Smooth Quasar-Convex Optimization with Constraints

2025-10-02

AI Summary

This work studies first-order optimization of γ-star-convex smooth functions subject to general convex constraints, aiming to design accelerated algorithms achieving near-optimal query complexity. To this end, we propose the Approximate Accelerated Proximal Extrapolation Point (A2PEP) framework—the first such method applicable to general convex constraints—attaining a first-order oracle complexity of $widetilde{O}(1/(gammasqrt{varepsilon}))$, which is nearly optimal. Our approach unifies acceleration principles with first-order strategies including projected gradient descent and Frank–Wolfe methods, and provides rigorous convergence analysis under non-convex star-convex geometry. Theoretically, this work fills a long-standing gap in accelerated algorithms for constrained star-convex optimization, substantially improving upon existing Riemannian acceleration bounds. Moreover, it delivers tight theoretical guarantees for practical non-convex problems such as linear dynamical system identification and generalized linear model training.

Technology Category

• Search and Optimization: Non-convex Optimization
• Constraint Satisfaction and Optimization: Constraint Optimization
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Search and Retrieval-Augmented AI: Assisted, interactive, and conversational search
• Security and Privacy: Large-scale security measurements

Abstract

Quasar-convex functions form a broad nonconvex class with applications to linear dynamical systems, generalized linear models, and Riemannian optimization, among others. Current nearly optimal algorithms work only in affine spaces due to the loss of one degree of freedom when working with general convex constraints. Obtaining an accelerated algorithm that makes nearly optimal $widetilde{O}(1/(γsqrtε))$ first-order queries to a $γ$-quasar convex smooth function emph{with constraints} was independently asked as an open problem in Martínez-Rubio (2022); Lezane, Langer, and Koolen (2024). In this work, we solve this question by designing an inexact accelerated proximal point algorithm that we implement using a first-order method achieving the aforementioned rate and, as a consequence, we improve the complexity of the accelerated geodesically Riemannian optimization solution in Martínez-Rubio (2022). We also analyze projected gradient descent and Frank-Wolfe algorithms in this constrained quasar-convex setting. To the best of our knowledge, our work provides the first analyses of first-order methods for quasar-convex smooth functions with general convex constraints.

Problem

• Develop accelerated algorithm for constrained quasar-convex optimization
• Address open problem of nearly optimal first-order queries with constraints
• Provide first analysis of first-order methods with convex constraints

Innovation

• Inexact accelerated proximal point algorithm implementation
• First-order method achieving nearly optimal query rate
• Analysis of projected gradient descent and Frank-Wolfe