The Geometry of Robust Value Functions

TL;DR

This paper explores the geometric structure of robust value functions in reinforcement learning, revealing how transition uncertainties shape the value space and providing insights for optimization in robust MDPs.

Contribution

It introduces a novel geometric perspective on the robust value space, characterizing it as a union of conic hypersurfaces and simplifying analysis by focusing on extreme points.

Findings

Robust value space is determined by conic hypersurfaces.

Extreme points in the uncertainty set suffice for characterization.

The robust value space exhibits non-convexity and policy agreement on multiple states.

Abstract

The space of value functions is a fundamental concept in reinforcement learning. Characterizing its geometric properties may provide insights for optimization and representation. Existing works mainly focus on the value space for Markov Decision Processes (MDPs). In this paper, we study the geometry of the robust value space for the more general Robust MDPs (RMDPs) setting, where transition uncertainties are considered. Specifically, since we find it hard to directly adapt prior approaches to RMDPs, we start with revisiting the non-robust case, and introduce a new perspective that enables us to characterize both the non-robust and robust value space in a similar fashion. The key of this perspective is to decompose the value space, in a state-wise manner, into unions of hypersurfaces. Through our analysis, we show that the robust value space is determined by a set of conic hypersurfaces, each of which contains the robust values of all policies that agree on one state. Furthermore, we find that taking only extreme points in the uncertainty set is sufficient to determine the robust value space. Finally, we discuss some other aspects about the robust value space, including its non-convexity and policy agreement on multiple states.