Global Convergence of SGD For Logistic Loss on Two Layer Neural Nets
This provides a theoretical guarantee for training neural networks, addressing a fundamental problem in machine learning, though it is incremental as it builds on existing analysis of Villani functions.
The authors proved that SGD converges to global minima for regularized logistic loss on two-layer neural nets with smooth bounded activations, and showed exponential convergence rates for continuous-time SGD on smooth unbounded activations.
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 and with any number of gates with adequately smooth and bounded activations like sigmoid and tanh. 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 the existence of Frobenius norm regularized logistic loss functions on constant-sized neural nets which are "Villani functions" and thus be able to build on recent progress with analyzing SGD on such objectives.