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 = {The key to generalization is controlling the complexity of

\ \ \ \ \ \ the network. However, there is no obvious control of

\ \ \ \ \ \ complexity -- such as an explicit regularization term --

\ \ \ \ \ \ in the training of deep networks for classification. We

\ \ \ \ \ \ will show that a classical form of norm control -- but

\ \ \ \ \ \ kind of hidden -- is present in deep networks trained with

\ \ \ \ \ \ gradient descent techniques on exponential-type losses. In

\ \ \ \ \ \ particular, gradient descent induces a dynamics of the

\ \ \ \ \ \ normalized weights which converge for $t \to \infty$ to an

\ \ \ \ \ \ equilibrium which corresponds to a minimum norm (or

\ \ \ \ \ \ maximum margin) solution. For sufficiently large but

\ \ \ \ \ \ finite $\rho$ -- and thus finite $t$ -- the dynamics

\ \ \ \ \ \ converges to one of several margin maximizers, with the

\ \ \ \ \ \ margin monotonically increasing towards a limit stationary

\ \ \ \ \ \ point of the flow. In the usual case of stochastic

\ \ \ \ \ \ gradient descent, most of the stationary points are likely

\ \ \ \ \ \ to be convex minima corresponding to a regularized,

\ \ \ \ \ \ constrained minimizer -- the network with normalized

\ \ \ \ \ \ weights-- which is stable and has asymptotic zero

\ \ \ \ \ \ generalization gap, asymptotically for $N \to \infty$,

\ \ \ \ \ \ where $N$ is the number of training examples. For finite,

\ \ \ \ \ \ fixed $N$ the generalizaton gap may not be zero but the

\ \ \ \ \ \ minimum norm property of the solution can provide, we

\ \ \ \ \ \ conjecture, good expected performance for suitable data

\ \ \ \ \ \ distributions. 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. We believe that the elusive complexity control we

\ \ \ \ \ \ describe is responsible for the puzzling empirical finding

\ \ \ \ \ \ of good predictive performance by deep networks, despite

\ \ \ \ \ \ overparametrization.\

[*formerly titled "Theory of Deep Learning III: Generalization Properties of SGD"*]

In Theory III we characterize with a mix of theory and experiments the generalization properties of Stochastic Gradient Descent in overparametrized deep convolutional networks. We show that Stochastic Gradient Descent (SGD) selects with high probability solutions that 1) have zero (or small) empirical error, 2) are degenerate as shown in Theory II and 3) have maximum generalization.

}, author = {Chiyuan Zhang and Qianli Liao and Alexander Rakhlin and Karthik Sridharan and Brando Miranda and Noah Golowich and Tomaso Poggio} } @article {3261, title = {Theory of Deep Learning IIb: Optimization Properties of SGD}, year = {2017}, month = {12/2017}, abstract = {In Theory IIb we characterize with a mix of theory and experiments the optimization of deep convolutional networks by Stochastic Gradient Descent. The main new result in this paper is theoretical and experimental evidence for the following conjecture about SGD: *SGD concentrates in probability - like the classical Langevin equation {\textendash} on large volume, {\textquotedblleft}flat{\textquotedblright} minima, selecting flat minimizers which are with very high probability also global minimizers*.

**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} } @article {2557, title = {Why and when can deep-but not shallow-networks avoid the curse of dimensionality: A review}, journal = {International Journal of Automation and Computing}, year = {2017}, month = {03/2017}, pages = {1-17}, abstract = {The paper reviews and extends an emerging body of theoretical results on deep learning including the conditions under which it can be exponentially better than shallow learning. A class of deep convolutional networks represent an important special case of these conditions, though weight sharing is not the main reason for their exponential advantage. Implications of a few key theorems are discussed, together with new results, open problems and conjectures.

}, keywords = {convolutional neural networks, deep and shallow networks, deep learning, function approximation, Machine Learning, Neural Networks}, doi = {10.1007/s11633-017-1054-2}, url = {http://link.springer.com/article/10.1007/s11633-017-1054-2?wt_mc=Internal.Event.1.SEM.ArticleAuthorOnlineFirst}, author = {Tomaso Poggio and Hrushikesh Mhaskar and Lorenzo Rosasco and Brando Miranda and Qianli Liao} } @article {2321, title = {Theory I: Why and When Can Deep Networks Avoid the Curse of Dimensionality?}, year = {2016}, month = {11/2016}, abstract = {[formerly titled "*Why and When Can Deep - but Not Shallow - Networks Avoid the Curse of Dimensionality: a Review*"]

The paper reviews and extends an emerging body of theoretical results on deep learning including the conditions under which it can be exponentially better than shallow learning. A class of deep convolutional networks represent an important special case of these conditions, though weight sharing is not the main reason for their exponential advantage. Implications of a few key theorems are discussed, together with new results, open problems and conjectures.

}, author = {Tomaso Poggio and Hrushikesh Mhaskar and Lorenzo Rosasco and Brando Miranda and Qianli Liao} }