A Novel Unified Parametric Assumption for Nonconvex Optimization

2025-02-17

AI Summary

Theoretical convergence guarantees for nonconvex optimization are often significantly weaker than observed empirical performance, exposing a substantial theory-practice gap. Method: We propose the first tunable, unified assumption framework for nonconvex functions—encompassing prominent structural conditions such as the Polyak–Łojasiewicz (PL) inequality and weak convexity—designed to balance generality with analytical tractability. Based on this framework, we derive unified convergence theorems for both deterministic and stochastic gradient methods, rigorously characterizing how algorithmic convergence rates depend on the tunable parameters. Contribution/Results: Our analysis identifies parameter regimes under which efficient optimization is theoretically feasible. Empirical trajectory validation on canonical nonconvex tasks—including neural network training—confirms that the assumed condition holds along actual optimization paths. This provides a new, verifiable theoretical foundation for understanding the practical efficacy of nonconvex optimization algorithms.

Technology Category

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

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

Abstract

Nonconvex optimization is central to modern machine learning, but the general framework of nonconvex optimization yields weak convergence guarantees that are too pessimistic compared to practice. On the other hand, while convexity enables efficient optimization, it is of limited applicability to many practical problems. To bridge this gap and better understand the practical success of optimization algorithms in nonconvex settings, we introduce a novel unified parametric assumption. Our assumption is general enough to encompass a broad class of nonconvex functions while also being specific enough to enable the derivation of a unified convergence theorem for gradient-based methods. Notably, by tuning the parameters of our assumption, we demonstrate its versatility in recovering several existing function classes as special cases and in identifying functions amenable to efficient optimization. We derive our convergence theorem for both deterministic and stochastic optimization, and conduct experiments to verify that our assumption can hold practically over optimization trajectories.

Problem

• Address weak convergence in nonconvex optimization
• Bridge gap between theory and practice
• Introduce unified parametric assumption for efficiency

Innovation

• Novel unified parametric assumption
• Derived convergence theorem
• Versatility in function classes