From Sampling to Optimization on Discrete Domains with Applications to Determinant Maximization

Nima Anari, Thuy-Duong Vuong
Proceedings of Thirty Fifth Conference on Learning Theory, PMLR 178:5596-5618, 2022.

Abstract

We establish a connection between sampling and optimization on discrete domains. For a family of distributions $\mu$ defined on size $k$ subsets of a ground set of elements, that is closed under external fields, we show that rapid mixing of natural local random walks implies the existence of simple approximation algorithms to find $\max \mu(\cdot)$. More precisely, we show that if $t$-step down-up random walks have spectral gap at least inverse polynomially large, then $t$-step local search finds $\max \mu(\cdot)$ within a factor of $k^{O(k)}$. As the main application of our result, we show that $2$-step local search achieves a nearly-optimal $k^{O(k)}$-factor approximation for MAP inference on nonsymmetric $k$-DPPs. This is the first nontrivial multiplicative approximation algorithm for this problem. In our main technical result, we show that an exchange inequality, a concept rooted in discrete convex analysis, can be derived from fast mixing of local random walks. We further advance the state of the art on the mixing of random walks for nonsymmetric DPPs and more generally sector-stable distributions, by obtaining the tightest possible bound on the step size needed for polynomial-time mixing of random walks. We bring the step size down by a factor of $2$ compared to prior works, and consequently get a quadratic improvement on the runtime of local search steps; this improvement is potentially of independent interest in sampling applications.

Cite this Paper

BibTeX
@InProceedings{pmlr-v178-anari22a, title = {From Sampling to Optimization on Discrete Domains with Applications to Determinant Maximization}, author = {Anari, Nima and Vuong, Thuy-Duong}, booktitle = {Proceedings of Thirty Fifth Conference on Learning Theory}, pages = {5596--5618}, year = {2022}, editor = {Loh, Po-Ling and Raginsky, Maxim}, volume = {178}, series = {Proceedings of Machine Learning Research}, month = {02--05 Jul}, publisher = {PMLR}, pdf = {https://proceedings.mlr.press/v178/anari22a/anari22a.pdf}, url = {https://proceedings.mlr.press/v178/anari22a.html}, abstract = {We establish a connection between sampling and optimization on discrete domains. For a family of distributions $\mu$ defined on size $k$ subsets of a ground set of elements, that is closed under external fields, we show that rapid mixing of natural local random walks implies the existence of simple approximation algorithms to find $\max \mu(\cdot)$. More precisely, we show that if $t$-step down-up random walks have spectral gap at least inverse polynomially large, then $t$-step local search finds $\max \mu(\cdot)$ within a factor of $k^{O(k)}$. As the main application of our result, we show that $2$-step local search achieves a nearly-optimal $k^{O(k)}$-factor approximation for MAP inference on nonsymmetric $k$-DPPs. This is the first nontrivial multiplicative approximation algorithm for this problem. In our main technical result, we show that an exchange inequality, a concept rooted in discrete convex analysis, can be derived from fast mixing of local random walks. We further advance the state of the art on the mixing of random walks for nonsymmetric DPPs and more generally sector-stable distributions, by obtaining the tightest possible bound on the step size needed for polynomial-time mixing of random walks. We bring the step size down by a factor of $2$ compared to prior works, and consequently get a quadratic improvement on the runtime of local search steps; this improvement is potentially of independent interest in sampling applications.} }
Endnote
%0 Conference Paper %T From Sampling to Optimization on Discrete Domains with Applications to Determinant Maximization %A Nima Anari %A Thuy-Duong Vuong %B Proceedings of Thirty Fifth Conference on Learning Theory %C Proceedings of Machine Learning Research %D 2022 %E Po-Ling Loh %E Maxim Raginsky %F pmlr-v178-anari22a %I PMLR %P 5596--5618 %U https://proceedings.mlr.press/v178/anari22a.html %V 178 %X We establish a connection between sampling and optimization on discrete domains. For a family of distributions $\mu$ defined on size $k$ subsets of a ground set of elements, that is closed under external fields, we show that rapid mixing of natural local random walks implies the existence of simple approximation algorithms to find $\max \mu(\cdot)$. More precisely, we show that if $t$-step down-up random walks have spectral gap at least inverse polynomially large, then $t$-step local search finds $\max \mu(\cdot)$ within a factor of $k^{O(k)}$. As the main application of our result, we show that $2$-step local search achieves a nearly-optimal $k^{O(k)}$-factor approximation for MAP inference on nonsymmetric $k$-DPPs. This is the first nontrivial multiplicative approximation algorithm for this problem. In our main technical result, we show that an exchange inequality, a concept rooted in discrete convex analysis, can be derived from fast mixing of local random walks. We further advance the state of the art on the mixing of random walks for nonsymmetric DPPs and more generally sector-stable distributions, by obtaining the tightest possible bound on the step size needed for polynomial-time mixing of random walks. We bring the step size down by a factor of $2$ compared to prior works, and consequently get a quadratic improvement on the runtime of local search steps; this improvement is potentially of independent interest in sampling applications.
APA
Anari, N. & Vuong, T.. (2022). From Sampling to Optimization on Discrete Domains with Applications to Determinant Maximization. Proceedings of Thirty Fifth Conference on Learning Theory, in Proceedings of Machine Learning Research 178:5596-5618 Available from https://proceedings.mlr.press/v178/anari22a.html.

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