The role of optimization geometry in single neuron learning

TL;DR

This paper investigates how the choice of optimization geometry affects the generalization performance in learning models, analyzing simplified models and providing theoretical bounds supported by experiments.

Contribution

It introduces a theoretical analysis of pseudogradient methods for generalized linear models, revealing how optimization geometry influences out-of-sample performance.

Findings

Optimization geometry impacts generalization error bounds.

Theoretical bounds characterize the interplay between geometry and feature space.

Experimental results show improved performance with geometry-informed methods.

Abstract

Recent numerical experiments have demonstrated that the choice of optimization geometry used during training can impact generalization performance when learning expressive nonlinear model classes such as deep neural networks. These observations have important implications for modern deep learning but remain poorly understood due to the difficulty of the associated nonconvex optimization problem. Towards an understanding of this phenomenon, we analyze a family of pseudogradient methods for learning generalized linear models under the square loss - a simplified problem containing both nonlinearity in the model parameters and nonconvexity of the optimization which admits a single neuron as a special case. We prove non-asymptotic bounds on the generalization error that sharply characterize how the interplay between the optimization geometry and the feature space geometry sets the out-of-sample performance of the learned model. Experimentally, selecting the optimization geometry as suggested by our theory leads to improved performance in generalized linear model estimation problems such as nonlinear and nonconvex variants of sparse vector recovery and low-rank matrix sensing.