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Conditioning analysis uncovers the landscape of an optimization objective byexploring the spectrum of its curvature matrix. This has been well exploredtheoretically for linear models. We extend this analysis to deep neuralnetworks (DNNs) in order to investigate their learning dynamics. To this end,we propose layer-wise conditioning analysis, which explores the optimizationlandscape with respect to each layer independently. Such an analysis istheoretically supported under mild assumptions that approximately hold inpractice. Based on our analysis, we show that batch normalization (BN) canstabilize the training, but sometimes result in the false impression of a localminimum, which has detrimental effects on the learning. Besides, weexperimentally observe that BN can improve the layer-wise conditioning of theoptimization problem. Finally, we find that the last linear layer of a verydeep residual network displays ill-conditioned behavior. We solve this problemby only adding one BN layer before the last linear layer, which achievesimproved performance over the original and pre-activation residual networks.

Layer-wise Conditioning Analysis in Exploring the Learning Dynamics of DNNs