Closing the Log Gap: Minimax-Optimal Regret for Episodic RL

We study regret minimization in finite-horizon episodic Markov Decision Processes (MDPs). While minimax-optimal algorithms are known, tractable approaches to instance-dependent optimality are still lacking. Motivated by this gap, we introduce $π_0$-PSRL, a variant of PSRL that uses posterior samples to decide when exploration is needed, while following a fixed reference policy $π_0$ during exploration episodes. This decouples the test for exploration, triggered when the sampled and empirical MDPs have different optimal policies, from the choice of the policy used to gather information. The resulting design addresses a limitation of standard PSRL, where the sampled optimal policy may not be the optimal choice for exploration. We prove instance-dependent regret bounds for $π_0$-PSRL, identifying the logarithmic exploration cost induced by $π_0$ and taking a step toward matching asymptotic lower bounds for episodic RL. Our proof techniques showcase a novel proof structure to derive problem-dependent regret bounds in episodic MDPs, and concentration results for Dirichlet random variables, that may be of independent interest.

We study reinforcement learning for episodic Markov Decision Processes (MDPs) whose transitions are modelled by a multinomial logistic (MNL) model. Existing algorithms for MNL mixture MDPs yield a regret of $\smash{\tilde{O}(dH^2\sqrt{T})}$ (Li et al., 2024), where $d$ is the feature dimension, $H$ the episode length, and $T$ the number of episodes. Inspired by the logistic bandit literature (Abeille et al., 2021; Faury et al., 2022; Boudart et al., 2026), we introduce a problem-dependent constant $\barσ\_T \leq 1/2$, measuring the normalised average variance of the optimal downstream value function along the learner's trajectory. We propose an algorithm achieving a regret of $\smash{\tilde{O}(dH^2\barσ\_T\sqrt{T})}$, which recovers the existing bound in the worst case and improves upon it for structured MDPs. For instance, for KL-constrained robust MDPs, $\barσ\_T = O(H^{-1})$, reducing the horizon dependence by a factor $H$. We further establish a matching $\smash{Ω(dH^2\barσ\_T\sqrt{T})}$ lower bound, proving minimax optimality (up to logarithmic factors) and fully characterising the regret complexity of MNL mixture MDPs for the first time.

Online reinforcement learning in infinite-horizon Markov decision processes (MDPs) remains less theoretically and algorithmically developed than its episodic counterpart, with many algorithms suffering from high ``burn-in'' costs and failing to adapt to benign instance-specific complexity. In this work, we address these shortcomings for two infinite-horizon objectives: the classical average-reward regret and the $γ$-regret. We develop a single tractable UCB-style algorithm applicable to both settings, which achieves the first optimal variance-dependent regret guarantees. Our regret bounds in both settings take the form $\tilde{O}( \sqrt{SA\,\text{Var}} + \text{lower-order terms})$, where $S,A$ are the state and action space sizes, and $\text{Var}$ captures cumulative transition variance. This implies minimax-optimal average-reward and $γ$-regret bounds in the worst case but also adapts to easier problem instances, for example yielding nearly constant regret in deterministic MDPs. Furthermore, our algorithm enjoys significantly improved lower-order terms for the average-reward setting. With prior knowledge of the optimal bias span $\Vert h^\star\Vert_\text{sp}$, our algorithm obtains lower-order terms scaling as $\Vert h^\star\Vert_\text{sp} S^2 A$, which we prove is optimal in both $\Vert h^\star\Vert_\text{sp}$ and $A$. Without prior knowledge, we prove that no algorithm can have lower-order terms smaller than $\Vert h^\star \Vert_\text{sp}^2 S A$, and we provide a prior-free algorithm whose lower-order terms scale as $\Vert h^\star\Vert_\text{sp}^2 S^3 A$, nearly matching this lower bound. Taken together, these results completely characterize the optimal dependence on $\Vert h^\star\Vert_\text{sp}$ in both leading and lower-order terms, and reveal a fundamental gap in what is achievable with and without prior knowledge.