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# Freeze and Chaos: NTK views on DNN Normalization, Checkerboard and Boundary Artifacts

*Proceedings of Mathematical and Scientific Machine Learning*, PMLR 190:257-270, 2022.

#### Abstract

We analyze architectural features of Deep Neural Networks (DNNs) using the so-called Neural Tangent Kernel (NTK), which describes the training and generalization of DNNs in the infinite-width setting. In this setting, we show that for fully-connected DNNs, as the depth grows, two regimes appear: freeze (or order), where the (scaled) NTK converges to a constant, and chaos, where it converges to a Kronecker delta. Extreme freeze slows down training while extreme chaos hinders generalization. Using the scaled ReLU as a nonlinearity, we end up in the frozen regime. In contrast, Layer Normalization brings the network into the chaotic regime. We observe a similar effect for Batch Normalization (BN) applied after the last nonlinearity. We uncover the same freeze and chaos modes in Deep Deconvolutional Networks (DC-NNs). Our analysis explains the appearance of so-called checkerboard patterns and border artifacts. Moving the network into the chaotic regime prevents checkerboard patterns; we propose a graph-based parametrization which eliminates border artifacts; finally, we introduce a new layer-dependent learning rate to improve the convergence of DC-NNs. We illustrate our findings on DCGANs: the frozen regime leads to a collapse of the generator to a checkerboard mode, which can be avoided by tuning the nonlinearity to reach the chaotic regime. As a result, we are able to obtain good quality samples for DCGANs without BN.