An agent acting under partial observability must retain a recursively updateable statistic of history that restores the Markov property, but the smallest such statistic is generally unknown. We characterize this minimal Markov sufficient statistic for holonomy-cover decision processes, a structured POMDP class in which the visible dynamics are Markov and every realized visible transition applies a fixed permutation to a hidden mode. In particular, we construct the stable quotient, the coarsest observation-wise abstraction preserving one-step rewards and quotient successors, and prove that the pair of the current observation and stable class forms an exact finite Markov state. When the current class is correctly initialized, exact class tracking requires exactly the minimal memory symbols, in the sense that under reachability and pairwise decision separation at a maximizing observation, no arbitrary finite-memory controller can use fewer. Under resettable diagnostics, nearest-prototype class inference has exponentially decaying error, and a calibrate-then-restart reduction transfers finite-MDP guarantees to the recovered state. The results enable \emph{Holonomy Memory Reinforcement Learning}. It represents memory by the current stable class, updates it through ordered edge transports, identifies local class coordinates when diagnostics are available, and applies a standard finite-MDP RL backbone after synchronization. Experiments recover an exact compression from raw states to quotient states and achieve perfect paired-order accuracy with three decision-time memory states, matching the quotient oracle and outperforming the non-oracle baselines.
Yivan Zhang, Ziyan Luo, Manuel Baltierics.LG cs.AI math.CT
State abstraction plays a key role in scaling reinforcement learning to complex but structured systems. In studying such systems, a wide range of behavioral structures have been studied in reinforcement learning, including value functions, invariants, bisimulation relations, and behavioral metrics. However, a general principle for determining what structures are provably preserved under state abstraction is still lacking. In this paper, we present a unified framework for defining and analyzing behavioral structures in reinforcement learning. Our framework provides a compositional way to specify behavioral semantics based on local, one-step descriptions of system dynamics. Using this framework, we establish results showing how behavioral structures can be safely transferred between abstract and concrete systems. We further show how to construct quantitative metrics from logical behavioral semantics with soundness guarantees. Together, these results provide a principled foundation for reasoning about behaviors under state abstraction in reinforcement learning and offer reusable definition and proof principles for a broad class of behavioral structures in reinforcement learning.
We study performance-driven environment abstraction for decision-making in large Markov decision processes. Rather than preserving geometric or topological structure, we seek abstractions that directly optimize decision quality. We model abstraction as a controlled approximation obtained by aggregating the state space and enforcing a shared action distribution within each aggregated state. For a fixed partition, we establish a performance guarantee that separates value-function approximation error from the loss introduced by action sharing. Guided by this analysis, we develop a multi-timescale reinforcement learning framework that jointly adapts the policy and a tree-structured environment abstraction. The resulting algorithm refines and coarsens regions of the state space based on Q-value discrepancies, balancing performance against abstraction size and complexity. Empirical results demonstrate substantial state compression, improved sample efficiency, and faster replanning compared to actor-critic baselines.
When learning to walk, infants seem to address a coarse version of the problem first - stay upright, reach the caregiver - and refine it only when further practice at that resolution stops paying off. Reinforcement learning offers multiple techniques for building simple versions of complex tasks, but lacks general principles for how to dynamically adjust the granularity of these abstractions during learning. This paper proposes one such principle: refine the abstraction as soon as the learning error within it becomes comparable to the error induced by the abstraction itself. Here, we investigate one way of formalising this principle via a performance certificate that decomposes value error into two terms: a learning error bound captured by a Bellman residual, and an abstraction error bound given by a bisimulation metric. The resulting switching strategy is implemented by soft state-action abstractions built from rate-distortion principles, whose resolution along state and action axes can be continuously adjusted. We validate this construction in a range of tabular settings, showing that near-optimal performance can be achieved under substantial lossy compression of state and action information.