Understanding the Loss Surface of Neural Networks for Binary Classification

TL;DR

This paper investigates the loss surface of single-layer neural networks for binary classification, showing conditions under which all local minima achieve zero training error with smooth hinge loss, and contrasting with other loss functions.

Contribution

It establishes conditions for zero training error at all local minima for single-layer networks with smooth hinge loss, highlighting the importance of the loss function choice.

Findings

All local minima have zero training error under certain convexity conditions.

Smooth hinge loss guarantees zero training error at local minima for single-layer networks.

Quadratic and logistic losses do not always lead to zero training error at local minima.

Abstract

It is widely conjectured that the reason that training algorithms for neural networks are successful because all local minima lead to similar performance, for example, see (LeCun et al., 2015, Choromanska et al., 2015, Dauphin et al., 2014). Performance is typically measured in terms of two metrics: training performance and generalization performance. Here we focus on the training performance of single-layered neural networks for binary classification, and provide conditions under which the training error is zero at all local minima of a smooth hinge loss function. Our conditions are roughly in the following form: the neurons have to be strictly convex and the surrogate loss function should be a smooth version of hinge loss. We also provide counterexamples to show that when the loss function is replaced with quadratic loss or logistic loss, the result may not hold.