Increasing Depth Leads to U-Shaped Test Risk in Over-parameterized Convolutional Networks
This addresses a fundamental question in deep learning theory for researchers, providing insights into depth optimization in neural networks, though it is incremental as it builds on prior work on width effects.
The paper tackles the problem of understanding how increasing depth affects test risk in over-parameterized convolutional networks, finding that test risk follows a U-shaped curve (decreasing then increasing) with depth, supported by empirical evidence from ResNets and CNTK experiments and theoretical analysis in simplified settings.
Recent works have demonstrated that increasing model capacity through width in over-parameterized neural networks leads to a decrease in test risk. For neural networks, however, model capacity can also be increased through depth, yet understanding the impact of increasing depth on test risk remains an open question. In this work, we demonstrate that the test risk of over-parameterized convolutional networks is a U-shaped curve (i.e. monotonically decreasing, then increasing) with increasing depth. We first provide empirical evidence for this phenomenon via image classification experiments using both ResNets and the convolutional neural tangent kernel (CNTK). We then present a novel linear regression framework for characterizing the impact of depth on test risk, and show that increasing depth leads to a U-shaped test risk for the linear CNTK. In particular, we prove that the linear CNTK corresponds to a depth-dependent linear transformation on the original space and characterize properties of this transformation. We then analyze over-parameterized linear regression under arbitrary linear transformations and, in simplified settings, provably identify the depths which minimize each of the bias and variance terms of the test risk.