Understanding the Loss Surface of Neural Networks for Binary Classification
This provides theoretical insights into neural network optimization for researchers, but it is incremental as it builds on existing conjectures and focuses on a narrow case.
The paper tackles the problem of understanding why neural network training algorithms succeed by analyzing the loss surface for binary classification, showing that under specific conditions (strictly convex neurons and smooth hinge loss), all local minima achieve zero training error, while counterexamples with quadratic or logistic loss demonstrate this does not always hold.
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.