GIST: Greedy Independent Set Thresholding for Max-Min Diversification with Submodular Utility

TL;DR

This paper introduces GIST, a greedy algorithm for max-min diversification with submodular utility, providing theoretical guarantees and demonstrating superior performance in data sampling tasks like ImageNet.

Contribution

The paper proposes GIST, a novel greedy algorithm with approximation guarantees for max-min diversification with submodular utility, and empirically outperforms existing methods.

Findings

GIST achieves a 1/2-approximation guarantee for MDMS.

It is NP-hard to approximate MDMS within a factor of 0.5584.

GIST outperforms state-of-the-art benchmarks on ImageNet data sampling.

Abstract

This work studies a novel subset selection problem called max-min diversification with monotone submodular utility ($\textsf{MDMS}$), which has a wide range of applications in machine learning, e.g., data sampling and feature selection. Given a set of points in a metric space, the goal of $\textsf{MDMS}$ is to maximize $f(S) = g(S) + \lambda \cdot \texttt{div}(S)$ subject to a cardinality constraint $|S| \le k$, where $g(S)$ is a monotone submodular function and $\texttt{div}(S) = \min_{u,v \in S : u \ne v} \text{dist}(u,v)$ is the max-min diversity objective. We propose the $\texttt{GIST}$ algorithm, which gives a $\frac{1}{2}$-approximation guarantee for $\textsf{MDMS}$ by approximating a series of maximum independent set problems with a bicriteria greedy algorithm. We also prove that it is NP-hard to approximate within a factor of $0.5584$. Finally, we show in our empirical study that $\texttt{GIST}$ outperforms state-of-the-art benchmarks for a single-shot data sampling task on ImageNet.