Nearly Optimal Sample Complexity of Offline KL-Regularized Contextual Bandits under Single-Policy Concentrability

• URL: http://arxiv.org/abs/2502.06051v1
• Date: Sun, 09 Feb 2025 22:14:45 GMT
• Title: Nearly Optimal Sample Complexity of Offline KL-Regularized Contextual Bandits under Single-Policy Concentrability
• Authors: Qingyue Zhao, Kaixuan Ji, Heyang Zhao, Tong Zhang, Quanquan Gu,
• Abstract summary: We propose the emphfirst algorithm with $tildeO(epsilon-1)$ sample complexity under single-policy concentrability for offline contextual bandits. Our proof leverages the strong convexity of the KL regularization, and the conditional non-negativity of the gap between the true reward and its pessimistic estimator. We extend our algorithm to contextual dueling bandits and achieve a similar nearly optimal sample complexity.
• Score: 49.96531901205305
• License:
• Abstract: KL-regularized policy optimization has become a workhorse in learning-based decision making, while its theoretical understanding is still very limited. Although recent progress has been made towards settling the sample complexity of KL-regularized contextual bandits, existing sample complexity bounds are either $\tilde{O}(\epsilon^{-2})$ under single-policy concentrability or $\tilde{O}(\epsilon^{-1})$ under all-policy concentrability. In this paper, we propose the \emph{first} algorithm with $\tilde{O}(\epsilon^{-1})$ sample complexity under single-policy concentrability for offline contextual bandits. Our algorithm is designed for general function approximation and based on the principle of \emph{pessimism in the face of uncertainty}. The core of our proof leverages the strong convexity of the KL regularization, and the conditional non-negativity of the gap between the true reward and its pessimistic estimator to refine a mean-value-type risk upper bound to its extreme. This in turn leads to a novel covariance-based analysis, effectively bypassing the need for uniform control over the discrepancy between any two functions in the function class. The near-optimality of our algorithm is demonstrated by an $\tilde{\Omega}(\epsilon^{-1})$ lower bound. Furthermore, we extend our algorithm to contextual dueling bandits and achieve a similar nearly optimal sample complexity.

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