Improved dimension dependence of a proximal algorithm for sampling

• URL: http://arxiv.org/abs/2302.10081v2
• Date: Wed, 28 Jun 2023 16:09:48 GMT
• Title: Improved dimension dependence of a proximal algorithm for sampling
• Authors: Jiaojiao Fan, Bo Yuan and Yongxin Chen
• Abstract summary: We propose a sampling algorithm that achieves superior complexity bounds in all the classical settings. Our algorithm is based on the proximal sampler introduced incitetlee 2021. For strongly log-concave distributions, our method has complexity bound $tildemathcalO(kappa d1/2)$ without warm start. For distributions satisfying the LSI, our bound is $tilde mathcalO(hat kappa d1/2)$ where $hat kappa$ is the ratio between smoothness and
• Score: 16.147290924171692
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: We propose a sampling algorithm that achieves superior complexity bounds in all the classical settings (strongly log-concave, log-concave, Logarithmic-Sobolev inequality (LSI), Poincar\'e inequality) as well as more general settings with semi-smooth or composite potentials. Our algorithm is based on the proximal sampler introduced in~\citet{lee2021structured}. The performance of this proximal sampler is determined by that of the restricted Gaussian oracle (RGO), a key step in the proximal sampler. The main contribution of this work is an inexact realization of RGO based on approximate rejection sampling. To bound the inexactness of RGO, we establish a new concentration inequality for semi-smooth functions over Gaussian distributions, extending the well-known concentration inequality for Lipschitz functions. Applying our RGO implementation to the proximal sampler, we achieve state-of-the-art complexity bounds in almost all settings. For instance, for strongly log-concave distributions, our method has complexity bound $\tilde\mathcal{O}(\kappa d^{1/2})$ without warm start, better than the minimax bound for MALA. For distributions satisfying the LSI, our bound is $\tilde \mathcal{O}(\hat \kappa d^{1/2})$ where $\hat \kappa$ is the ratio between smoothness and the LSI constant, better than all existing bounds.

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