SBEED: Convergent Reinforcement Learning with Nonlinear Function Approximation

TL;DR

This paper introduces SBEED, a new reinforcement learning algorithm that guarantees convergence with nonlinear function approximation by reformulating the Bellman equation into a primal-dual optimization problem.

Contribution

It presents the first convergence guarantee for nonlinear function approximation in reinforcement learning through a novel primal-dual formulation and a new algorithm, SBEED.

Findings

Favorable empirical performance on benchmark control problems

First convergence guarantee for nonlinear function approximation

Analysis of sample complexity

Abstract

When function approximation is used, solving the Bellman optimality equation with stability guarantees has remained a major open problem in reinforcement learning for decades. The fundamental difficulty is that the Bellman operator may become an expansion in general, resulting in oscillating and even divergent behavior of popular algorithms like Q-learning. In this paper, we revisit the Bellman equation, and reformulate it into a novel primal-dual optimization problem using Nesterov's smoothing technique and the Legendre-Fenchel transformation. We then develop a new algorithm, called Smoothed Bellman Error Embedding, to solve this optimization problem where any differentiable function class may be used. We provide what we believe to be the first convergence guarantee for general nonlinear function approximation, and analyze the algorithm's sample complexity. Empirically, our algorithm compares favorably to state-of-the-art baselines in several benchmark control problems.