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

Matthew Fahrbach, Srikumar Ramalingam, Morteza Zadimoghaddam, Sara Ahmadian, Gui Citovsky, Giulia DeSalvo

Keywords: approximation algorithm, submodular maximization, max-min diversification, data sampling, subset selection

Abstract:

This work studies a novel subset selection problem called *max-min diversification with monotone submodular utility* (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 MDMS is to maximize $f(S) = g(S) + \lambda \cdot \text{div}(S)$
subject to a cardinality constraint $|S| \le k$,
where
$g(S)$ is a monotone submodular function
and
$\text{div}(S) = \min_{u,v \in S : u \ne v} \text{dist}(u,v)$ is the *max-min diversity* objective.
We propose the `GIST` algorithm, which gives a $\frac{1}{2}$-approximation guarantee for 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 `GIST` outperforms state-of-the-art benchmarks
for a single-shot data sampling task on ImageNet.

Primary Area: Optimization (e.g., convex and non-convex, stochastic, robust)

Submission Number: 10383