Non-asymptotic convergence bounds of modified EM schemes for non-dissipative SDEs

2025/12/08 by Bao, Jianhai, Hao, Jiaqing, Ren, Panpan
Computer Science · Mathematics · #FOS: Mathematics #Gas Dynamics and Kinetic Theory #Markov Chains and Monte Carlo Methods #Probability (math.PR) #Stochastic Gradient Optimization Techniques

paper · doi:10.48550/arxiv.2512.07298

openalex publication_date 2025/12/08 · openalex created_date 2025/12/10 · openalex updated_date 2026/07/28

Abstract

In this paper, we address the issue on non-asymptotic convergence bounds of Euler-type schemes associated with non-dissipative SDEs. On the one hand, for non-degenerate SDEs with super-linear drifts, we propose a novel modified Euler scheme and establish the corresponding non-asymptotic convergence bound under the multiplicative type quasi-Wasserstein distance by the aid of the asymptotic reflection by coupling. As a direct application of the theory derived, we explore the non-asymptotic convergence bound of the modified tamed/truncated Euler scheme and, as a byproduct, furnish the associated non-asymptotic convergence rate under the L¹-Wasserstein distance although the dissipativity at infinity is not in force. On the other hand, we tackle the non-asymptotic convergence analysis of the Euler scheme corresponding to a kind of degenerate SDEs, where the underdamped Langevin SDE is a typical candidate. To handle such setting, we also appeal to a carefully tailored coupling approach, where the ingredient in the coupling construction lies in that a proper metric and a suitable substitute in the cut-off function and the reflection matrix need to be chosen appropriately. In addition, as a consequent application, the non-asymptotic convergence bound and the L¹-Wasserstein convergence rate are revealed for the kinetic Langevin sampler.

Citations

• kTULA: A Langevin sampling algorithm with improved KL bounds under super-linear log-gradients
• Geometric ergodicity of modified Euler schemes for SDEs with super-linearity
• L²-Wasserstein contraction of modified Euler schemes for SDEs with high diffusivity and applications
• Error estimates between SGD with momentum and underdamped Langevin diffusion
• Convergence of kinetic Langevin samplers for non-convex potentials
• Uniform-in-time estimates for mean-field type SDEs and applications
• Limit theorems for SDEs with irregular drifts
• Taming under isoperimetry
• L²-Wasserstein contraction for Euler schemes of elliptic diffusions and interacting particle systems
• Non-asymptotic convergence bounds for modified tamed unadjusted Langevin algorithm in non-convex setting
• Weak Convergence of Approximate reflection coupling and its Application to Non-convex Optimization
• Strict Kantorovich contractions for Markov chains and Euler schemes with general noise
• Non-asymptotic estimates for TUSLA algorithm for non-convex learning with applications to neural networks with ReLU activation function
• Discrete sticky couplings of functional autoregressive processes
• Exponential Ergodicity for Non-Dissipative McKean-Vlasov SDEs
• Exponential ergodicity for SDEs and McKean-Vlasov processes with Lévy noise
• Quantitative contraction rates for Markov chains on general state spaces
• An Elementary Approach To Uniform In Time Propagation Of Chaos
• Ergodicity for Neutral Type SDEs with Infinite Length of Memory
• The Tamed Unadjusted Langevin Algorithm
• Couplings and quantitative contraction rates for Langevin dynamics
• Quantitative Harris type theorems for diffusions and McKean-Vlasov\n processes
• Refined basic couplings and Wasserstein-type distances for SDEs with Lévy noises
• The truncated Euler–Maruyama method for stochastic differential equations
• Asymptotic Couplings by Reflection and Applications for Non-Linear Monotone SPDES
• Asymptotic coupling and a weak form of Harris' theorem with applications to stochastic delay equations

Related

• L²-Wasserstein contraction of modified Euler schemes for SDEs with high diffusivity and applications
• Geometric ergodicity of modified Euler schemes for SDEs with super-linearity
• A Tamed Euler Scheme for SDEs with Non-Locally Integrable Drift Coefficient
• Non-asymptotic bounds for sampling algorithms without log-concavity
• Non-asymptotic Error Analysis of Explicit Modified Euler Methods for Superlinear and Non-contractive SODEs