Research map / Optimization / Nonconvex and smooth optimization
Langevin Monte Carlo Beyond Lipschitz Gradient Continuity
Read the paperCheck your idea against 108,967 papers
AI-extracted summary. Written by AI from the paper's full text. Numbers and claims can be wrong or missing context, so check the paper before you cite or build on it.
Proposes the Inexact Proximal Langevin Algorithm (IPLA), a Langevin Monte Carlo variant using inexact proximal steps that gives KL and Wasserstein convergence guarantees for convex potentials with polynomial growth and non-Lipschitz gradients.
Problem
Langevin Monte Carlo theory and practice assume the potential has an L-Lipschitz gradient (growth at most quadratic), which excludes many classical super-quadratic potentials such as the Ginzburg-Landau potential.
Approach
Splits the Wasserstein gradient flow of KL-functional into a proximal step for the potential V (computed inexactly with bounded error) followed by Gaussian noise injection, yielding a Langevin-type sampler whose analysis needs only convexity, tail strong convexity and polynomial growth of V.
Key results
- On the quartic-potential example (d=1000), IPLA gives lower relative error than TULA for E|Y|^2, E|Y|^4, E|Y|^6 in both starting scenarios (e.g., 0.0027 vs 0.0047 for E|Y|^2 starting in tail).
- ULA diverges (NaN) when started in the tail for both Example 1 and Example 2, while IPLA and TULA remain stable.
- IPLA with KL error bound gives d^{(q_V+1)/2} O(eps^-2) iteration complexity for alpha>=1 under warm start; Wasserstein bound gives O(eps^-2) when alpha>=1/2.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use IPLA instead of TULA | Quartic light-tail distribution, start in tail · E|Y|^2, E|Y|^4, and E|Y|^6 | RE: 0.0047 -> 0.0027; 0.0095 -> 0.0054; 0.0144 -> 0.0081 | Helped |
| Use IPLA instead of TULA | Quartic light-tail distribution, start at x0=0 · E|Y|^2, E|Y|^4, and E|Y|^6 | RE: 0.0030 -> 0.0006; 0.0073 -> 0.0025; 0.0120 -> 0.0047 | Helped |
| Use IPLA instead of ULA | Quartic light-tail distribution, start at x0=0 · E|Y|^2, E|Y|^4, and E|Y|^6 | RE: 0.0020 -> 0.0006; 0.0028 -> 0.0025; 0.0032 -> 0.0047 | Mixed |
| Use IPLA instead of ULA | Quartic light-tail distribution, start in tail · E|Y|^2, E|Y|^4, and E|Y|^6 | RE | Helped |
| Use IPLA instead of TULA | Ginzburg–Landau model, start in tail · E|Y|^2, E|Y|^4, and E|Y|^6 | RE: 0.0067 -> 0.0025; 0.0134 -> 0.0053; 0.0199 -> 0.0083 | Helped |
| Use IPLA instead of TULA | Ginzburg–Landau model, start at x0=0 · E|Y|^2, E|Y|^4, and E|Y|^6 | RE: 0.0859 -> 0.0748; 0.1635 -> 0.1425; 0.2337 -> 0.2040 | Helped |
| Use IPLA instead of ULA | Ginzburg–Landau model, start at x0=0 · E|Y|^2, E|Y|^4, and E|Y|^6 | RE: 0.0727 -> 0.0748; 0.1385 -> 0.1425; 0.1980 -> 0.2040 | Mixed |
| Use IPLA instead of ULA | Ginzburg–Landau model, start in tail · E|Y|^2, E|Y|^4, and E|Y|^6 | RE | Helped |
Surprising findings
- ULA with a fixed stepsize blows up in a few iterations on non-Lipschitz-gradient potentials even when started at the minimizer in some regimes, whereas its results are only stable for small tau.
Limitations
- Theoretical guarantees require convexity of V and an inexact proximal error of order tau^{1+alpha}, and the bounds scale as d^{(q_V+1)/2}, worsening with growth power.
- Example 3 (TV prior, non-smooth) has no theoretical guarantees; the authors only show empirical performance.
- The moment and error constants depend on initial moments and on the lambda_V and R_V parameters, with explicit formulas given only in appendix.
Experimental setup
Datasets: synthetic target distributions (quartic potential, Ginzburg-Landau model, Bayesian TV image deconvolution)
Benchmarks: Example 1 quartic potential moment estimation Example 2 Ginzburg-Landau moment estimation Example 3 Bayesian image deconvolution
Metrics: relative error coefficient of variation
Baselines: ULA TULA Metropolis-Hastings
Builds on: Unadjusted Langevin Algorithm Tamed Unadjusted Langevin Algorithm Proximal Langevin Monte Carlo JKO scheme
Closest papers by meaning
- Stochastic Proximal Langevin Algorithm: Potential Splitting and Nonasymptotic Rates (NeurIPS 2019 · similarity 0.90)
- Provable Benefit of Annealed Langevin Monte Carlo for Non-log-concave Sampling (ICLR 2025 · similarity 0.86)
- The Poisson Midpoint Method for Langevin Dynamics: Provably Efficient Discretization for Diffusion Models (NeurIPS 2024 · similarity 0.86)
- Fast Non-Log-Concave Sampling under Nonconvex Equality and Inequality Constraints with Landing (NeurIPS 2025 · similarity 0.86)
- 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 (ICML 2025 · similarity 0.85)
- Mean-field Underdamped Langevin Dynamics and its Spacetime Discretization (ICML 2024 · similarity 0.85)
Writing a paper in this area? scime draft is a free LaTeX editor with AI included, real-time co-authors and a check of your references and venue rules before you submit. It also imports OpenAI Prism projects.