Vanishing L2 regularization for the softmax Multi Armed Bandit

Algorithms for the Multi-Armed Bandit (MAB) problem play a central role in sequential decision-making and have been extensively explored both theoretically and numerically. While most classical approaches aim to identify the arm with the highest expected reward, we focus on a risk-aware setting where the goal is to select the arm with the lowest variance, favoring stability over potentially high but uncertain returns. To model the decision process, we consider a softmax parameterization of the policy; we propose a new algorithm to select the minimal variance (or minimal risk) arm and prove its convergence under natural conditions. The algorithm constructs an unbiased estimate of the objective by using two independent draws from the selected arm's distribution. We provide numerical experiments that illustrate the practical behavior of these algorithms and offer guidance on implementation choices. The setting also covers general risk-aware problems where there is a trade-off between maximizing the average reward and minimizing its variance.

Reinforcement learning with verifiable rewards (RLVR) and group-based policy optimization methods such as GRPO update a stochastic policy by sampling multiple completions per prompt and increasing the policy's probability on those with higher reward, regularized by a KL penalty toward a reference policy. These updates do not include explicit mechanisms that track epistemic uncertainty. This paper studies a stylized explanation for why such uncertainty-agnostic updates can nevertheless be effective. We analyze an annealed softmax (Boltzmann) policy that selects actions according to a softmax of empirical mean rewards in a many-armed Bayesian Bernoulli bandit. Under a linear upper-tail condition on the prior (the $β=1$ case of $β$-regularity), which implies an abundance of near-optimal arms, we prove that annealed softmax greedy achieves Bayes regret $\tilde{O}(m + T/m)$, and in particular $\tilde{O}(\sqrt{T})$ when the number of arms scales as $m = Θ(\sqrt{T})$. This is the near-optimal Bayes regret rate in this regime, attained also by empirical-mean greedy. Under $β$-regularity, many arms maintain empirical means close to the optimum throughout learning, so when softmax samples an arm other than the empirically best, that arm tends to be another near-optimal one rather than a clearly inferior one. By contrast, with a small number of arms, the same kind of softmax policy can suffer linear regret. The result also provides a structural analogy to RLVR, where a base policy with a non-negligible probability of producing a correct completion plays the role of $β$-regularity.

Kullback-Leibler (KL) regularization is widely used in offline decision-making and offers several benefits, motivating recent work on the sample complexity of offline learning with respect to KL-regularized performance metrics. Nevertheless, the exact sample complexity of KL-regularized offline learning remains largely from fully characterized. In this paper, we study this question in the setting of multi-armed bandits (MABs). We provide a sharp analysis of KL-PCB (Zhao et al., 2026), showing that it achieves a sample complexity of $\tilde{O}(ηSAC^{π^*}/ε)$ under large regularization $η= \tilde{O}(ε^{-1})$, and a sample complexity of $\tildeΩ(SAC^{π^*}/ε^2)$ under small regularization $η= \tildeΩ(ε^{-1})$, where $η$ is the regularization parameter, $S$ is the number of contexts, $A$ is the number of arms, $C^{π^*}$ policy coverage coefficient at the optimal policy $π^*$, $ε$ is the desired sub-optimality, and $\tilde{O}$ and $\tildeΩ$ hide all poly-logarithmic factors. We further provide a pair of sharper sample complexity lower bounds, which matches the upper bounds over the entire range of regularization strengths. Overall, our results provide a nearly complete characterization of offline multi-armed bandits with KL regularization.