Online Statistical Inference of Constant Sample-averaged Q-Learning

In this paper, we study online statistical inference for quantile-based distributional policy evaluation, which aims to characterize the distribution of discounted cumulative rewards under a given policy. To obtain a finite-dimensional representation of the return distribution, we consider the fixed point $\bmη_m$ of the quantile-projected distributional Bellman equation, parameterized by the quantile vector $\bmθ_m$. We develop two online inference procedures for $\bmθ_m$, covering synchronous and asynchronous sampling from a generative model as well as Markovian data. The first method uses the trajectory of quantile temporal-difference learning (QTD). We establish functional central limit theorems for the partial sums of QTD iterates and construct a recursively computable random-scaling statistic. The resulting confidence intervals have asymptotically exact coverage at deterministic horizons, with length of order $T^{-1/2}$ in probability. The second method constructs anytime-valid confidence intervals directly from samples. Exploiting the monotonicity of the quantile Bellman map, we develop a computationally explicit coordinate-shaving algorithm with finite-sample simultaneous coverage. Under positive limiting state-visit frequencies, the resulting confidence intervals have lengths of order $\sqrt{\log\log T/T}$, which we establish as classwise optimal through a matching lower bound. Together, these results provide complementary guarantees for online inference in quantile distributional reinforcement learning, combining asymptotically exact fixed-horizon inference with finite-sample anytime validity.

We investigate the statistical properties of Temporal Difference (TD) learning with Polyak-Ruppert averaging, arguably one of the most widely used algorithms in reinforcement learning, for the task of estimating the parameters of the optimal linear approximation to the value function. Assuming independent samples, we make three theoretical contributions that improve upon the current state-of-the-art results: (i) we establish refined high-dimensional Berry-Esseen bounds over the class of convex sets, achieving faster rates than the best known results, and (ii) we propose and analyze a novel, computationally efficient online plug-in estimator of the asymptotic covariance matrix; (iii) we derive sharper high probability convergence guarantees that depend explicitly on the asymptotic variance and hold under weaker conditions than those adopted in the literature. These results enable the construction of confidence regions and simultaneous confidence intervals for the linear parameters of the value function approximation, with guaranteed finite-sample coverage. We demonstrate the applicability of our theoretical findings through numerical experiments.

We study the finite-sample convergence of Q-learning algorithms for the average-reward formulation of Reinforcement Learning (RL) with options. While options allow temporarily extended and hierarchical actions, the average-reward formulation incorporates continual tasks beyond discounted and episodic RL. We establish the first $\tilde{O}(1/\text{iterations})$ convergence guarantees, in the mean-square sense, for average-reward option Q-learning algorithms in both the inter- and intra-option settings. First, we prove that both inter- and intra-option Bellman operators satisfy multi-step span contractions under unichain assumptions. Then, we propose synchronous estimators of these multi-step option Bellman operators leading to two variants of option Q-learning. Finally, we extend the stochastic approximation framework to prove convergence rates of these two algorithms under the minimal assumptions used in the literature.