# SBEED: Convergent Reinforcement Learning with Nonlinear Function Approximation

arXiv: https://arxiv.org/abs/1712.10285 · HTML: https://mlsanity.com/paper/1712.10285/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 29 Dec 2017.

## Built on by (most cited first, 4 of 4)

- [Offline Reinforcement Learning: Tutorial, Review, and Perspectives on Open Problems](https://mlsanity.com/paper/2005.01643/) (2020)
- [On Computation and Generalization of Generative Adversarial Imitation Learning](https://mlsanity.com/paper/2001.02792/) (2020)
- [Overcoming the Curse of Dimensionality in Reinforcement Learning Through Approximate Factorization](https://mlsanity.com/paper/2411.07591/) (2024)
- [Multi-level Monte-Carlo Gradient Methods for Stochastic Optimization with Biased Oracles](https://mlsanity.com/paper/2408.11084/) (2024)

## Abstract (arXiv)

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.