[Paper Review] Resilient Monotone Submodular Function Maximization

Vasileios Tzoumas, Konstantinos Gatsis+2 more|arXiv (Cornell University)|Mar 21, 2017

Complexity and Algorithms in Graphs32 references4 citations

TL;DR

This paper proposes a scalable approximation algorithm for resilient monotone submodular function maximization under adversarial failures or attacks, leveraging function curvature to achieve superior approximation guarantees. It ensures valid performance for any number of failures β and selections α, with O(|V|(α−β)) function evaluations and improved approximation ratios for low-curvature functions.

ABSTRACT

In this paper, we focus on applications in machine learning, optimization, and control that call for the resilient selection of a few elements, e.g. features, sensors, or leaders, against a number of adversarial denial-of-service attacks or failures. In general, such resilient optimization problems are hard, and cannot be solved exactly in polynomial time, even though they often involve objective functions that are monotone and submodular. Notwithstanding, in this paper we provide the first scalable, curvature-dependent algorithm for their approximate solution, that is valid for any number of attacks or failures, and which, for functions with low curvature, guarantees superior approximation performance. Notably, the curvature has been known to tighten approximations for several non-resilient maximization problems, yet its effect on resilient maximization had hitherto been unknown. We complement our theoretical analyses with supporting empirical evaluations.

Motivation & Objective

• Address the challenge of selecting a resilient set of elements (e.g., sensors, features, leaders) that maintains high utility even after β adversarial removals or failures.
• Overcome limitations of prior algorithms that are either computationally infeasible (exponential in β) or restricted to small β (≤√(2α)).
• Introduce curvature as a key parameter to tighten approximation bounds in resilient submodular optimization, previously unexplored in this context.
• Develop an algorithm that is both scalable and universally valid for any α and β, without restricting the number of failures.
• Provide theoretical guarantees on approximation performance that improve with decreasing function curvature.

Proposed method

• Propose a novel algorithm (Algorithm 1) that iteratively builds a set A by selecting elements with the highest marginal gain, while maintaining resilience against β worst-case removals.
• Use the concept of curvature (κ_f) to refine the approximation bound, where lower curvature leads to better performance guarantees.
• Introduce a key inequality (ineq. 24) that relates the marginal gain of a subset B*₂ given a set A⁺₁ to the value of A⁺₁, enabling tighter analysis.
• Define η as a normalization factor to relate the marginal gain of B*₂ to the value of A₂, which is used to derive the final approximation bound.
• Establish a theoretical approximation bound of (1−κ_f)⋅(1−e^{−(1−κ_f)⋅(1−η)})/((1−κ_f)⋅(1−η)) for κ_f ≠ 0, which improves with lower curvature.
• Prove that the algorithm runs in O(|V|(α−β)τ_f) time, where τ_f is the cost of one function evaluation, making it scalable for large-scale problems.

Experimental results

Research questions

• RQ1 Can curvature be used to improve approximation guarantees in resilient submodular optimization under adversarial failures?
• RQ2 Is it possible to design a scalable algorithm for resilient submodular maximization that works for any β and α, without restricting β to be small?
• RQ3 How does the curvature of the submodular function affect the approximation performance in the resilient setting?
• RQ4 Can a theoretical approximation bound be derived for resilient submodular maximization that depends on curvature and is tighter than existing bounds?
• RQ5 What is the computational complexity of the proposed algorithm, and does it scale to large problem instances with thousands of elements and failures?

Key findings

• The proposed algorithm guarantees an approximation ratio of at least (1−κ_f)⋅(1−e^{−(1−κ_f)⋅(1−η)})/((1−κ_f)⋅(1−η)) for any α and β, with the bound improving as curvature κ_f decreases.
• For low-curvature functions (e.g., κ_f < 0.5), the approximation ratio significantly exceeds the 29% bound of prior work, which is valid only for β ≤ √(2α).
• The algorithm runs in O(|V|(α−β)τ_f) time, making it scalable for large-scale problems with |V| in the thousands and β up to α/2.
• Theoretical analysis shows that the approximation bound tightens as curvature decreases, validating curvature as a critical factor in resilient optimization.
• The algorithm is universally valid for any α and β, unlike prior methods that restrict β to be sublinear in α.
• Empirical evaluations complement theoretical analysis, demonstrating the algorithm’s effectiveness in practical resilient selection tasks.