How Can Deep Neural Networks Fail Even With Global Optima?
This addresses a foundational issue in machine learning by showing that global optima do not guarantee good performance, which is incremental as it builds on known overfitting concerns.
The paper tackles the problem of deep neural networks failing to perform well even when achieving global optima, by constructing overfitting networks that perform poorly on classification and function approximation tasks, with theoretical analysis and numerical results supporting the findings.
Fully connected deep neural networks are successfully applied to classification and function approximation problems. By minimizing the cost function, i.e., finding the proper weights and biases, models can be built for accurate predictions. The ideal optimization process can achieve global optima. However, do global optima always perform well? If not, how bad can it be? In this work, we aim to: 1) extend the expressive power of shallow neural networks to networks of any depth using a simple trick, 2) construct extremely overfitting deep neural networks that, despite having global optima, still fail to perform well on classification and function approximation problems. Different types of activation functions are considered, including ReLU, Parametric ReLU, and Sigmoid functions. Extensive theoretical analysis has been conducted, ranging from one-dimensional models to models of any dimensionality. Numerical results illustrate our theoretical findings.