Stability and Generalization for Bellman Residuals

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.

Gradients can train reward models: An Empirical Risk Minimization Approach for Offline Inverse RL and Dynamic Discrete Choice Model

We study the problem of estimating Dynamic Discrete Choice (DDC) models, also known as offline Maximum Entropy-Regularized Inverse Reinforcement Learning (offline MaxEnt-IRL) in machine learning. The objective is to recover reward or $Q^*$ functions that govern agent behavior from offline behavior data. In this paper, we propose a globally convergent gradient-based method for solving these problems without the restrictive assumption of linearly parameterized rewards. The novelty of our approach lies in introducing the Empirical Risk Minimization (ERM) based IRL/DDC framework, which circumvents the need for explicit state transition probability estimation in the Bellman equation. Furthermore, our method is compatible with non-parametric estimation techniques such as neural networks. Therefore, the proposed method has the potential to be scaled to high-dimensional, infinite state spaces. A key theoretical insight underlying our approach is that the Bellman residual satisfies the Polyak-Lojasiewicz (PL) condition -- a property that, while weaker than strong convexity, is sufficient to ensure fast global convergence guarantees. Through a series of synthetic experiments, we demonstrate that our approach consistently outperforms benchmark methods and state-of-the-art alternatives.