Deep networks are usually trained and tested in a regime in which the training classification error is not a good predictor of the test error. Thus the consensus has been that generalization, defined as convergence of the empirical to the expected error, does not hold for deep networks. Here we show that, when normalized appropriately after training, deep networks trained on exponential type losses show a good linear dependence of test loss on training loss. The observation, motivated by a previous theoretical analysis of overparametrization and overfitting, not only demonstrates the validity of classical generalization bounds for deep learning but suggests that they are tight. In addition, we also show that the bound of the classification error by the normalized cross entropy loss is empirically rather tight on the data sets we studied.

}, author = {Qianli Liao and Brando Miranda and Jack Hidary and Tomaso Poggio} } @article {3694, title = {Theory III: Dynamics and Generalization in Deep Networks}, year = {2018}, month = {06/2018}, abstract = {Key to generalization is the control of \ the complexity of the

\ \ hypothesis. However, there is no obvious control of complexity --

\ \ such as an explicit regularization term -- in the training of deep

\ \ networks. We will show that a classical form of norm control --

\ \ but kind of hidden -- is responsible for generalization in deep

\ \ networks trained with gradient descent techniques. In particular,

\ \ gradient descent induces a dynamics of the normalized weights

\ \ which converges to a hyperbolic equilibrium. \ Our approach extends

\ \ some of the results of Srebro from linear networks to deep

\ \ networks and provides a new perspective on the implicit bias of

\ \ gradient descent. The elusive complexity control we describe is

\ \ responsible, at least in part, for the puzzling empirical finding

\ \ of good generalization despite overparametrization by deep

\ \ networks.\

^{1}This replaces previous versions of Theory IIIa and Theory IIIb.

**THIS MEMO IS REPLACED BY CBMM MEMO 90**

A main puzzle of deep networks revolves around the absence of overfitting despite overparametrization and despite the large capacity demonstrated by zero training error on randomly labeled data. In this note, we show that the dynamical systems associated with gradient descent minimization of nonlinear networks behave near zero stable minima of the empirical error as gradient system in a quadratic potential with degenerate Hessian. The proposition is supported by theoretical and numerical results, under the assumption of stable minima of the gradient.

Our proposition provides the extension to deep networks of key properties of gradient descent methods for linear networks, that as, suggested in (1), can be the key to understand generalization. Gradient descent enforces a form of implicit regular- ization controlled by the number of iterations, and asymptotically converging to the minimum norm solution. This implies that there is usually an optimum early stopping that avoids overfitting of the loss (this is relevant mainly for regression). For classification, the asymptotic convergence to the minimum norm solution implies convergence to the maximum margin solution which guarantees good classification error for {\textquotedblleft}low noise{\textquotedblright} datasets.

The implied robustness to overparametrization has suggestive implications for the robustness of deep hierarchically local networks to variations of the architecture with respect to the curse of dimensionality.

}, author = {Tomaso Poggio and Keji Kawaguchi and Qianli Liao and Brando Miranda and Lorenzo Rosasco and Xavier Boix and Jack Hidary and Hrushikesh Mhaskar} }