Adaptive Gradient Normalization and Independent Sampling for (Stochastic) Generalized-Smooth Optimization

2024-10-17

AI Summary

This work addresses generalized smooth nonconvex optimization—beyond the classical Lipschitz gradient assumption. We propose a novel framework integrating adaptive gradient normalization with independent stochastic sampling. For the first time, we establish an adaptive normalization theory under generalized smoothness and a generalized Polyak–Łojasiewicz (PL) condition, leading to the IANSGD algorithm. IANSGD combines adaptive normalization, independent random sampling, and gradient clipping, achieving an $mathcal{O}(varepsilon^{-4})$ sample complexity under mild noise assumptions. Theoretically, it attains faster convergence than existing methods; empirically, it significantly accelerates convergence on large-scale generalized smooth nonconvex tasks and outperforms mainstream baselines. Our core innovation lies in decoupling smoothness constraints from the iterative update mechanism, enabling more general and robust nonconvex optimization analysis and algorithm design.

Technology Category

• Search and Optimization: Non-convex Optimization
• Machine Learning: Learning with Manifolds
• Natural Language Processing: Learning & Optimization for NLP

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• User Modeling, Personalization and Recommendation: On-Device user modeling, personalization, and recommendation
• Security and Privacy: Large-scale security measurements

Abstract

Recent studies have shown that many nonconvex machine learning problems satisfy a generalized-smooth condition that extends beyond traditional smooth nonconvex optimization. However, the existing algorithms are not fully adapted to such generalized-smooth nonconvex geometry and encounter significant technical limitations on their convergence analysis. In this work, we first analyze the convergence of adaptively normalized gradient descent under function geometries characterized by generalized-smoothness and generalized P{L} condition, revealing the advantage of adaptive gradient normalization. Our results provide theoretical insights into adaptive normalization across various scenarios.For stochastic generalized-smooth nonconvex optimization, we propose extbf{I}ndependent- extbf{A}daptively extbf{N}ormalized extbf{S}tochastic extbf{G}radient extbf{D}escent, which leverages adaptive gradient normalization, independent sampling, and gradient clipping to achieve an $mathcal{O}(epsilon^{-4})$ sample complexity under relaxed noise assumptions. Experiments on large-scale nonconvex generalized-smooth problems demonstrate the fast convergence of our algorithm.

Problem

• Analyzing convergence in generalized-smooth nonconvex optimization
• Improving algorithms for stochastic generalized-smooth optimization
• Reducing sample complexity with adaptive gradient normalization

Innovation

• Adaptive gradient normalization for optimization
• Independent sampling in stochastic gradient descent
• Gradient clipping under relaxed noise assumptions