Contraction-Aligned Analysis of Soft Bellman Residual Minimization with Weighted Lp-Norm for Markov Decision Problem

TL;DR

This paper introduces a soft Bellman residual minimization approach using weighted Lp-norms that aligns with the contraction properties of the Bellman operator, improving error control in Markov decision processes.

Contribution

It extends residual minimization to a generalized weighted Lp-norm, connecting it with Bellman contraction geometry for better optimization and error bounds.

Findings

Alignment of residual minimization with Bellman contraction as p increases

Derivation of performance error bounds for the proposed method

Enhanced control of error propagation in policy evaluation

Abstract

The problem of solving Markov decision processes under function approximation remains a fundamental challenge, even under linear function approximation settings. A key difficulty arises from a geometric mismatch: while the Bellman optimality operator is contractive in the Linfty-norm, commonly used objectives such as projected value iteration and Bellman residual minimization rely on L2-based formulations. To enable gradient-based optimization, we consider a soft formulation of Bellman residual minimization and extend it to a generalized weighted Lp -norm. We show that this formulation aligns the optimization objective with the contraction geometry of the Bellman operator as p increases, and derive corresponding performance error bounds. Our analysis provides a principled connection between residual minimization and Bellman contraction, leading to improved control of error propagation while remaining compatible with gradient-based optimization.