Towards a Sharp Analysis of Offline Policy Learning for $f$-Divergence-Regularized Contextual Bandits
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866914351103868928 |
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| author | Zhao, Qingyue Ji, Kaixuan Zhao, Heyang Zhang, Tong Gu, Quanquan |
| author_facet | Zhao, Qingyue Ji, Kaixuan Zhao, Heyang Zhang, Tong Gu, Quanquan |
| contents | Many offline reinforcement learning algorithms are underpinned by $f$-divergence regularization, but their sample complexity *defined with respect to regularized objectives* still lacks tight analyses, especially in terms of concrete data coverage conditions. In this paper, we study the exact concentrability requirements to achieve the $\tildeΘ(ε^{-1})$ sample complexity for offline $f$-divergence-regularized contextual bandits. For reverse Kullback-Leibler (KL) divergence, arguably the most commonly used one, we achieve an $\tilde{O}(ε^{-1})$ sample complexity under single-policy concentrability for the first time via a novel pessimism-based analysis, surpassing existing $\tilde{O}(ε^{-1})$ bound under all-policy concentrability and $\tilde{O}(ε^{-2})$ bound under single-policy concentrability. We also propose a near-matching lower bound, demonstrating that a multiplicative dependency on single-policy concentrability is necessary to maximally exploit the curvature property of reverse KL. Moreover, for $f$-divergences with strongly convex $f$, to which reverse KL *does not* belong, we show that the sharp sample complexity $\tildeΘ(ε^{-1})$ is achievable even without pessimistic estimation or single-policy concentrability. We further corroborate our theoretical insights with numerical experiments and extend our analysis to contextual dueling bandits. We believe these results take a significant step towards a comprehensive understanding of objectives with $f$-divergence regularization. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2502_06051 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Towards a Sharp Analysis of Offline Policy Learning for $f$-Divergence-Regularized Contextual Bandits Zhao, Qingyue Ji, Kaixuan Zhao, Heyang Zhang, Tong Gu, Quanquan Machine Learning Artificial Intelligence Statistics Theory Many offline reinforcement learning algorithms are underpinned by $f$-divergence regularization, but their sample complexity *defined with respect to regularized objectives* still lacks tight analyses, especially in terms of concrete data coverage conditions. In this paper, we study the exact concentrability requirements to achieve the $\tildeΘ(ε^{-1})$ sample complexity for offline $f$-divergence-regularized contextual bandits. For reverse Kullback-Leibler (KL) divergence, arguably the most commonly used one, we achieve an $\tilde{O}(ε^{-1})$ sample complexity under single-policy concentrability for the first time via a novel pessimism-based analysis, surpassing existing $\tilde{O}(ε^{-1})$ bound under all-policy concentrability and $\tilde{O}(ε^{-2})$ bound under single-policy concentrability. We also propose a near-matching lower bound, demonstrating that a multiplicative dependency on single-policy concentrability is necessary to maximally exploit the curvature property of reverse KL. Moreover, for $f$-divergences with strongly convex $f$, to which reverse KL *does not* belong, we show that the sharp sample complexity $\tildeΘ(ε^{-1})$ is achievable even without pessimistic estimation or single-policy concentrability. We further corroborate our theoretical insights with numerical experiments and extend our analysis to contextual dueling bandits. We believe these results take a significant step towards a comprehensive understanding of objectives with $f$-divergence regularization. |
| title | Towards a Sharp Analysis of Offline Policy Learning for $f$-Divergence-Regularized Contextual Bandits |
| topic | Machine Learning Artificial Intelligence Statistics Theory |
| url | https://arxiv.org/abs/2502.06051 |