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A Unified Approach to Submodular Maximization Under Noise

Kshipra Bhawalkar, Yang Cai, Zhe Feng, Christopher Liaw, Tao Lin

TL;DR

This work tackles maximizing submodular functions when only a noisy, persistent value oracle is available. It introduces a universal meta-algorithm that reuses any robust exact-submodular maximization algorithm as a black box by smoothing with a random surrogate $F^{H,t}$ and evaluating it via a contracted matroid $\\mathcal{I}_H$, obtaining $(\alpha - o(1))$-approximation guarantees under noise. When instantiated with the measured Continuous Greedy method, it achieves $(1-1/e)$ for monotone and $(1/e)$ for non-monotone submodular maximization subject to matroid constraints under noise; with the Double Greedy method, it yields a $1/2$-approximation for unconstrained non-monotone maximization under noise. The results extend prior work by providing a unified, algorithm-agnostic approach that preserves near-optimal guarantees in noisy settings, including for non-monotone objectives, and include both matroid-constrained and unconstrained cases along with high-level probabilistic analyses and simulations demonstrating practical performance improvements.

Abstract

We consider the problem of maximizing a submodular function with access to a noisy value oracle for the function instead of an exact value oracle. Similar to prior work, we assume that the noisy oracle is persistent in that multiple calls to the oracle for a specific set always return the same value. In this model, Hassidim and Singer (2017) design a $(1-1/e)$-approximation algorithm for monotone submodular maximization subject to a cardinality constraint, and Huang et al (2022) design a $(1-1/e)/2$-approximation algorithm for monotone submodular maximization subject to any arbitrary matroid constraint. In this paper, we design a meta-algorithm that allows us to take any "robust" algorithm for exact submodular maximization as a black box and transform it into an algorithm for the noisy setting while retaining the approximation guarantee. By using the meta-algorithm with the measured continuous greedy algorithm, we obtain a $(1-1/e)$-approximation (resp. $1/e$-approximation) for monotone (resp. non-monotone) submodular maximization subject to a matroid constraint under noise. Furthermore, by using the meta-algorithm with the double greedy algorithm, we obtain a $1/2$-approximation for unconstrained (non-monotone) submodular maximization under noise.

A Unified Approach to Submodular Maximization Under Noise

TL;DR

This work tackles maximizing submodular functions when only a noisy, persistent value oracle is available. It introduces a universal meta-algorithm that reuses any robust exact-submodular maximization algorithm as a black box by smoothing with a random surrogate and evaluating it via a contracted matroid , obtaining -approximation guarantees under noise. When instantiated with the measured Continuous Greedy method, it achieves for monotone and for non-monotone submodular maximization subject to matroid constraints under noise; with the Double Greedy method, it yields a -approximation for unconstrained non-monotone maximization under noise. The results extend prior work by providing a unified, algorithm-agnostic approach that preserves near-optimal guarantees in noisy settings, including for non-monotone objectives, and include both matroid-constrained and unconstrained cases along with high-level probabilistic analyses and simulations demonstrating practical performance improvements.

Abstract

We consider the problem of maximizing a submodular function with access to a noisy value oracle for the function instead of an exact value oracle. Similar to prior work, we assume that the noisy oracle is persistent in that multiple calls to the oracle for a specific set always return the same value. In this model, Hassidim and Singer (2017) design a -approximation algorithm for monotone submodular maximization subject to a cardinality constraint, and Huang et al (2022) design a -approximation algorithm for monotone submodular maximization subject to any arbitrary matroid constraint. In this paper, we design a meta-algorithm that allows us to take any "robust" algorithm for exact submodular maximization as a black box and transform it into an algorithm for the noisy setting while retaining the approximation guarantee. By using the meta-algorithm with the measured continuous greedy algorithm, we obtain a -approximation (resp. -approximation) for monotone (resp. non-monotone) submodular maximization subject to a matroid constraint under noise. Furthermore, by using the meta-algorithm with the double greedy algorithm, we obtain a -approximation for unconstrained (non-monotone) submodular maximization under noise.
Paper Structure (31 sections, 27 theorems, 65 equations, 2 tables, 3 algorithms)

This paper contains 31 sections, 27 theorems, 65 equations, 2 tables, 3 algorithms.

Table of Contents

  1. Introduction
  2. Main question.
  3. Our contributions.
  4. Other related works.
  5. Preliminaries
  6. Submodular set function.
  7. Noisy value oracle.
  8. Matroid constraints.
  9. A Unified Approach to Noisy Submodular Maximization
  10. Surrogate Function
  11. A Meta-Algorithm for Noisy Submodular Maximization
  12. Proof of Theorem \ref{['theorem:meta']}
  13. Noisy Submodular Maximization Under Specific Settings
  14. Matroid Constraints
  15. Robustness of Measured Continuous Greedy.
  16. ...and 16 more sections

Key Result

Theorem 3.1

Let $\mathcal{A}$ be a "robust" algorithm that obtains an $\alpha$-approximation ratio to the problem $\max_{S \in \mathcal{I}} f(S)$ with exact value oracle. Then, $\mathcal{A}$ can be converted into an algorithm achieving an $(\alpha - o(1))$-approximation to the problem $\max_{S \in \mathcal{I}}

Theorems & Definitions (50)

  • Definition 2.1: see, e.g., wainwright_high-dimensional_2019
  • Theorem 3.1: Informal
  • Remark 3.1
  • Claim 3.2
  • Lemma 3.3
  • Definition 3.1: Robustness
  • Theorem 3.4
  • Lemma 3.5: Smoothing lemma
  • proof : Proof of Theorem \ref{['theorem:meta']}
  • proof : Proof of Lemma \ref{['lem:smoothing']}
  • ...and 40 more