Memo No . 90 March 3 , 2019 Theory III : Dynamics and Generalization in Deep Networks 1

We review recent observations on the dynamical systems induced by gradient descent methods used for training deep networks and summarize properties of the solutions they converge to. Recent results by [1] illuminate the puzzle in the special case of linear networks for binary classification. They prove that minimization of loss functions such as the logistic, the crossentropy and the exponential loss yields asymptotic convergence to the maximum margin solution for linearly separable datasets, independently of the initial conditions. Here we discuss the case of nonlinear multilayer DNNs near zero minima of the empirical loss, under exponential-type losses and square loss, for several variations of the basic gradient descent algorithm, including a new NMGD (norm minimizing gradient descent) version that converges to the minimum norm fixed points of the gradient descent iteration. Our main results are: • gradient descent algorithms with weight normalization constraint achieve generalization; • the fundamental reason for the effectiveness of existing weight normalization and batch normalization techniques is that they are approximate implementations of maximizing the margin under unit norm constraint; • without unit norm constraints some level of generalization can still be obtained for not-too-deep networks because the balance of the weights across different layers, if present at initialization, is maintained by the gradient flow [2]. In the perspective of these theoretical results, we discuss experimental evidence around the apparent absence of “overfitting”, that is the observation that the expected classification error does not get worse when increasing the number of parameters. Our explanation focuses on the implicit normalization enforced by algorithms such as batch normalization, since the control of the norm of the weights is related to Halpern iterations for minimum norm solutions which are equivalent to regularization with vanishing λ(t). This material is based upon work supported by the Center for Brains, Minds and Machines (CBMM), funded by NSF STC award CCF-1231216. 1This replaces previous versions of Theory IIIa and TheoryIIIb. Theory III: Dynamics and Generalization in Deep Networks∗ Andrzej Banburski 1, Qianli Liao1, Brando Miranda1, Tomaso Poggio1, Lorenzo Rosasco1, Bob Liang1, and Jack Hidary2 1Center for Brains, Minds and Machines, MIT 1CSAIL, MIT 2Alphabet (Google) X Abstract We review recent observations on the dynamical systems induced by gradient descent methods used for training deep networks and summarize properties of the solutions they converge to. Recent results by [1] illuminate the puzzle in the special case of linear networks for binary classification. They prove that minimization of loss functions such as the logistic, the cross-entropy and the exponential loss yields asymptotic convergence to the maximum margin solution for linearly separable datasets, independently of the initial conditions. Here we discuss the case of nonlinear multilayer DNNs near zero minima of the empirical loss, under exponential-type losses and square loss, for several variations of the basic gradient descent algorithm, including a new NMGD (norm minimizing gradient descent) version that converges to the minimum norm fixed points of the gradient descent iteration. Our main results are: • gradient descent algorithms with weight normalization constraint achieve generalization; • the fundamental reason for the effectiveness of existing weight normalization and batch normalization techniques is that they are approximate implementations of maximizing the margin under unit norm constraint; • without unit norm constraints some level of generalization can still be obtained for not-too-deep networks because the balance of the weights across different layers, if present at initialization, is maintained by the gradient flow[2]. In the perspective of these theoretical results, we discuss experimental evidence around the apparent absence of “overfitting”, that is the observation that the expected classification error does not get worse when increasing the number of parameters. Our explanation focuses on the implicit normalization enforced by algorithms such as batch normalization, since the control of the norm of the weights is related to Halpern iterations for minimum norm solutions which are equivalent to regularization with vanishing λ(t). ∗This replaces previous versions of Theory IIIa and TheoryIIIb. The basic analysis is reformulated with comments on work that appeared after the original version of our memos.We review recent observations on the dynamical systems induced by gradient descent methods used for training deep networks and summarize properties of the solutions they converge to. Recent results by [1] illuminate the puzzle in the special case of linear networks for binary classification. They prove that minimization of loss functions such as the logistic, the cross-entropy and the exponential loss yields asymptotic convergence to the maximum margin solution for linearly separable datasets, independently of the initial conditions. Here we discuss the case of nonlinear multilayer DNNs near zero minima of the empirical loss, under exponential-type losses and square loss, for several variations of the basic gradient descent algorithm, including a new NMGD (norm minimizing gradient descent) version that converges to the minimum norm fixed points of the gradient descent iteration. Our main results are: • gradient descent algorithms with weight normalization constraint achieve generalization; • the fundamental reason for the effectiveness of existing weight normalization and batch normalization techniques is that they are approximate implementations of maximizing the margin under unit norm constraint; • without unit norm constraints some level of generalization can still be obtained for not-too-deep networks because the balance of the weights across different layers, if present at initialization, is maintained by the gradient flow[2]. In the perspective of these theoretical results, we discuss experimental evidence around the apparent absence of “overfitting”, that is the observation that the expected classification error does not get worse when increasing the number of parameters. Our explanation focuses on the implicit normalization enforced by algorithms such as batch normalization, since the control of the norm of the weights is related to Halpern iterations for minimum norm solutions which are equivalent to regularization with vanishing λ(t). ∗This replaces previous versions of Theory IIIa and TheoryIIIb. The basic analysis is reformulated with comments on work that appeared after the original version of our memos.