Geometric Theory of Decision Boundaries in Structured MDPs
A Geometric Theory of Decision Boundaries in Structured Markov Decision Processes
This paper provides a new theoretical foundation for understanding and improving the efficiency of decision-making algorithms in structured environments, which is relevant for fields like robotics and reinforcement learning.
This academic paper develops a geometric theory for analyzing structured Markov Decision Processes (MDPs).
It proposes that the geometry of decision boundaries, rather than the size of the state space, governs the complexity of policy reconstruction.
The paper introduces intrinsic notions of boundary and decision complexity, and derives information-theoretic measures for decision compression.
Controlled numerical experiments validate the theoretical framework.
A Geometric Theory of Decision Boundaries in Structured Markov Decision Processes
Classical dynamic programming represents optimal sequential decisions through value functions and policies. While this functional representation is natural for computing optimal decisions, it does not directly identify the mathematical object governing policy reconstruction, representation complexity, or oracle-query complexity once an optimal policy is fixed. This paper addresses this question by developing a geometric theory of structured optimal policies in which the decision-boundary geometry induced by the policy becomes the primary object of analysis. We show that, under suitable structural regularity conditions, this geometry provides the minimal representation required for policy reconstruction and determines the statistical and computational complexity of the reconstruction problem. Building upon this representation, we establish structural properties of policy-induced decision geometry, introduce intrinsic notions of boundary and decision complexity, derive information-theoretic measures of decision compression, and obtain statistical guarantees for boundary estimation and policy reconstruction from black-box policy queries. Collectively, these results demonstrate that, for the structured decision problems considered here, the complexity of policy reconstruction is governed by the geometry of the decision boundary rather than by the cardinality of the ambient state space. Controlled numerical experiments examine the principal theoretical predictions and provide empirical evidence consistent with the proposed framework.
This report is categorized under AITOP's "Papers" desk, focusing on US frontier AI labs and Silicon Valley technology leaders. Within our cross-regional analytical framework, it directly maps to AI developer toolchains, coding agents, and engineering workflows. Key entities and technical topics under continuous tracking include structured MDPs, decision boundaries, reinforcement learning, geometric theory.
Source attribution: curated from arXiv cs.LG (@Fredy Pokou).