Convex Dominance in Deep Learning I: A Scaling Law of Loss and Learning Rate
This work provides a theoretical framework for optimizing deep learning models, potentially improving training efficiency across various tasks, though it is incremental in applying convexity concepts to non-convex settings.
The paper tackled the challenge of analyzing and controlling deep learning optimization dynamics by showing that the loss landscape becomes weakly convex early in training, enabling predictable loss bounds and optimal learning rate scaling. They derived scaling laws for learning rates and losses that extrapolate up to 80X across training horizons and 70X across model sizes.
Deep learning has non-convex loss landscape and its optimization dynamics is hard to analyze or control. Nevertheless, the dynamics can be empirically convex-like across various tasks, models, optimizers, hyperparameters, etc. In this work, we examine the applicability of convexity and Lipschitz continuity in deep learning, in order to precisely control the loss dynamics via the learning rate schedules. We illustrate that deep learning quickly becomes weakly convex after a short period of training, and the loss is predicable by an upper bound on the last iterate, which further informs the scaling of optimal learning rate. Through the lens of convexity, we build scaling laws of learning rates and losses that extrapolate as much as 80X across training horizons and 70X across model sizes.