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Chained Information-Theoretic bounds and Tight Regret Rate for Linear Bandit Problems (2403.03361v1)
Published 5 Mar 2024 in stat.ML and cs.LG
Abstract: This paper studies the Bayesian regret of a variant of the Thompson-Sampling algorithm for bandit problems. It builds upon the information-theoretic framework of [Russo and Van Roy, 2015] and, more specifically, on the rate-distortion analysis from [Dong and Van Roy, 2020], where they proved a bound with regret rate of $O(d\sqrt{T \log(T)})$ for the $d$-dimensional linear bandit setting. We focus on bandit problems with a metric action space and, using a chaining argument, we establish new bounds that depend on the metric entropy of the action space for a variant of Thompson-Sampling. Under suitable continuity assumption of the rewards, our bound offers a tight rate of $O(d\sqrt{T})$ for $d$-dimensional linear bandit problems.
- D. Russo and B. Van Roy, “An Information-Theoretic Analysis of Thompson Sampling,” Jun. 2015, number: arXiv:1403.5341 arXiv:1403.5341 [cs]. [Online]. Available: http://arxiv.org/abs/1403.5341
- S. Dong and B. Van Roy, “An Information-Theoretic Analysis for Thompson Sampling with Many Actions,” Jul. 2020, arXiv:1805.11845 [cs, math, stat]. [Online]. Available: http://arxiv.org/abs/1805.11845
- W. R. Thompson, “On the likelihood that one unknown probability exceeds another in view of the evidence of two samples,” Biometrika, vol. 25, no. 3-4, pp. 285–294, 1933.
- D. J. Russo, B. Van Roy, A. Kazerouni, I. Osband, Z. Wen et al., “A tutorial on thompson sampling,” Foundations and Trends® in Machine Learning, vol. 11, no. 1, pp. 1–96, 2018.
- D. Russo and B. Van Roy, “Learning to Optimize via Information-Directed Sampling,” Jul. 2017, arXiv:1403.5556 [cs]. [Online]. Available: http://arxiv.org/abs/1403.5556
- O. Chapelle and L. Li, “An empirical evaluation of Thompson sampling,” Advances in neural information processing systems, vol. 24, 2011.
- V. Dani, T. P. Hayes, and S. M. Kakade, “Stochastic Linear Optimization under Bandit Feedback,” 21st Annual Conference on Learning Theory, vol. 21st Annual Conference on Learning Theory, pp. 355–366, 2008.
- G. Neu, I. Olkhovskaia, M. Papini, and L. Schwartz, “Lifting the Information Ratio: An Information-Theoretic Analysis of Thompson Sampling for Contextual Bandits,” Advances in Neural Information Processing Systems, vol. 35, pp. 9486–9498, 2022.
- A. Gouverneur, B. Rodríguez-Gálvez, T. J. Oechtering, and M. Skoglund, “Thompson Sampling Regret Bounds for Contextual Bandits with sub-Gaussian rewards,” Apr. 2023, arXiv:2304.13593 [cs, stat]. [Online]. Available: http://arxiv.org/abs/2304.13593
- J. Negrea, M. Haghifam, G. K. Dziugaite, A. Khisti, and D. M. Roy, “Information-Theoretic Generalization Bounds for SGLD via Data-Dependent Estimates,” arXiv:1911.02151 [cs, math, stat], Jan. 2020, arXiv: 1911.02151. [Online]. Available: http://arxiv.org/abs/1911.02151