On the different regimes of stochastic gradient descent.

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Abstract

Modern deep networks are trained with stochastic gradient descent (SGD) whose key hyperparameters are the number of data considered at each step or batch size [Formula: see text], and the step size or learning rate [Formula: see text]. For small [Formula: see text] and large [Formula: see text], SGD corresponds to a stochastic evolution of the parameters, whose noise amplitude is governed by the “temperature” [Formula: see text]. Yet this description is observed to break down for sufficiently large batches [Formula: see text], or simplifies to gradient descent (GD) when the temperature is sufficiently small. Understanding where these cross-overs take place remains a central challenge. Here, we resolve these questions for a teacher-student perceptron classification model and show empirically that our key predictions still apply to deep networks. Specifically, we obtain a phase diagram in the [Formula: see text]-[Formula: see text] plane that separates three dynamical phases: i) a noise-dominated SGD governed by temperature, ii) a large-first-step-dominated SGD and iii) GD. These different phases also correspond to different regimes of generalization error. Remarkably, our analysis reveals that the batch size [Formula: see text] separating regimes (i) and (ii) scale with the size [Formula: see text] of the training set, with an exponent that characterizes the hardness of the classification problem.