Error estimates for tamed Euler and Randomized Euler schemes for SDEs with locally Lipschitz drift with applications to non-logconcave sampling and optimization

2026-05-24

AI Summary

This work addresses the numerical instability of discretizing stochastic differential equations with locally Lipschitz drift exhibiting superlinear growth, and the associated challenges in non-log-concave sampling and non-convex optimization. The authors propose a truncated stochastic midpoint scheme (tRLMC) and analyze a KL-accelerated truncated unadjusted Langevin algorithm (kTULA). Leveraging a shift-composition technique, they develop two novel local error analysis frameworks, establishing—for the first time—non-asymptotic convergence guarantees in total variation for truncated Langevin-type algorithms under superlinear drift. The theoretical results show that kTULA achieves a near-optimal iteration complexity of $\widetilde{O}(\varepsilon^{-1/2})$ in KL divergence, while tRLMC attains $\widetilde{O}(\varepsilon^{-1})$ complexity in both total variation and Wasserstein distance, yielding non-asymptotic bounds on excess risk for non-convex optimization.

Abstract

In this paper, we study the numerical discretization of stochastic differential equations with locally Lipschitz, super-linearly growing drift, and the resulting implications for sampling from non-log-concave distributions satisfying a logarithmic Sobolev inequality. In this regime, the classical Euler--Maruyama scheme underlying the unadjusted Langevin algorithm (ULA) is known to be unstable. We analyze the KL-accelerated tamed unadjusted Langevin algorithm (kTULA) and introduce a new tamed randomized midpoint scheme, termed tRLMC. Building on the shifted-composition approach of \cite{chewi2024local}, we develop two new local-error frameworks that yield finite-time, non-asymptotic error estimates against the underlying SDE -- in KL divergence for kTULA, and in total variation for tRLMC -- valid for general locally Lipschitz drift. Specializing these frameworks to the sampling problem under a logarithmic Sobolev inequality, we obtain a near-optimal $\widetilde{O}(\varepsilon^{-1/2})$ iteration complexity for kTULA in KL divergence, with corresponding guarantees in total variation and Wasserstein distance. We further establish, for the first time, a non-asymptotic guarantee in total variation for a tamed randomized Langevin scheme under super-linear drift growth, together with the corresponding Wasserstein-distance bound, both with $\widetilde{O}(\varepsilon^{-1})$ complexity for tRLMC. As a consequence, both schemes yield non-asymptotic bounds for a non-convex excess-risk optimization problem.

Problem

• stochastic differential equations
• locally Lipschitz drift
• non-logconcave sampling
• numerical stability
• error estimates

Innovation

• tamed Euler scheme
• randomized midpoint
• non-logconcave sampling
• local error analysis
• logarithmic Sobolev inequality