Single-loop Algorithms for Stochastic Non-convex Optimization with Weakly-Convex Constraints
This addresses a crucial subset of constrained optimization problems in machine learning, offering a more efficient alternative to existing methods that are often slow or rely on double-loop designs.
The paper tackles stochastic non-convex optimization with weakly-convex constraints by introducing a single-loop penalty-based algorithm, achieving state-of-the-art complexity for approximate KKT solutions and demonstrating improved performance in fair learning and continual learning applications.
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 {\bf hinge-based penalty}, which permits the use of a constant penalty parameter, enabling us to achieve a {\bf 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.