Scalable Greedy Feature Selection via Weak Submodularity

TL;DR

This paper extends the theoretical understanding of greedy feature selection algorithms by showing they work well under weak submodularity, not just submodularity, and demonstrates their effectiveness through empirical validation.

Contribution

It proves that weak submodularity suffices for approximation guarantees, broadening the applicability of fast greedy algorithms beyond submodular functions.

Findings

Fast greedy algorithms outperform baselines on artificial datasets.

Weak submodularity guarantees approximation performance.

Data-dependent bounds can be tighter than traditional submodular bounds.

Abstract

Greedy algorithms are widely used for problems in machine learning such as feature selection and set function optimization. Unfortunately, for large datasets, the running time of even greedy algorithms can be quite high. This is because for each greedy step we need to refit a model or calculate a function using the previously selected choices and the new candidate. Two algorithms that are faster approximations to the greedy forward selection were introduced recently ([Mirzasoleiman et al. 2013, 2015]). They achieve better performance by exploiting distributed computation and stochastic evaluation respectively. Both algorithms have provable performance guarantees for submodular functions. In this paper we show that divergent from previously held opinion, submodularity is not required to obtain approximation guarantees for these two algorithms. Specifically, we show that a generalized concept of weak submodularity suffices to give multiplicative approximation guarantees. Our result extends the applicability of these algorithms to a larger class of functions. Furthermore, we show that a bounded submodularity ratio can be used to provide data dependent bounds that can sometimes be tighter also for submodular functions. We empirically validate our work by showing superior performance of fast greedy approximations versus several established baselines on artificial and real datasets.