Dimension-Free Correlated Sampling for the Hypersimplex

TL;DR

This paper introduces a dimension-free correlated sampling algorithm for the hypersimplex, achieving improved overlap guarantees and efficient computation, with applications in online paging, metric multi-labeling, and submodular welfare.

Contribution

We develop a new correlated sampling method for the hypersimplex with an overlap factor independent of ambient dimension, improving upon previous bounds.

Findings

Achieves an $O( ext{log} k)$ overlap factor, independent of $n$

Provides input-sparsity sampling time and efficient parallel and dynamic updates

Demonstrates applications in paging, multi-labeling, and submodular welfare

Abstract

Sampling from multiple distributions so as to maximize overlap has been studied by statisticians since the 1950s. Since the 2000s, such correlated sampling from the probability simplex has been a powerful building block in disparate areas of theoretical computer science. We study a generalization of this problem to sampling sets from given vectors in the hypersimplex, i.e., outputting sets of size (at most) some $k$ in $[n]$, while maximizing the sampled sets' overlap. Specifically, the expected difference between two output sets should be at most $\alpha$ times their input vectors' $\ell_1$ distance. A value of $\alpha=O(\log n)$ is known to be achievable, due to Chen et al.~(ICALP'17). We improve this factor to $O(\log k)$, independent of the ambient dimension~$n$. Our algorithm satisfies other desirable properties, including (up to a $\log^* n$ factor) input-sparsity sampling time, logarithmic parallel depth and dynamic update time, as well as preservation of submodular objectives. Anticipating broader use of correlated sampling algorithms for the hypersimplex, we present applications of our algorithm to online paging, offline approximation of metric multi-labeling and swift multi-scenario submodular welfare approximating reallocation.