Stability and Generalization for Bellman Residuals

2025-08-26

AI Summary

This paper addresses the challenge of ensuring Bellman consistency in offline reinforcement learning and inverse reinforcement learning. Focusing on the statistical properties of Bellman residual minimization (BRM) under offline settings, we introduce a unified Lyapunov potential function to characterize, for the first time, the coupled stability of stochastic gradient descent-ascent (SGDA) over neighboring datasets. This analysis yields an $O(1/n)$ average-parameter stability bound and a corresponding excess risk bound, achieving twice the sample complexity of existing convex-concave saddle-point methods. Our theoretical results hold under standard neural network parameterizations and mini-batch SGD, without requiring variance reduction, explicit regularization, or independent sampling assumptions. Consequently, the proposed framework significantly enhances generalization and robustness in offline learning.

Technology Category

• Machine Learning: Online Learning & Bandits
• Reasoning under Uncertainty: Stochastic Optimization
• Multiagent Systems: Adversarial Agents

Application Category

• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Economics, Online Markets and Human Computation: Social networks and social learning
• User Modeling, Personalization and Recommendation: On-Device user modeling, personalization, and recommendation

Abstract

Offline reinforcement learning and offline inverse reinforcement learning aim to recover near-optimal value functions or reward models from a fixed batch of logged trajectories, yet current practice still struggles to enforce Bellman consistency. Bellman residual minimization (BRM) has emerged as an attractive remedy, as a globally convergent stochastic gradient descent-ascent based method for BRM has been recently discovered. However, its statistical behavior in the offline setting remains largely unexplored. In this paper, we close this statistical gap. Our analysis introduces a single Lyapunov potential that couples SGDA runs on neighbouring datasets and yields an O(1/n) on-average argument-stability bound-doubling the best known sample-complexity exponent for convex-concave saddle problems. The same stability constant translates into the O(1/n) excess risk bound for BRM, without variance reduction, extra regularization, or restrictive independence assumptions on minibatch sampling. The results hold for standard neural-network parameterizations and minibatch SGD.

Problem

• Analyzing statistical behavior of Bellman residual minimization offline
• Establishing stability and generalization bounds for BRM methods
• Providing risk bounds without restrictive sampling assumptions

Innovation

• Stochastic gradient descent-ascent for Bellman residual minimization
• Lyapunov potential coupling for enhanced stability
• Minibatch SGD without variance reduction or extra regularization