Performance Bounds with Curvature for Batched Greedy Optimization

TL;DR

This paper establishes performance bounds for the batched greedy algorithm in maximizing set functions under matroid constraints, introducing the total batched curvature as a key parameter and analyzing how bounds change with batch size.

Contribution

It develops new bounds on the performance of batched greedy strategies using total batched curvature, applicable to general and uniform matroids, and studies the effect of batch size on these bounds.

Findings

Harmonic bound for general matroids in terms of total batched curvature.

Exponential bound for uniform matroids in terms of total batched curvature.

Bounds improve or remain nondecreasing as batch size increases.

Abstract

The batched greedy strategy is an approximation algorithm to maximize a set function subject to a matroid constraint. Starting with the empty set, the batched greedy strategy iteratively adds to the current solution set a batch of elements that results in the largest gain in the objective function while satisfying the matroid constraints. In this paper, we develop bounds on the performance of the batched greedy strategy relative to the optimal strategy in terms of a parameter called the total batched curvature. We show that when the objective function is a polymatroid set function, the batched greedy strategy satisfies a harmonic bound for a general matroid constraint and an exponential bound for a uniform matroid constraint, both in terms of the total batched curvature. We also study the behavior of the bounds as functions of the batch size. Specifically, we prove that the harmonic bound for a general matroid is nondecreasing in the batch size and the exponential bound for a uniform matroid is nondecreasing in the batch size under the condition that the batch size divides the rank of the uniform matroid. Finally, we illustrate our results by considering a task scheduling problem and an adaptive sensing problem.