Global Convergence of SGD For Logistic Loss on Two Layer Neural Nets

Published: 25 Feb 2024, Last Modified: 25 Feb 2024Accepted by TMLREveryoneRevisionsBibTeX
Abstract: In this note, we demonstrate a first-of-its-kind provable convergence of SGD to the global minima of appropriately regularized logistic empirical risk of depth $2$ nets -- for arbitrary data with any number of gates with adequately smooth and bounded activations, like sigmoid and tanh, and for a class of distributions from which the initial weight is sampled. We also prove an exponentially fast convergence rate for continuous time SGD that also applies to smooth unbounded activations like SoftPlus. Our key idea is to show that the logistic loss function on any size neural net can be Frobenius norm regularized by a width-independent parameter such that the regularized loss is a ``Villani function'' -- and thus be able to build on recent progress with analyzing SGD on such objectives.
Submission Length: Regular submission (no more than 12 pages of main content)
Changes Since Last Submission: We have rewritten the conclusion section to give a more clear comparison to this work -- and have delineated some of the next research questions that are motivated by comparing our work and theirs.
Assigned Action Editor: ~Atsushi_Nitanda1
License: Creative Commons Attribution 4.0 International (CC BY 4.0)
Submission Number: 1589