Efficiently learning and sampling multimodal distributions with data-based initialization

Frederic Koehler, Holden Lee, Thuy-Duong Vuong
Proceedings of Thirty Eighth Conference on Learning Theory, PMLR 291:3264-3326, 2025.

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 Poincaré 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.

Cite this Paper

BibTeX
@InProceedings{pmlr-v291-koehler25a, title = {Efficiently learning and sampling multimodal distributions with data-based initialization}, author = {Koehler, Frederic and Lee, Holden and Vuong, Thuy-Duong}, booktitle = {Proceedings of Thirty Eighth Conference on Learning Theory}, pages = {3264--3326}, year = {2025}, editor = {Haghtalab, Nika and Moitra, Ankur}, volume = {291}, series = {Proceedings of Machine Learning Research}, month = {30 Jun--04 Jul}, publisher = {PMLR}, pdf = {https://raw.githubusercontent.com/mlresearch/v291/main/assets/koehler25a/koehler25a.pdf}, url = {https://proceedings.mlr.press/v291/koehler25a.html}, 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 Poincaré 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.} }
Endnote
%0 Conference Paper %T Efficiently learning and sampling multimodal distributions with data-based initialization %A Frederic Koehler %A Holden Lee %A Thuy-Duong Vuong %B Proceedings of Thirty Eighth Conference on Learning Theory %C Proceedings of Machine Learning Research %D 2025 %E Nika Haghtalab %E Ankur Moitra %F pmlr-v291-koehler25a %I PMLR %P 3264--3326 %U https://proceedings.mlr.press/v291/koehler25a.html %V 291 %X 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 Poincaré 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.
APA
Koehler, F., Lee, H. & Vuong, T.. (2025). Efficiently learning and sampling multimodal distributions with data-based initialization. Proceedings of Thirty Eighth Conference on Learning Theory, in Proceedings of Machine Learning Research 291:3264-3326 Available from https://proceedings.mlr.press/v291/koehler25a.html.

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