How Can Deep Neural Networks Fail Even With Global Optima?
Qingguang Guan
TL;DR
The work addresses the paradox that global optima in fully connected deep nets can still yield poor generalization. It introduces a simple depth-extension trick that preserves the target function while adding arbitrary depth via extra layers, supported by precise constructions for ReLU, Parametric ReLU, and Sigmoid activations. The authors provide explicit, high-dimensional 1D, 2D, and HD counterexamples where zero training loss coexists with localized, unreliable predictions, demonstrating severe overfitting despite global optima. The findings highlight fundamental limits of optimization-driven generalization in DNNs and point to the need for regularization and architectural considerations, with potential extensions to CNNs and RNNs.
Abstract
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.