Quotient-Categorical Representations for Bellman-Compatible Average-Reward Distributional Reinforcement Learning

Recent non-asymptotic analyses have substantially advanced the theory of distributional policy evaluation, but they largely concern synchronous full-state updates under a generative model, model-based estimators, accelerated variants, or different approximation architectures. Standard categorical temporal-difference learning is typically used in a different regime. It asynchronously performs a single-state update at each iteration and, in online settings, is driven by a Markovian trajectory. This leaves an important gap between existing finite-iteration theory and the categorical recursions most closely aligned with practical distributional temporal-difference implementations. We bridge this gap for two categorical policy-evaluation methods: scalar categorical temporal-difference learning in the Cramér geometry and multivariate signed-categorical temporal-difference learning in the maximum mean discrepancy geometry. After suitable isometric embeddings, both algorithms take the form of asynchronous single-state stochastic-approximation recursions that contract in a statewise supremum norm. This permits finite-iteration guarantees in discounted problems under both i.i.d. and Markovian state sampling, and in undiscounted fixed-horizon problems under i.i.d. episodic sampling.

We study model-free methods for distributionally robust infinite-horizon average-reward Markov decision processes (MDPs). We present non-asymptotic convergence analyses of Q-learning and actor-critic algorithms for robust average-reward MDPs under contamination, total-variation distance, and Wasserstein uncertainty sets. A key ingredient of our analysis is showing that the optimal robust Bellman operator is a strict contraction with respect to a carefully designed semi-norm. This property enables a stochastic approximation update that learns the optimal robust $Q$-function with $\tilde{\mathcal{O}}(ε^{-2})$ dependence on the target accuracy. We also establish robust TD convergence bounds whose constants are uniform over all stationary policies, yielding an efficient data-driven routine for robust critic estimation. Building on this, we introduce an actor-critic algorithm that learns an $ε$-optimal robust policy with $\tilde{\mathcal{O}}(ε^{-2})$ dependence on the target accuracy. We provide numerical simulations to illustrate the qualitative behavior of the proposed algorithms. Our results contribute to the theoretical foundations of robust planning under model misspecification and to model-free approaches for building robust long-run policies directly from simulation data.

Average-reward reinforcement-learning regret is known up to logarithmic factors, but the numerical content of published guarantees is difficult to compare because probability mode, structural parameter, logarithmic normalization, prior information, and planning assumptions differ. We introduce a constant-aware comparison protocol and derive an explicit finite lower certificate for communicating MDPs. The construction is a binary tree of two-state blocks; its proof uses exact trajectory-level Bernoulli KL divergence and keeps action budget, diameter, occupancy, navigation cost, and terminal bias explicit. A common closed-form envelope improves the published coefficient $0.015$ across a finite frontier: $0.0200$ in a moderate regime and up to $0.0291$ under stronger action, diameter, and horizon conditions, a $94\%$ increase. The limiting coefficient is $\frac1{32}\sqrt{(A-3)/A}$. For upper bounds, we give an auditable composition rule for a span-constrained optimistic learner, but do not claim a coefficient while adaptive directional-variance and planning certificates remain open. We also formalize valid expectation conversion and constant comparability. Controlled diagnostics test diameter dependence, bonus-by-width interactions, span misspecification, and the finite lower certificate on its exact family.