A Minimalist Example of Edge-of-Stability and Progressive Sharpening
This work addresses a fundamental problem in deep learning optimization theory, offering new insights that could inform more effective strategies, though it is incremental as it builds on existing minimalist and generalist approaches.
The paper tackles the challenge of explaining Edge of Stability and Progressive Sharpening phenomena in deep learning optimization under large learning rates by introducing a two-layer network with a two-dimensional input, rigorously proving the existence of progressive sharpening and self-stabilization, and providing non-asymptotic analysis of training dynamics and sharpness.
Recent advances in deep learning optimization have unveiled two intriguing phenomena under large learning rates: Edge of Stability (EoS) and Progressive Sharpening (PS), challenging classical Gradient Descent (GD) analyses. Current research approaches, using either generalist frameworks or minimalist examples, face significant limitations in explaining these phenomena. This paper advances the minimalist approach by introducing a two-layer network with a two-dimensional input, where one dimension is relevant to the response and the other is irrelevant. Through this model, we rigorously prove the existence of progressive sharpening and self-stabilization under large learning rates, and establish non-asymptotic analysis of the training dynamics and sharpness along the entire GD trajectory. Besides, we connect our minimalist example to existing works by reconciling the existence of a well-behaved ``stable set" between minimalist and generalist analyses, and extending the analysis of Gradient Flow Solution sharpness to our two-dimensional input scenario. These findings provide new insights into the EoS phenomenon from both parameter and input data distribution perspectives, potentially informing more effective optimization strategies in deep learning practice.