Understanding Optimization in Deep Learning with Central Flows

2024-10-31

arXiv.org

AI Summary

Classical optimization theory fails to characterize the actual training dynamics of deterministic deep learning, which often exhibit complex oscillatory behavior near the “edge of stability.” Method: We propose the “central flow” differential equation—a time-averaged dynamical system that explicitly models the smoothed optimization trajectory. Our approach integrates differential equation modeling, temporal averaging analysis, and numerical simulation to dissect the intrinsic mechanisms of adaptive optimizers such as RMSProp. Contribution/Results: We uncover a previously unrecognized “regularized acceleration” effect: adaptive step sizes implicitly bias optimization toward low-curvature regions, thereby enhancing convergence efficiency. The central flow framework enables high-fidelity prediction of long-term optimization behavior across general neural networks. To our knowledge, this work provides the first principled, quantitative explanation for the empirical effectiveness of adaptive optimization algorithms.

Technology Category

• Machine Learning: Optimization
• Search and Optimization: Learning to Search
• Natural Language Processing: Learning & Optimization for NLP

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web
• User Modeling, Personalization and Recommendation: On-Device user modeling, personalization, and recommendation

Abstract

Optimization in deep learning remains poorly understood, even in the simple setting of deterministic (i.e. full-batch) training. A key difficulty is that much of an optimizer's behavior is implicitly determined by complex oscillatory dynamics, referred to as the"edge of stability."The main contribution of this paper is to show that an optimizer's implicit behavior can be explicitly captured by a"central flow:"a differential equation which models the time-averaged optimization trajectory. We show that these flows can empirically predict long-term optimization trajectories of generic neural networks with a high degree of numerical accuracy. By interpreting these flows, we reveal for the first time 1) the precise sense in which RMSProp adapts to the local loss landscape, and 2) an"acceleration via regularization"mechanism, wherein adaptive optimizers implicitly navigate towards low-curvature regions in which they can take larger steps. This mechanism is key to the efficacy of these adaptive optimizers. Overall, we believe that central flows constitute a promising tool for reasoning about optimization in deep learning.

Problem

• Traditional theories fail to describe deep learning optimization dynamics
• Optimizers operate in complex oscillatory edge of stability regime
• Analyzing time-averaged trajectories instead of exact oscillatory paths

Innovation

• Central flows model time-averaged optimizer trajectories
• Differential equations describe smoothed optimization dynamics
• Theory explains edge of stability regime behavior