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arXiv · 2410.24206

Understanding Optimization in Deep Learning with Central Flows

Jeremy M. Cohen·Alex Damian·Ameet Talwalkar·J. Zico Kolter·Jason D. Lee
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Abstract

Traditional theories of optimization cannot describe the dynamics of optimization in deep learning, even in the simple setting of deterministic training. The challenge is that optimizers typically operate in a complex, oscillatory regime called the "edge of stability." In this paper, we develop theory that can describe the dynamics of optimization in this regime. Our key insight is that while the *exact* trajectory of an oscillatory optimizer may be challenging to analyze, the *time-averaged* (i.e. smoothed) trajectory is often much more tractable. To analyze an optimizer, we derive a differential equation called a "central flow" that characterizes this time-averaged trajectory. We empirically show that these central flows can predict long-term optimization trajectories for generic neural networks with a high degree of numerical accuracy. By interpreting these central flows, we are able to understand how gradient descent makes progress even as the loss sometimes goes up; how adaptive optimizers "adapt" to the local loss landscape; and how adaptive optimizers implicitly navigate towards regions where they can take larger steps. Our results suggest that central flows can be a valuable theoretical tool for reasoning about optimization in deep learning.

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Jeremy M. Cohen, Alex Damian, Ameet Talwalkar, J. Zico Kolter, Jason D. Lee. 2025-09-25. Understanding Optimization in Deep Learning with Central Flows. https://arxiv.org/abs/2410.24206

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