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

• URL: http://arxiv.org/abs/2605.11289v1
• Date: Mon, 11 May 2026 22:17:09 GMT
• Title: Quotient-Categorical Representations for Bellman-Compatible Average-Reward Distributional Reinforcement Learning
• Abstract summary: We introduce a quotient-space formulation in which state-indexed bias laws are identified up to a common translation.<n>We show that synchronous exact updates are gain-independent at the quotient-law level.
• Score: 8.988097534171995
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: Average-reward reinforcement learning requires estimating the gain and the bias, which is defined only up to an additive constant. This makes direct distributional analogues ill-posed on the real line. We introduce a quotient-space formulation in which state-indexed bias laws are identified up to a common translation, together with a categorical parameterization that respects this symmetry. On this quotient-categorical space, we define a projected average-reward distributional operator and show that it is well-defined, non-expansive in a coordinate Cramér metric, and admits fixed points. We then study sampled recursions whose mean-field maps are asynchronous relaxations of this operator. In an idealized centered-reward setting, a one-state temporal-difference update enjoys almost sure convergence together with finite-iteration residual bounds under both i.i.d. and Markovian sampling. When the gain is unknown, we augment the recursion with an online gain estimator, and prove non-expansiveness and Markovian convergence of the resulting coupled scheme. Finally, we show that synchronous exact updates are gain-independent at the quotient-law level, isolating a structural contrast between ideal quotient distributions and practical fixed-grid categorical representations.

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