routable LLMs, and scalability to multi-model routing.

Abstract

LLM Routing

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