Minimisation of Submodular Functions Using Gaussian Zeroth-Order Random Oracles

2025-10-16

AI Summary

This work addresses submodular function minimization in black-box optimization settings where explicit gradients are unavailable. We propose the first algorithmic framework integrating Gaussian smoothing with zero-order stochastic estimation for submodular minimization, enabling both offline and online solutions via gradient-free stochastic approximation. Theoretically, our method converges to an ε-approximate global optimum in polynomial time in the offline setting; in the online setting, it achieves Hannan consistency and attains a dynamic regret bound of O(√(N P_N^*)), where P_N^* denotes the path-length. Our approach unifies Gaussian smoothing, zero-order optimization, and static/dynamic regret analysis. Numerical experiments validate its effectiveness and robustness across both offline and online scenarios. The key contribution lies in establishing, for the first time, a rigorous theoretical connection between zero-order stochastic optimization and submodular structure—overcoming classical limitations requiring either gradient access or restrictive structural assumptions.

Technology Category

• Reasoning under Uncertainty: Stochastic Optimization
• Search and Optimization: Non-convex Optimization
• Machine Learning: Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

Abstract

We consider the minimisation problem of submodular functions and investigate the application of a zeroth-order method to this problem. The method is based on exploiting a Gaussian smoothing random oracle to estimate the smoothed function gradient. We prove the convergence of the algorithm to a global $ε$-approximate solution in the offline case and show that the algorithm is Hannan-consistent in the online case with respect to static regret. Moreover, we show that the algorithm achieves $O(sqrt{NP_N^ast})$ dynamic regret, where $N$ is the number of iterations and $P_N^ast$ is the path length. The complexity analysis and hyperparameter selection are presented for all the cases. The theoretical results are illustrated via numerical examples.

Problem

• Minimizing submodular functions via zeroth-order optimization methods
• Proving convergence to global approximate solutions with guarantees
• Analyzing static and dynamic regret bounds for online cases

Innovation

• Uses Gaussian smoothing for gradient estimation
• Achieves convergence to global approximate solution
• Provides dynamic regret bounds for optimization