An Information-Theoretic Analysis of Thompson Sampling for Logistic Bandits

• URL: http://arxiv.org/abs/2412.02861v2
• Date: Thu, 20 Feb 2025 18:24:53 GMT
• Title: An Information-Theoretic Analysis of Thompson Sampling for Logistic Bandits
• Authors: Amaury Gouverneur, Borja Rodríguez-Gálvez, Tobias J. Oechtering, Mikael Skoglund,
• Abstract summary: We study the performance of the Thompson Sampling algorithm for logistic bandit problems.<n>We derive a bound order $O(d/alphaqrtT log(beta T/d))$ of regret incurred after $T$ expected of Sampling steps.
• Score: 36.37704574907495
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: We study the performance of the Thompson Sampling algorithm for logistic bandit problems. In this setting, an agent receives binary rewards with probabilities determined by a logistic function, $\exp(\beta \langle a, \theta \rangle)/(1+\exp(\beta \langle a, \theta \rangle))$, with slope parameter $\beta>0$, and where both the action $a\in \mathcal{A}$ and parameter $\theta \in \mathcal{O}$ lie within the $d$-dimensional unit ball. Adopting the information-theoretic framework introduced by Russo and Van Roy (2016), we analyze the information ratio, a statistic that quantifies the trade-off between the immediate regret incurred and the information gained about the optimal action. We improve upon previous results by establishing that the information ratio is bounded by $\tfrac{9}{2}d\alpha^{-2}$, where $\alpha$ is a minimax measure of the alignment between the action space $\mathcal{A}$ and the parameter space $\mathcal{O}$, and is independent of $\beta$. Using this result, we derive a bound of order $O(d/\alpha\sqrt{T \log(\beta T/d)})$ on the Bayesian expected regret of Thompson Sampling incurred after $T$ time steps. To our knowledge, this is the first regret bound for logistic bandits that depends only logarithmically on $\beta$ while being independent of the number of actions. In particular, when the action space contains the parameter space, the bound on the expected regret is of order $\tilde{O}(d \sqrt{T})$.

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