Approximation Guarantees of Local Search Algorithms via Localizability of Set Functions

TL;DR

This paper introduces a new framework based on localizability to analyze and improve local search algorithms for set function maximization, with applications in sparse optimization.

Contribution

It defines localizability for set functions and links it to approximation guarantees of local search algorithms under various constraints.

Findings

Localizability effectively measures local improvement quality.

Restricted strong concavity and smoothness imply localizability.

Accelerated local search algorithms perform well in experiments.

Abstract

This paper proposes a new framework for providing approximation guarantees of local search algorithms. Local search is a basic algorithm design technique and is widely used for various combinatorial optimization problems. To analyze local search algorithms for set function maximization, we propose a new notion called localizability of set functions, which measures how effective local improvement is. Moreover, we provide approximation guarantees of standard local search algorithms under various combinatorial constraints in terms of localizability. The main application of our framework is sparse optimization, for which we show that restricted strong concavity and restricted smoothness of the objective function imply localizability, and further develop accelerated versions of local search algorithms. We conduct experiments in sparse regression and structure learning of graphical models to confirm the practical efficiency of the proposed local search algorithms.