Efficient Submodular Optimization under Noise: Local Search is Robust

TL;DR

This paper introduces robust local search algorithms for monotone submodular maximization under noisy evaluations, achieving near-optimal approximation guarantees with polynomial query complexity for both cardinality and matroid constraints.

Contribution

It develops a novel local search framework and smoothing techniques to handle noise, providing the first constant approximation for general matroid constraints under noise.

Findings

Achieves near-optimal approximation with polynomial queries for cardinality constraints.

First constant approximation algorithm for submodular maximization under noise with general matroid constraints.

Introduces noise-reduction smoothing surrogate functions.

Abstract

The problem of monotone submodular maximization has been studied extensively due to its wide range of applications. However, there are cases where one can only access the objective function in a distorted or noisy form because of the uncertain nature or the errors involved in the evaluation. This paper considers the problem of constrained monotone submodular maximization with noisy oracles introduced by [Hassidim et al., 2017]. For a cardinality constraint, we propose an algorithm achieving a near-optimal $\left(1-\frac{1}{e}-O(\varepsilon)\right)$-approximation guarantee (for arbitrary $\varepsilon > 0$) with only a polynomial number of queries to the noisy value oracle, which improves the exponential query complexity of [Singer et al., 2018]. For general matroid constraints, we show the first constant approximation algorithm in the presence of noise. Our main approaches are to design a novel local search framework that can handle the effect of noise and to construct certain smoothing surrogate functions for noise reduction.