Structuring Value Representations via Geometric Coherence in Markov Decision Processes

2026-02-03

AI Summary

This work addresses the challenges of instability and low sample efficiency in value function estimation within reinforcement learning by introducing, for the first time, an order-theoretic perspective. The authors formulate value learning as a partially ordered set (poset) learning problem and propose the GCR-RL framework, which progressively refines a hyper-poset structure guided by temporal difference signals to ensure geometric consistency in value representations. Building on this foundation, they develop two novel algorithms compatible with both Q-learning and Actor-Critic architectures, accompanied by theoretical convergence guarantees. Empirical evaluations demonstrate that the proposed approach significantly improves sample efficiency and training stability across a variety of tasks, outperforming several strong baselines.

Technology Category

Machine Learning: Reinforcement LearningGame Theory and Economic Paradigms: Adversarial LearningSearch and Optimization: Learning to Search

Application Category

Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for rankingGraph Algorithms and Modeling for the Web: Graph embeddings and representation learning for Web-related graphsEconomics, Online Markets and Human Computation: Architectures and workflows that use LLMs for crowd work

Abstract

Geometric properties can be leveraged to stabilize and speed reinforcement learning. Existing examples include encoding symmetry structure, geometry-aware data augmentation, and enforcing structural restrictions. In this paper, we take a novel view of RL through the lens of order theory and recast value function estimates into learning a desired poset (partially ordered set). We propose \emph{GCR-RL} (Geometric Coherence Regularized Reinforcement Learning) that computes a sequence of super-poset refinements -- by refining posets in previous steps and learning additional order relationships from temporal difference signals -- thus ensuring geometric coherence across the sequence of posets underpinning the learned value functions. Two novel algorithms by Q-learning and by actor--critic are developed to efficiently realize these super-poset refinements. Their theoretical properties and convergence rates are analyzed. We empirically evaluate GCR-RL in a range of tasks and demonstrate significant improvements in sample efficiency and stable performance over strong baselines.

Problem

• value representation
• geometric coherence
• Markov Decision Processes
• reinforcement learning
• partially ordered set

Innovation

• Geometric Coherence
• Partially Ordered Set (Poset)
• Super-poset Refinement
• Order Theory
• Reinforcement Learning