Faster Sampling Algorithms for Polytopes with Small Treewidth
Yekun Ke, Xiaoyu Li, Zhao Song, Tianyi Zhou · 2024
Sampling is a fundamental problem in optimization, machine learning and theoretical computer science. A common region of interest for sampling is the polytope, which is defined by a set of linear inequalities. The algorithm that is sampling from polytopes usually requires heavy matrix algebra, including matrix multiplication, matrix inversion and matrix determinant. In this work, we show how to implement the heavy matrix algebra in the area of sampling in nearly linear time for the polytope that has small treewidth. In particular, given a polytope defined by a matrix A ∈ ℝn×dwith treewidth τ, we improve the running time of each iteration for three typical sampling algorithms for polytopes such as Dikin Walk, Soft-Threshold Dikin Walk and Vaidya Walk from O(nd2) to O(nτ2) by exploiting the small treewidth structures of the matrices.