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Deep neural networks (DNNs) are powerful tools for compressing and distillinginformation. Their scale and complexity, often involving billions ofinter-dependent parameters, render direct microscopic analysis difficult. Undersuch circumstances, a common strategy is to identify slow variables thataverage the erratic behavior of the fast microscopic variables. Here, weidentify a similar separation of scales occurring in fully trained finitelyover-parameterized deep convolutional neural networks (CNNs) and fullyconnected networks (FCNs). Specifically, we show that DNN layers couple onlythrough the second moment (kernels) of their activations and pre-activations.Moreover, the latter fluctuates in a nearly Gaussian manner. For infinite widthDNNs, these kernels are inert, while for finite ones they adapt to the data andyield a tractable data-aware Gaussian Process. The resulting thermodynamictheory of deep learning yields accurate predictions in various settings. Inaddition, it provides new ways of analyzing and understanding DNNs in general.

Separation of Scales and a Thermodynamic Description of Feature Learning in Some CNNs