Faster Sampling from Log-Concave Densities over Polytopes via Efficient Linear Solvers

TL;DR

This paper introduces a faster method for sampling from log-concave distributions over polytopes by leveraging efficient linear solvers and matrix update techniques, significantly reducing per-step computational complexity.

Contribution

It presents a nearly-optimal implementation of the Dikin walk that reduces per-step complexity by exploiting slow matrix changes and advanced linear algebra techniques.

Findings

Per-step complexity is roughly proportional to the number of non-zero entries in A.

The method maintains the same number of Markov chain steps as previous algorithms.

Speedups are achieved through matrix update strategies and randomized estimators.

Abstract

We consider the problem of sampling from a log-concave distribution $\pi(\theta) \propto e^{-f(\theta)}$ constrained to a polytope $K:=\{\theta \in \mathbb{R}^d: A\theta \leq b\}$, where $A\in \mathbb{R}^{m\times d}$ and $b \in \mathbb{R}^m$.The fastest-known algorithm \cite{mangoubi2022faster} for the setting when $f$ is $O(1)$-Lipschitz or $O(1)$-smooth runs in roughly $O(md \times md^{\omega -1})$ arithmetic operations, where the $md^{\omega -1}$ term arises because each Markov chain step requires computing a matrix inversion and determinant (here $\omega \approx 2.37$ is the matrix multiplication constant). We present a nearly-optimal implementation of this Markov chain with per-step complexity which is roughly the number of non-zero entries of $A$ while the number of Markov chain steps remains the same. The key technical ingredients are 1) to show that the matrices that arise in this Dikin walk change slowly, 2) to deploy efficient linear solvers that can leverage this slow change to speed up matrix inversion by using information computed in previous steps, and 3) to speed up the computation of the determinantal term in the Metropolis filter step via a randomized Taylor series-based estimator.