Rationalizing Boltzmann Rationality: An Axiomatic Characterization of Entropy-Regularized Policies

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.

Entropy regularization is widely used in continuous-time reinforcement learning (RL) to reduce sensitivity to environmental perturbations, yet its robustness benefits lack a rigorous theoretical foundation. This paper establishes the first robustness guarantees for entropy-regularized continuous-time Markov decision processes. We show that maximizing an entropy-regularized objective yields a lower bound on a worst-case robust RL problem with joint reward and transition perturbations. We analytically characterize the induced robust sets and prove that they expand monotonically with the regularization strength, justifying the empirical observation that stronger entropy improves robustness. In contrast to prior discrete-time analyses, our results remove the intractable state-distribution entropy term and provide guarantees invariant to action frequency. Experiments on queueing network control and market making confirm our theory, showing that entropy-regularized policies outperform greedy and $ε$-greedy baselines under dynamics perturbations.

Inverse reinforcement learning aims to infer the reward function that explains expert behavior observed through trajectories of state--action pairs. A long-standing difficulty in classical IRL is the non-uniqueness of the recovered reward: many reward functions can induce the same optimal policy, rendering the inverse problem ill-posed. In this paper, we develop a statistical framework for Inverse Entropy-regularized Reinforcement Learning that resolves this ambiguity by combining entropy regularization with a least-squares reconstruction of the reward from the soft Bellman residual. This combination yields a unique and well-defined so-called least-squares reward consistent with the expert policy. We model the expert demonstrations as a Markov chain with the invariant distribution defined by an unknown expert policy $π^\star$ and estimate the policy by a penalized maximum-likelihood procedure over a class of conditional distributions on the action space. We establish high-probability bounds for the excess Kullback--Leibler divergence between the estimated policy and the expert policy, accounting for statistical complexity through covering numbers of the policy class. These results lead to non-asymptotic minimax optimal convergence rates for the least-squares reward function, revealing the interplay between smoothing (entropy regularization), model complexity, and sample size. Our analysis bridges the gap between behavior cloning, inverse reinforcement learning, and modern statistical learning theory.