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Non-asymptotic Error Bounds in $\mathcal{W}_2$-Distance with Sqrt(d) Dimension Dependence and First Order Convergence for Langevin Monte Carlo beyond Log-Concavity
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Gives a non-asymptotic W2 error bound O(sqrt(d) h) for Langevin Monte Carlo without log-concavity, under log-Sobolev inequality and smoothness assumptions, plus a projected LMC variant for superlinear gradients.
Problem
Non-asymptotic W2 error and mixing time of Langevin Monte Carlo for non-log-concave targets, where prior W2 bounds with sqrt(d) dependence required strong log-concavity.
Approach
A uniform-in-time convergence framework for SDE discretizations that combines a finite-time mean-square convergence theorem, uniform-in-time moment bounds of the scheme, and exponential W2 ergodicity of the Langevin SDE under dissipativity and LSI, plus a projected LMC (pLMC) for superlinearly growing gradients.
Key results
- Proves W2 error bound O(sqrt(d) h) for LMC under LSI and smoothness, giving mixing time O~(sqrt(d) eps^-1) without strong log-concavity.
- Mixing time for pLMC with superlinear gradients is O~(d^((11gamma+2)/4) / eps).
- Numerical experiments at d=100 show strong convergence of order 1 in step size for LMC and pLMC on Gaussian mixture and double-well targets.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use LMC under the stated log-smoothness and log-Sobolev assumptions without strong log-concavity | LMC · Non-convex targets | W2 mixing time: eO(dϵ−4) -> eO(d1/2ϵ−1) | Helped |
| Use LMC under the stated log-smoothness and log-Sobolev assumptions without strong log-concavity | LMC · Non-convex targets | W2 mixing time: eO(dϵ−1) -> eO(d1/2ϵ−1) | Helped |
| Use the projected LMC sampler for a potential with a non-globally Lipschitz, superlinear-growth gradient | pLMC · Non-globally Lipschitz gradient targets | W2 error bound: d(11γ+2)/4h | Mixed |
| Apply LMC to the Gaussian Mixture example | LMC · Gaussian Mixture | strong convergence rate: order 1.0 | Helped |
| Apply pLMC to the Double-well potential example | pLMC · Double-well potential | strong convergence rate: order 1.0 | Helped |
Limitations
- Requires log-Sobolev inequality and third-order derivative linear growth assumptions
- Constants are only given implicitly (in appendix), dependence on other parameters not fully explicit
- Numerical validation is limited to two low-complexity potentials
Experimental setup
Benchmarks: Gaussian mixture target double-well potential target
Metrics: strong error in W2 (L2 path error) vs step size empirical density
Baselines: Li et al. 2022 Durmus and Moulines 2019 Majka et al. 2020 Mou et al. 2022
Builds on: Li et al. 2022 LMC W2 bound Wang 2020 W2 exponential ergodicity Milstein finite-time mean-square convergence Pang et al. 2025 projected LMC
Closest papers by meaning
- Sqrt(d) Dimension Dependence of Langevin Monte Carlo (ICLR 2022 · similarity 0.93)
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- Fast Non-Log-Concave Sampling under Nonconvex Equality and Inequality Constraints with Landing (NeurIPS 2025 · similarity 0.89)
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- Generalization of noisy SGD in unbounded non-convex settings (ICML 2025 · similarity 0.88)
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