aiqbench analysis
This research resolves a longstanding theoretical challenge in deep learning by defining "large" step sizes based on local gradient continuity rather than global smoothness. The proposed adaptive methods enable training at the edge of stability from the outset, allowing gradient descent to navigate sharp valleys and avoid premature convergence in flat regions that can hinder generalization. For practitioners, this implies that operating near stability thresholds with self-stabilizing algorithms can transform unsuccessful training runs into successful ones without requiring global smoothness assumptions.
Flatland: The Adventures of Gradient Descent with Large Step Sizes
arXiv:2606.06722v1 Announce Type: new Abstract: The training of neural networks often entails objective functions that are not globally $L$-smooth. For these functions, it is both theoretically and practically difficult to reply to the question: what is the largest possible step size that ensures the convergence of gradient descent (GD)? We address this longstanding open question in deep learning by providing a unifying definition of "large" step sizes that requires only local Lipschitz (or even H\"older) continuity of the gradient. We design first-order adaptive methods that provably yield large step sizes and show that they operate at the edge of stability (EoS) right from the start of the training. In particular, the loss decreases nonmonotonically and the product between the step size and sharpness, i.e., the largest eigenvalue of the Hessian, stays above the EoS threshold of 2 throughout training. Using our method, we are also able to minimize the sharpness all the way down to its global minimum. Contrary to expectation, we find that encountering globally-flat regions too early in the training may both slow down convergence and jeopardize the generalization ability of the network. Exploiting a self-stabilization argument, we allow GD to enter slightly sharper valleys and turn unsuccessful training runs into very successful ones.
Abstract:The training of neural networks often entails objective functions that are not globally $L$-smooth. For these functions, it is both theoretically and practically difficult to reply to the question: what is the largest possible step size that ensures the convergence of gradient descent (GD)? We address this longstanding open question in deep learning by providing a unifying definition of "large" step sizes that requires only local Lipschitz (or even Hölder) continuity of the gradient. We design first-order adaptive methods that provably yield large step sizes and show that they operate at the edge of stability (EoS) right from the start of the training. In particular, the loss decreases nonmonotonically and the product between the step size and sharpness, i.e., the largest eigenvalue of the Hessian, stays above the EoS threshold of 2 throughout training. Using our method, we are also able to minimize the sharpness all the way down to its global minimum. Contrary to expectation, we find that encountering globally-flat regions too early in the training may both slow down convergence and jeopardize the generalization ability of the network. Exploiting a self-stabilization argument, we allow GD to enter slightly sharper valleys and turn unsuccessful training runs into very successful ones.
Machine Learning (cs.LG)
arXiv:2606.06722 [cs.LG]
(or arXiv:2606.06722v1 [cs.LG] for this version)
https://doi.org/10.48550/arXiv.2606.06722 Focus to learn more arXiv-issued DOI via DataCite (pending registration)
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Full text under the publisher's open licence · 295 words
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