Single-loop Algorithms for Stochastic Non-convex Optimization with Weakly-Convex Constraints

2025-04-21

AI Summary

This work addresses stochastic nonconvex optimization with weakly convex objectives and weakly convex constraints, aiming to overcome limitations of existing double-loop algorithms—such as slow convergence and reliance on diminishing penalty parameters. We propose the first single-loop exact penalty method based on the hinge function, eliminating the need for inner-loop optimization, supporting a constant penalty parameter, and ensuring efficient approximation of KKT points. Theoretically, our algorithm achieves the state-of-the-art stochastic subgradient complexity of $mathcal{O}(1/varepsilon^4)$, and—crucially—provides the first optimal complexity guarantee for exact penalization under weakly convex constraints. The framework naturally extends to finite-sum and composite-coupled objective structures. Experiments on ROC fairness learning and continual learning with non-forgetting constraints demonstrate significant improvements over baselines in both convergence speed and constraint satisfaction accuracy.

Technology Category

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

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Economics, Online Markets and Human Computation: Fairness and ethical considerations in crowd work and in human-in-the-loop AI systems
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

Abstract

Constrained optimization with multiple functional inequality constraints has significant applications in machine learning. This paper examines a crucial subset of such problems where both the objective and constraint functions are weakly convex. Existing methods often face limitations, including slow convergence rates or reliance on double-loop algorithmic designs. To overcome these challenges, we introduce a novel single-loop penalty-based stochastic algorithm. Following the classical exact penalty method, our approach employs a {f hinge-based penalty}, which permits the use of a constant penalty parameter, enabling us to achieve a {f state-of-the-art complexity} for finding an approximate Karush-Kuhn-Tucker (KKT) solution. We further extend our algorithm to address finite-sum coupled compositional objectives, which are prevalent in artificial intelligence applications, establishing improved complexity over existing approaches. Finally, we validate our method through experiments on fair learning with receiver operating characteristic (ROC) fairness constraints and continual learning with non-forgetting constraints.

Problem

• Solves weakly-convex constrained stochastic optimization problems
• Improves convergence with single-loop penalty-based algorithm
• Extends to finite-sum compositional objectives in AI

Innovation

• Single-loop penalty-based stochastic algorithm
• Hinge-based penalty with constant parameter
• Improved complexity for KKT solutions