Composable Coresets for Determinant Maximization: Greedy is Almost Optimal

TL;DR

This paper demonstrates that the greedy algorithm for determinant maximization produces nearly optimal composable coresets with improved approximation guarantees, supported by theoretical analysis and empirical evidence.

Contribution

It proves that the greedy algorithm achieves an almost optimal approximation factor for composable coresets in determinant maximization, improving previous bounds and confirming practical effectiveness.

Findings

Greedy algorithm yields an approximation factor of O(k)^{3k}.

Swapping one point in the greedy solution increases volume by at most (1+√k).

Empirical results show the local optimality bound is even lower on real data.

Abstract

Given a set of $n$ vectors in $\mathbb{R}^d$, the goal of the \emph{determinant maximization} problem is to pick $k$ vectors with the maximum volume. Determinant maximization is the MAP-inference task for determinantal point processes (DPP) and has recently received considerable attention for modeling diversity. As most applications for the problem use large amounts of data, this problem has been studied in the relevant \textit{composable coreset} setting. In particular, [Indyk-Mahabadi-OveisGharan-Rezaei--SODA'20, ICML'19] showed that one can get composable coresets with optimal approximation factor of $\tilde O(k)^k$ for the problem, and that a local search algorithm achieves an almost optimal approximation guarantee of $O(k)^{2k}$. In this work, we show that the widely-used Greedy algorithm also provides composable coresets with an almost optimal approximation factor of $O(k)^{3k}$, which improves over the previously known guarantee of $C^{k^2}$, and supports the prior experimental results showing the practicality of the greedy algorithm as a coreset. Our main result follows by showing a local optimality property for Greedy: swapping a single point from the greedy solution with a vector that was not picked by the greedy algorithm can increase the volume by a factor of at most $(1+\sqrt{k})$. This is tight up to the additive constant $1$. Finally, our experiments show that the local optimality of the greedy algorithm is even lower than the theoretical bound on real data sets.