Efficiently learning and sampling multimodal distributions with data-based initialization

TL;DR

This paper demonstrates that with a small number of samples, a Markov chain can be efficiently initialized to approximate a multimodal distribution, enabling effective sampling even when mixing is slow, especially for mixtures satisfying certain inequalities.

Contribution

It introduces a data-based initialization method for Markov chains that achieves efficient sampling of multimodal distributions with linear dependence on the number of modes, extending previous results.

Findings

Efficient sampling from multimodal distributions with few samples.

Stable bounds applicable to Langevin diffusion and Glauber dynamics.

First demonstration of learning low-complexity Ising models from samples.

Abstract

We consider the problem of sampling a multimodal distribution with a Markov chain given a small number of samples from the stationary measure. Although mixing can be arbitrarily slow, we show that if the Markov chain has a $k$th order spectral gap, initialization from a set of $\tilde O(k/\varepsilon^2)$ samples from the stationary distribution will, with high probability over the samples, efficiently generate a sample whose conditional law is $\varepsilon$-close in TV distance to the stationary measure. In particular, this applies to mixtures of $k$ distributions satisfying a Poincar\'e inequality, with faster convergence when they satisfy a log-Sobolev inequality. Our bounds are stable to perturbations to the Markov chain, and in particular work for Langevin diffusion over $\mathbb R^d$ with score estimation error, as well as Glauber dynamics combined with approximation error from pseudolikelihood estimation. This justifies the success of data-based initialization for score matching methods despite slow mixing for the data distribution, and improves and generalizes the results of Koehler and Vuong (2023) to have linear, rather than exponential, dependence on $k$ and apply to arbitrary semigroups. As a consequence of our results, we show for the first time that a natural class of low-complexity Ising measures can be efficiently learned from samples.