Calibration-Gated LLM Pseudo-Observations for Online Contextual Bandits

We study contextual bandit problems with correlated arms and access to surrogate reward signals produced by a machine learning model, motivated by applications such as large language model (LLM) routing. Unlike classical contextual bandits that rely solely on bandit feedback and assume conditional independence across arms, our setting allows context-dependent inter-arm correlations and auxiliary reward information that may be noisy or misspecified. We propose algorithms that leverage such surrogate rewards through two complementary designs. A coupled reward-mixing approach pools true and surrogate rewards to accelerate learning when surrogate signals are reliable, while a decoupled prediction-mixing approach maintains separate estimators for bandit feedback and surrogate rewards and adaptively combines their predictions. This decoupling yields robustness to surrogate misspecification, recovering regret guarantees comparable to reward-only bandit methods in the worst case, while achieving improved regret when surrogate predictions are sufficiently informative. We provide theoretical regret analyses for both approaches and evaluate them on LLM routing benchmarks under varying accuracy versus cost trade-offs. The results demonstrate improved sample efficiency and consistently better accuracy-cost trade-offs compared to standard contextual bandit baselines and strong static routing methods.

Iterative refinement has significantly enhanced Large Language Model (LLM) performance; however, existing methods ranging from feedback-based Self-Refine to traditional bandit approaches often rely on static options or overlook the saturation effect. This neglect leads to over-exploitation, where the continuous use of identical prompts or arms results in diminishing rewards over time. To address this challenge, we propose a novel contextual bandit algorithm that explicitly incorporates reward decay modeling. Utilizing an Expectation-Maximization (EM) algorithm, our method simultaneously estimates both arm-specific and decay parameters. Furthermore, by embedding prompts as arms, we facilitate the joint learning of arm values, distinguishing our approach from the traditional disjoint Linear Upper Confidence Bound (LinUCB) framework. Experimental results on Sentiment Reversal and GSM8K benchmarks demonstrate that our method achieves significant performance gains over strong baselines. Finally, our ablation study confirms that the integration of reward decay modeling within the bandit framework is crucial for mitigating over-exploitation and optimizing the iterative refinement process.

We study linear contextual bandits with paid observations, where at each round the learner observes a context, selects an action, and may pay a fixed cost to observe feedback from a subset of arms. We propose two Follow-the-Regularized-Leader algorithms with Best-of-Both-Worlds guarantees. The first, Agg-SPB, extends the SPB-matching framework of Tsuchiya and Ito (2024) by aggregating context-dependent stability terms, achieving the characteristic $T^{2/3}$ adversarial regret rate and logarithmic dependence on $T$ in stochastic environments. The second, CE-SPB, combines arm-dependent observation probabilities with an entropy-adaptive learning rate inspired by Kuroki et al. (2024). It achieves an entropy-adaptive $\widetilde{O}(T^{2/3})$ adversarial guarantee and polylogarithmic stochastic regret, while avoiding the minimum-context-mass dependence arising in the stochastic analysis of Agg-SPB. Both algorithms further extend to corrupted stochastic environments with explicit corruption-dependent guarantees. These results establish that logarithmic stochastic regret is compatible with the $T^{2/3}$ adversarial regime for linear contextual bandits with paid observations, while highlighting a tradeoff between sharper horizon dependence in stochastic settings and path-dependent matching without explicit minimum-context-mass dependence.