A Contractive Feedback Semantics for Reinforcement Learning

A new theoretical paper recasts discounted reinforcement learning as a compositional system, treating one-step decision processes as open components that yield infinite-horizon policy evaluation when closed through a contractive feedback loop [1]. The work, posted to arXiv on 23 May 2026, departs from the standard presentation of Bellman equations on closed Markov decision processes [1]. Instead, it assigns typed Bellman transformers to open stochastic components and interprets series and parallel wiring as composition and tensoring of those transformers [2]. Feedback is modeled as an admissible guarded Banach trace realized by a unique fixed point [2]. The authors state that their central claim is not that all reinforcement-learning morphisms form a global traced monoidal category, but that discounted Bellman evaluation admits a contractive feedback semantics on the admissible class of guarded circuits [2]. The framework yields three theoretical consequences [1]. First, approximate component equivalence is shown to be a contextual congruence for well-typed guarded one-hole contexts: local operator error remains controlled after a component is plugged into a surrounding circuit whose feedback nodes have certified uniform guardedness [2]. Second, exact and approximate state abstractions become commuting or near-commuting coalgebraic diagrams, providing value-preservation and explicit sup-norm distortion bounds [2]. Third, under monotone ω-continuous contract-transformer semantics, safety, risk, and resource specifications can be represented as quantale-valued contracts, where local inductive bounds lift through wiring and feedback by least-fixed-point reasoning [2]. The paper does not include empirical benchmarks or numerical results; its contribution is strictly mathematical [1]. The compositional perspective builds on a long line of work that treats learning algorithms through the lens of category theory and coalgebra, though the authors limit their scope to the admissible class of guarded circuits rather than claiming a universal categorical structure [2]. The preprint has been submitted to the Machine Learning section of arXiv and is available in both PDF and experimental HTML formats [1].