A Contractive Feedback Semantics for Reinforcement Learning

2026-05-23

AI Summary

This work addresses the limitations of traditional reinforcement learning in semantically characterizing open component composition and feedback loops, which hinders modular analysis and error control. The paper proposes a compositional semantic framework grounded in contractive feedback, modeling single-step decisions as typed open stochastic components endowed with Bellman transformer semantics. This framework supports sequential, parallel, and feedback compositions. By leveraging the Banach fixed-point theorem, coalgebraic state abstraction, and quantitative contract mechanisms, the approach ensures bounded propagation of local errors within composite environments, guaranteeing value preservation and a sup-norm distortion bound. Furthermore, it enables safety and resource specification reasoning based on least fixed points.

Abstract

Discounted reinforcement learning is usually presented through Bellman equations on closed Markov decision processes. This paper develops a compositional view: a one-step decision process is treated as an open stochastic component, and infinite-horizon policy evaluation is obtained by closing a contractive feedback loop. The resulting semantics assigns typed Bellman transformers to open components, interprets series and parallel wiring as composition and tensoring of transformers, and interprets feedback as an admissible guarded Banach trace realized by a unique fixed point. This perspective yields three theoretical consequences. First, approximate component equivalence is a contextual congruence for admitted well-typed guarded one-hole contexts: local operator error remains controlled after plugging the component into a surrounding circuit that uses the hole once and whose feedback nodes have certified uniform guardedness. Second, exact and approximate state abstractions become commuting or near-commuting coalgebraic diagrams, giving value-preservation and explicit sup-norm distortion bounds. 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. Its central claim is not that all RL morphisms form a global traced monoidal category, but that discounted Bellman evaluation admits a contractive feedback semantics on the admissible class of guarded circuits.

Problem

• reinforcement learning
• compositional semantics
• contractive feedback
• Markov decision processes
• Bellman equations

Innovation

• contractive feedback
• compositional semantics
• Bellman transformer
• guarded trace
• quantale-valued contracts