# Non-asymptotic bounds for sampling algorithms without log-concavity

arXiv: https://arxiv.org/abs/1808.07105 · HTML: https://mlsanity.com/paper/1808.07105/

## Key facts

- Cited by 35 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 21 Aug 2018.

## Built on by (most cited first, 30 of 35)

- [Convergence for score-based generative modeling with polynomial complexity](https://mlsanity.com/paper/2206.06227/) (2022)
- [Towards a Theory of Non-Log-Concave Sampling: First-Order Stationarity Guarantees for Langevin Monte Carlo](https://mlsanity.com/paper/2202.05214/) (2022)
- [Fisher information lower bounds for sampling](https://mlsanity.com/paper/2210.02482/) (2022)
- [Asymptotic bias of inexact Markov Chain Monte Carlo methods in high dimension](https://mlsanity.com/paper/2108.00682/) (2021)
- [Wasserstein distance estimates for the distributions of numerical approximations to ergodic stochastic differential equations](https://mlsanity.com/paper/2104.12384/) (2021)
- [Taming under isoperimetry](https://mlsanity.com/paper/2311.09003/) (2023)
- [Tamed Langevin sampling under weaker conditions](https://mlsanity.com/paper/2405.17693/) (2024)
- [Implicit Regularisation in Diffusion Models: An Algorithm-Dependent Generalisation Analysis](https://mlsanity.com/paper/2507.03756/) (2025)
- [Heavy-tailed Sampling via Transformed Unadjusted Langevin Algorithm](https://mlsanity.com/paper/2201.08349/) (2022)
- [Constrained Langevin Algorithms with L-mixing External Random Variables](https://mlsanity.com/paper/2205.14192/) (2022)
- [Convergence of the Inexact Langevin Algorithm in KL Divergence with Application to Score-based Generative Models](https://mlsanity.com/paper/2211.01512/) (2022)
- [Fast Conditional Mixing of MCMC Algorithms for Non-log-concave Distributions](https://mlsanity.com/paper/2306.10506/) (2023)
- [Federated Learning with a Sampling Algorithm under Isoperimetry](https://mlsanity.com/paper/2206.00920/) (2022)
- [A sharp uniform-in-time error estimate for Stochastic Gradient Langevin Dynamics](https://mlsanity.com/paper/2207.09304/) (2022)
- [NF-ULA: Langevin Monte Carlo with Normalizing Flow Prior for Imaging Inverse Problems](https://mlsanity.com/paper/2304.08342/) (2023)
- [The Performance Of The Unadjusted Langevin Algorithm Without Smoothness Assumptions](https://mlsanity.com/paper/2502.03458/) (2025)
- [Unadjusted Langevin algorithm for sampling a mixture of weakly smooth potentials](https://mlsanity.com/paper/2112.09311/) (2021)
- [Reflection coupling for unadjusted generalized Hamiltonian Monte Carlo in the nonconvex stochastic gradient case](https://mlsanity.com/paper/2310.18774/) (2023)
- [Simultaneous Approximation of the Score Function and Its Derivatives by Deep Neural Networks](https://mlsanity.com/paper/2512.23643/) (2025)
- [Weak Convergence of Approximate reflection coupling and its Application to Non-convex Optimization](https://mlsanity.com/paper/2205.11970/) (2022)
- [A Dynamical System View of Langevin-Based Non-Convex Sampling](https://mlsanity.com/paper/2210.13867/) (2022)
- [Approximating Langevin Monte Carlo with ResNet-like Neural Network architectures](https://mlsanity.com/paper/2311.03242/) (2023)
- [Random Reshuffling for Stochastic Gradient Langevin Dynamics](https://mlsanity.com/paper/2501.16055/) (2025)
- [Error estimates for tamed Euler and Randomized Euler schemes for SDEs with locally Lipschitz drift with applications to non-logconcave sampling and optimization](https://mlsanity.com/paper/2605.24937/) (2026)
- [Geometric ergodicity of SGLD via reflection coupling](https://mlsanity.com/paper/2301.06769/) (2023)
- [Subsampling Error in Stochastic Gradient Langevin Diffusions](https://mlsanity.com/paper/2305.13882/) (2023)
- [Contractive kinetic Langevin samplers beyond global Lipschitz continuity](https://mlsanity.com/paper/2509.12031/) (2025)
- [When Langevin Monte Carlo Meets Randomization: New Sampling Algorithms with Non-asymptotic Error Bounds beyond Log-Concavity and Gradient Lipschitzness](https://mlsanity.com/paper/2509.25630/) (2025)
- [Load--Reserve Wasserstein Propagation for Isotropic Diffusion Samplers](https://mlsanity.com/paper/2603.19670/) (2026)
- [Uniform Generalization Bound on Time and Inverse Temperature for Gradient Descent Algorithm and its Application to Analysis of Simulated Annealing](https://mlsanity.com/paper/2205.12959/) (2022)

## Abstract (arXiv)

Discrete time analogues of ergodic stochastic differential equations (SDEs) are one of the most popular and flexible tools for sampling high-dimensional probability measures. Non-asymptotic analysis in the L2 Wasserstein distance of sampling algorithms based on Euler discretisations of SDEs has been recently developed by several authors for log-concave probability distributions. In this work we replace the log-concavity assumption with a log-concavity at infinity condition. We provide novel L2 convergence rates for Euler schemes, expressed explicitly in terms of problem parameters. From there we derive non-asymptotic bounds on the distance between the laws induced by Euler schemes and the invariant laws of SDEs, both for schemes with standard and with randomised (inaccurate) drifts. We also obtain bounds for the hierarchy of discretisation, which enables us to deploy a multi-level Monte Carlo estimator. Our proof relies on a novel construction of a coupling for the Markov chains that can be used to control both the L1 and L2 Wasserstein distances simultaneously. Finally, we provide a weak convergence analysis that covers both the standard and the randomised (inaccurate) drift case. In particular, we reveal that the variance of the randomised drift does not influence the rate of weak convergence of the Euler scheme to the SDE.