[Paper Review] Beyond Log-concavity: Provable Guarantees for Sampling Multi-modal Distributions using Simulated Tempering Langevin Monte Carlo

Holden Lee, Andrej Risteski+1 more|arXiv (Cornell University)|Jan 1, 2018

Bayesian Methods and Mixture Models19 citations

TL;DR

This paper proposes a novel sampling method combining Langevin diffusion with simulated tempering to achieve fast mixing for multi-modal distributions, particularly mixtures of Gaussians. By dynamically adjusting temperature levels, the algorithm overcomes the torpid mixing of standard Langevin diffusion in multi-modal settings, providing the first provable polynomial-time sampling guarantees for such distributions using only gradient access to the log-pdf.

ABSTRACT

A key task in Bayesian machine learning is sampling from distributions that are only specified up to a partition function (i.e., constant of proportionality). One prevalent example of this is sampling posteriors in parametric distributions, such as latent-variable generative models. However sampling (even very approximately) can be #P-hard. Classical results (going back to Bakry and Emery) on sampling focus on log-concave distributions, and show a natural Markov chain called Langevin diffusion mix in polynomial time. However, all log-concave distributions are uni-modal, while in practice it is very common for the distribution of interest to have multiple modes. In this case, Langevin diffusion suffers from torpid mixing. We address this problem by combining Langevin diffusion with simulated tempering. The result is a Markov chain that mixes more rapidly by transitioning between different temperatures of the distribution. We analyze this Markov chain for a mixture of (strongly) log-concave distributions of the same shape. In particular, our technique applies to the canonical multi-modal distribution: a mixture of gaussians (of equal variance). Our algorithm efficiently samples from these distributions given only access to the gradient of the log-pdf. To the best of our knowledge, this is the first result that proves fast mixing for multimodal distributions.

Motivation & Objective

• Address the challenge of efficient sampling from multi-modal distributions in Bayesian machine learning, where standard methods fail due to slow mixing.
• Overcome the limitation of classical Langevin diffusion, which only guarantees fast mixing for log-concave (uni-modal) distributions.
• Develop a provably efficient sampling algorithm for multi-modal posteriors, particularly mixtures of Gaussians with equal variance.
• Ensure the method requires only access to the gradient of the log-posterior, making it practical for complex models.
• Provide theoretical guarantees on mixing time that are polynomial in relevant problem parameters, even in the presence of multiple modes.

Proposed method

• Introduce a hybrid Markov chain that combines Langevin diffusion with simulated tempering to enable transitions between different temperature levels of the target distribution.
• Use a temperature schedule that allows the chain to escape local modes by exploring higher-temperature versions of the distribution.
• Apply a Metropolis-Hastings correction at each temperature level to ensure detailed balance and correct stationary distribution.
• Leverage the strong log-concavity of individual components in a Gaussian mixture to bound mixing time at each temperature.
• Design a transition mechanism between temperatures that ensures rapid exploration across modes while maintaining convergence to the correct equilibrium distribution.
• Prove that the overall chain mixes in polynomial time by analyzing the spectral gap and conductance of the joint state space across temperatures.

Experimental results

Research questions

• RQ1 Can simulated tempering be rigorously combined with Langevin diffusion to achieve fast mixing for multi-modal distributions?
• RQ2 Does the resulting algorithm maintain polynomial mixing time when sampling from a mixture of strongly log-concave components, such as Gaussians?
• RQ3 Can the method achieve provable efficiency with only gradient access to the log-posterior, without requiring full density evaluations?
• RQ4 Is the mixing time of the combined chain polynomial in the dimension and inverse temperature range, even when the target has multiple modes?
• RQ5 How does the algorithm’s performance compare to standard Langevin diffusion in terms of convergence speed on multi-modal posteriors?

Key findings

• The proposed algorithm achieves polynomial mixing time for multi-modal distributions, specifically mixtures of Gaussians with equal variance.
• The method provably overcomes the torpid mixing problem of standard Langevin diffusion in multi-modal settings.
• Mixing time is bounded by a polynomial in the dimension and the inverse temperature range, under mild assumptions on the component distributions.
• The algorithm requires only access to the gradient of the log-pdf, making it applicable to high-dimensional Bayesian inference problems.
• The theoretical analysis establishes the first provable fast mixing guarantee for a sampling method on multi-modal distributions using Langevin dynamics with simulated tempering.
• The result holds for any number of modes, provided the components are strongly log-concave and of the same shape (e.g., same covariance).