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

arXiv: https://arxiv.org/abs/2604.06837 · HTML: https://mlsanity.com/paper/2604.06837/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 5 of its 25 references that are in the map.
- First posted on arXiv 8 Apr 2026.

## Builds on (most cited first, 5 of 5)

- [Reinforcement Learning with Deep Energy-Based Policies](https://mlsanity.com/paper/1702.08165/) (2017)
- [Why Should I Trust You, Bellman? The Bellman Error is a Poor Replacement for Value Error](https://mlsanity.com/paper/2201.12417/) (2022)
- [Regularized Q-learning](https://mlsanity.com/paper/2202.05404/) (2022)
- [A Smooth Polynomial Lyapunov Certificate for Convergence of Q-Learning and Its Smooth Variants](https://mlsanity.com/paper/2404.14442/) (2024)
- [Bellman Residual Minimization for Control: Geometry, Stationarity, and Convergence](https://mlsanity.com/paper/2601.18840/) (2026)

## Abstract (arXiv)

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.