首次给出对数几率老虎机中汤普森采样的对数β依赖性理论分析

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, $\exp(β\langle a, θ\rangle)/(1+\exp(β\langle a, θ\rangle))$, with slope parameter $β>0$, and where both the action $a\in \mathcal{A}$ and parameter $θ\in \mathcal{O}$ 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 $\tfrac{9}{2}dα^{-2}$, where $α$ is a minimax measure of the alignment between the action space $\mathcal{A}$ and the parameter space $\mathcal{O}$, and is independent of $β$. Using this result, we derive a bound of order $O(d/α\sqrt{T \log(βT/d)})$ 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 $β$ 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 $\tilde{O}(d \sqrt{T})$.