|Title||Theory III: Dynamics and Generalization in Deep Networks|
|Publication Type||CBMM Memos|
|Year of Publication||2018|
|Authors||Banburski, A, Liao, Q, Miranda, B, Poggio, T, Rosasco, L, Liang, B, Hidary, J|
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 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:
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).
1This replaces previous versions of Theory IIIa and Theory IIIb.
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