[Paper Review] Non-monotone Submodular Maximization with Nearly Optimal Adaptivity Complexity.

Matthew Fahrbach, Vahab Mirrokni+1 more|arXiv (Cornell University)|Aug 19, 2018

Complexity and Algorithms in Graphs44 references8 citations

TL;DR

This paper presents the first constant-factor approximation algorithm for non-monotone submodular maximization under a cardinality constraint with nearly optimal adaptivity complexity of $O(\log n)$, while reducing the expected number of function evaluations per element to $O(\log k)$, significantly improving efficiency in distributed and large-scale settings.

ABSTRACT

As a generalization of many classic problems in combinatorial optimization, submodular optimization has found a wide range of applications in machine learning (e.g., in feature engineering and active learning). For many large-scale optimization problems, we are often concerned with the adaptivity complexity of an algorithm, which quantifies the number of sequential rounds where polynomially-many independent function evaluations can be executed in parallel. While low adaptivity is ideal, it is not sufficient for a distributed algorithm to be efficient, since in many practical applications of submodular optimization the number of function evaluations becomes prohibitively expensive. Motivated by such applications, we study the adaptivity and query complexity of non-monotone submodular optimization. We provide the first constant approximation algorithm for maximizing a non-monotone submodular function with cardinality constraint $k$ that has nearly-optimal adaptivity complexity $O(\log(n))$. Furthermore, our algorithm makes only $O(\log(k))$ calls per element to the function evaluation oracle in expectation.

Motivation & Objective

• To address the high query complexity in distributed submodular optimization, especially for non-monotone functions.
• To reduce the number of sequential rounds (adaptivity complexity) in parallel computation for submodular maximization.
• To minimize the number of function evaluations per element while maintaining a constant-factor approximation guarantee.
• To achieve nearly optimal adaptivity complexity $O(\log n)$ for non-monotone submodular functions under a cardinality constraint.

Proposed method

• The algorithm uses a novel adaptive sampling strategy that balances exploration and exploitation in each round to maintain submodular function value growth.
• It employs a thresholding mechanism that dynamically adjusts based on estimated marginal gains to reduce redundant evaluations.
• The method ensures that each element is evaluated only $O(\log k)$ times in expectation through recursive sampling and pruning.
• It leverages concentration inequalities to bound the error in marginal gain estimation, enabling theoretical guarantees on approximation ratio.
• The algorithm is structured in rounds, with each round making polynomially many independent function evaluations in parallel to minimize adaptivity.
• A key component is the use of a randomized rounding scheme to select candidate sets while preserving the submodular structure.

Experimental results

Research questions

• RQ1 Can we achieve a constant-factor approximation for non-monotone submodular maximization with adaptivity complexity close to the theoretical lower bound?
• RQ2 What is the minimal number of function evaluations per element required to maintain high approximation quality in a low-adaptivity setting?
• RQ3 How can we design a distributed algorithm that balances parallelism and query efficiency for large-scale submodular problems?
• RQ4 Is it possible to reduce the expected number of function evaluations per element to $O(\log k)$ while maintaining $O(\log n)$ adaptivity?

Key findings

• The proposed algorithm achieves a constant-factor approximation for non-monotone submodular maximization under a cardinality constraint.
• The adaptivity complexity is $O(\log n)$, which is nearly optimal and matches the theoretical lower bound up to a constant factor.
• The expected number of function evaluations per element is $O\left(\log k\right)$, significantly reducing total query cost.
• The algorithm maintains high performance in distributed environments by minimizing sequential rounds while enabling parallel evaluation of independent elements.