Benefits of Overparameterized Convolutional Residual Networks: Function Approximation under Smoothness Constraint
This work addresses the need for smooth neural networks to enhance generalization and robustness, particularly in adversarial settings, though it is incremental by extending existing approximation theories to include smoothness constraints.
The paper tackles the problem of ensuring smoothness in overparameterized neural networks, proving that large convolutional residual networks can approximate target functions while maintaining sufficient first-order smoothness, with numerical experiments showing improved adversarial robustness in image classification.
Overparameterized neural networks enjoy great representation power on complex data, and more importantly yield sufficiently smooth output, which is crucial to their generalization and robustness. Most existing function approximation theories suggest that with sufficiently many parameters, neural networks can well approximate certain classes of functions in terms of the function value. The neural network themselves, however, can be highly nonsmooth. To bridge this gap, we take convolutional residual networks (ConvResNets) as an example, and prove that large ConvResNets can not only approximate a target function in terms of function value, but also exhibit sufficient first-order smoothness. Moreover, we extend our theory to approximating functions supported on a low-dimensional manifold. Our theory partially justifies the benefits of using deep and wide networks in practice. Numerical experiments on adversarial robust image classification are provided to support our theory.