# Improved Langevin Monte Carlo for stochastic optimization via landscape modification

arXiv: https://arxiv.org/abs/2302.03973 · HTML: https://mlsanity.com/paper/2302.03973/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 11 of its 24 references that are in the map.
- First posted on arXiv 8 Feb 2023.

## Builds on (most cited first, 11 of 11)

- [Non-convex learning via Stochastic Gradient Langevin Dynamics: a nonasymptotic analysis](https://mlsanity.com/paper/1702.03849/) (2017)
- [Rapid Convergence of the Unadjusted Langevin Algorithm: Isoperimetry Suffices](https://mlsanity.com/paper/1903.08568/) (2019)
- [Underdamped Langevin MCMC: A non-asymptotic analysis](https://mlsanity.com/paper/1707.03663/) (2017)
- [User-friendly guarantees for the Langevin Monte Carlo with inaccurate gradient](https://mlsanity.com/paper/1710.00095/) (2017)
- [A Hitting Time Analysis of Stochastic Gradient Langevin Dynamics](https://mlsanity.com/paper/1702.05575/) (2017)
- [The promises and pitfalls of Stochastic Gradient Langevin Dynamics](https://mlsanity.com/paper/1811.10072/) (2018)
- [Wasserstein Control of Mirror Langevin Monte Carlo](https://mlsanity.com/paper/2002.04363/) (2020)
- [Beyond Log-concavity: Provable Guarantees for Sampling Multi-modal Distributions using Simulated Tempering Langevin Monte Carlo](https://mlsanity.com/paper/1710.02736/) (2017)
- [Faster Convergence of Stochastic Gradient Langevin Dynamics for Non-Log-Concave Sampling](https://mlsanity.com/paper/2010.09597/) (2020)
- [Improved Convergence Rate of Stochastic Gradient Langevin Dynamics with Variance Reduction and its Application to Optimization](https://mlsanity.com/paper/2203.16217/) (2022)
- [On Stationary-Point Hitting Time and Ergodicity of Stochastic Gradient Langevin Dynamics](https://mlsanity.com/paper/1904.13016/) (2019)

## Built on by (most cited first, 1 of 1)

- [Accelerating Convergence in Global Non-Convex Optimization with Reversible Diffusion](https://mlsanity.com/paper/2305.11493/) (2023)

## Abstract (arXiv)

Given a target function H to minimize or a target Gibbs distribution πβ0 propto e-β H to sample from in the low temperature, in this paper we propose and analyze Langevin Monte Carlo (LMC) algorithms that run on an alternative landscape as specified by Hfβ,c,1 and target a modified Gibbs distribution πfβ,c,1 propto e^-β Hfβ,c,1, where the landscape of Hfβ,c,1 is a transformed version of that of H which depends on the parameters f,β and c. While the original Log-Sobolev constant affiliated with π0β exhibits exponential dependence on both β and the energy barrier M in the low temperature regime, with appropriate tuning of these parameters and subject to assumptions on H, we prove that the energy barrier of the transformed landscape is reduced which consequently leads to polynomial dependence on both β and M in the modified Log-Sobolev constant associated with πfβ,c,1. This yield improved total variation mixing time bounds and improved convergence toward a global minimum of H. We stress that the technique developed in this paper is not only limited to LMC and is broadly applicable to other gradient-based optimization or sampling algorithms.