Near-Optimal Regret for KL-Regularized Multi-Armed Bandits
This work provides a more complete theoretical understanding of KL-regularized multi-armed bandits, which is important for researchers and practitioners working on reinforcement learning with regularized objectives.
This paper addresses the statistical efficiency of online learning with KL-regularized objectives in multi-armed bandits (MABs). The authors provide a $\tilde{O}(\eta K\log^2T)$ upper bound for KL-UCB, which is the first high-probability regret bound with linear dependence on $K$, and a matching $\Omega(\eta K \log T)$ lower bound. In the low-regularization regime, they show the regret scales as $\tilde{\Theta}(\sqrt{KT})$.
Recent studies have shown that reinforcement learning with KL-regularized objectives can enjoy faster rates of convergence or logarithmic regret, in contrast to the classical $\sqrt{T}$-type regret in the unregularized setting. However, the statistical efficiency of online learning with respect to KL-regularized objectives remains far from completely characterized, even when specialized to multi-armed bandits (MABs). We address this problem for MABs via a sharp analysis of KL-UCB using a novel peeling argument, which yields a $\tilde{O}(ηK\log^2T)$ upper bound: the first high-probability regret bound with linear dependence on $K$. Here, $T$ is the time horizon, $K$ is the number of arms, $η^{-1}$ is the regularization intensity, and $\tilde{O}$ hides all logarithmic factors except those involving $\log T$. The near-tightness of our analysis is certified by the first non-constant lower bound $Ω(ηK \log T)$, which follows from subtle hard-instance constructions and a tailored decomposition of the Bayes prior. Moreover, in the low-regularization regime (i.e., large $η$), we show that the KL-regularized regret for MABs is $η$-independent and scales as $\tildeΘ(\sqrt{KT})$. Overall, our results provide a thorough understanding of KL-regularized MABs across all regimes of $η$ and yield nearly optimal bounds in terms of $K$, $η$, and $T$.