Restricted Strong Convexity Implies Weak Submodularity

TL;DR

This paper establishes that restricted strong convexity ensures weak submodularity, enabling strong performance guarantees for greedy subset selection algorithms across various objective functions without relying on statistical modeling assumptions.

Contribution

It extends the connection between convexity and submodularity from linear regression to general objectives, providing new performance bounds and recovery guarantees for greedy algorithms.

Findings

Greedy algorithms perform within a constant factor of the optimal subset.

The methods allow direct control over the number of features selected.

A novel proof technique links convex analysis with submodular set function theory.

Abstract

We connect high-dimensional subset selection and submodular maximization. Our results extend the work of Das and Kempe (2011) from the setting of linear regression to arbitrary objective functions. For greedy feature selection, this connection allows us to obtain strong multiplicative performance bounds on several methods without statistical modeling assumptions. We also derive recovery guarantees of this form under standard assumptions. Our work shows that greedy algorithms perform within a constant factor from the best possible subset-selection solution for a broad class of general objective functions. Our methods allow a direct control over the number of obtained features as opposed to regularization parameters that only implicitly control sparsity. Our proof technique uses the concept of weak submodularity initially defined by Das and Kempe. We draw a connection between convex analysis and submodular set function theory which may be of independent interest for other statistical learning applications that have combinatorial structure.