Improved Bounds for Discretization of Langevin Diffusions: Near-Optimal Rates without Convexity
This provides a near-optimal theoretical foundation for sampling and learning algorithms, improving methods based on Dalayan's approach, though it is incremental in nature.
The paper tackles the problem of analyzing the Euler-Maruyama discretization of Langevin diffusions without requiring global contractivity, achieving an improved rate from O(η) to O(η^2) in KL divergence with polynomial time dependence.
We present an improved analysis of the Euler-Maruyama discretization of the Langevin diffusion. Our analysis does not require global contractivity, and yields polynomial dependence on the time horizon. Compared to existing approaches, we make an additional smoothness assumption, and improve the existing rate from $O(η)$ to $O(η^2)$ in terms of the KL divergence. This result matches the correct order for numerical SDEs, without suffering from exponential time dependence. When applied to algorithms for sampling and learning, this result simultaneously improves all those methods based on Dalayan's approach.