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Benefits of Deep and Wide Convolutional Residual Networks: Function Approximation under Smoothness Constraint
Hao Liu · Minshuo Chen · Siawpeng Er · Wenjing Liao · Tong Zhang · Tuo Zhao

Large (deep and wide) neural networks enjoy great representation power on complex data, and more importantly yield sufficiently smooth output, which is crucial to their generalization and robustness. Existing function approximation theories only 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.

Author Information

Hao Liu (Hong Kong Baptist University)
Minshuo Chen (Georgia Tech)
Siawpeng Er (Georgia Institute of Technology)
Wenjing Liao (Georgia Tech)
Tong Zhang (HKUST)
Tong Zhang

Tong Zhang is a professor of Computer Science and Mathematics at the Hong Kong University of Science and Technology. His research interests are machine learning, big data and their applications. He obtained a BA in Mathematics and Computer Science from Cornell University, and a PhD in Computer Science from Stanford University. Before joining HKUST, Tong Zhang was a professor at Rutgers University, and worked previously at IBM, Yahoo as research scientists, Baidu as the director of Big Data Lab, and Tencent as the founding director of AI Lab. Tong Zhang was an ASA fellow and IMS fellow, and has served as the chair or area-chair in major machine learning conferences such as NIPS, ICML, and COLT, and has served as associate editors in top machine learning journals such as PAMI, JMLR, and Machine Learning Journal.

Tuo Zhao (Georgia Tech)

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