[Paper Review] Analysis of Langevin Monte Carlo from Poincaré to Log-Sobolev

Sinho Chewi, Murat A. Erdogdu+3 more|arXiv (Cornell University)|Dec 23, 2021

Markov Chains and Monte Carlo Methods8 citations

TL;DR

This paper provides the first non-asymptotic convergence guarantees for the discrete-time Langevin Monte Carlo (LMC) algorithm under Lataà–Oleszkiewicz (LO) and modified log-Sobolev inequalities, which interpolate between Poincaré and log-Sobolev conditions. It establishes exponential convergence in Rényi divergence under Hölder-continuous gradients without requiring convexity, dissipativity, or strong log-concavity, extending sampling theory to broader non-log-concave targets.

ABSTRACT

Classically, the continuous-time Langevin diffusion converges exponentially fast to its stationary distribution $π$ under the sole assumption that $π$ satisfies a Poincaré inequality. Using this fact to provide guarantees for the discrete-time Langevin Monte Carlo (LMC) algorithm, however, is considerably more challenging due to the need for working with chi-squared or Rényi divergences, and prior works have largely focused on strongly log-concave targets. In this work, we provide the first convergence guarantees for LMC assuming that $π$ satisfies either a Latała--Oleszkiewicz or modified log-Sobolev inequality, which interpolates between the Poincaré and log-Sobolev settings. Unlike prior works, our results allow for weak smoothness and do not require convexity or dissipativity conditions.

Motivation & Objective

• To extend non-asymptotic sampling guarantees for LMC beyond strongly log-concave targets.
• To remove the need for dissipativity or convexity assumptions in LMC convergence analysis.
• To establish convergence under functional inequalities that interpolate between Poincaré and log-Sobolev, including LO and modified log-Sobolev.
• To analyze LMC in Rényi divergence using chi-squared convergence from continuous-time diffusion.
• To handle weak smoothness via s-Hölder continuous gradients, avoiding Lipschitz assumptions.

Proposed method

• Analyzes the continuous-time Langevin diffusion under LO and modified log-Sobolev inequalities to derive exponential ergodicity in Rényi divergence.
• Uses a novel moment bound for the supremum of Brownian motion increments in L2s norm via tail estimates and exponential moment control.
• Applies Grönwall's inequality to control the growth of the SDE solution under s-Hölder continuous gradients.
• Establishes a moment bound for the increment of the Langevin process using a decomposition into drift and diffusion terms.
• Derives a bound on the exponential moment of the maximum displacement over time steps, enabling Rényi divergence control.
• Combines functional inequality assumptions with moment bounds to derive non-asymptotic convergence rates for discrete-time LMC.

Experimental results

Research questions

• RQ1 Can LMC be guaranteed to converge under functional inequalities weaker than log-Sobolev, such as Lataà–Oleszkiewicz?
• RQ2 Does the absence of dissipativity or convexity assumptions still allow for non-asymptotic convergence in LMC?
• RQ3 Can exponential convergence be established in Rényi divergence under LO or modified log-Sobolev inequalities?
• RQ4 How does s-Hölder continuity of the gradient affect the convergence analysis compared to Lipschitz assumptions?
• RQ5 Can the analysis be extended to non-log-concave targets with polynomial growth potentials?

Key findings

• The paper establishes exponential convergence of LMC in Rényi divergence under the Lataà–Oleszkiewicz inequality with parameter α ∈ [1,2], interpolating between Poincaré and log-Sobolev.
• Convergence is achieved without requiring convexity, dissipativity, or gradient Lipschitz continuity, only s-Hölder continuity of the gradient.
• The analysis provides non-asymptotic bounds on the Rényi divergence between the LMC output and the target distribution.
• A key technical contribution is the exponential moment bound for the supremum of Brownian motion increments in L2s norm, which controls the discretization error.
• The method applies to targets with polynomial growth V(x) ≈ ||x||^α, including non-log-concave distributions such as Gaussian mixtures with bounded support.
• The results extend to modified log-Sobolev inequalities, which are useful when the LO constant depends on dimension.