On the different regimes of Stochastic Gradient Descent

TL;DR

This paper characterizes the different dynamical regimes of stochastic gradient descent in deep learning, identifying phase transitions based on batch size and learning rate, and relates these regimes to generalization performance.

Contribution

The paper provides a phase diagram of SGD regimes in the batch size and learning rate plane, revealing how these regimes depend on problem hardness and training set size.

Findings

Identifies three SGD regimes: noise-dominated, large-first-step, and gradient descent.

Shows the boundary between regimes scales with training set size and problem difficulty.

Empirically confirms phase diagram predictions in deep neural networks.

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 $B$, and the step size or learning rate $\eta$. For small $B$ and large $\eta$, SGD corresponds to a stochastic evolution of the parameters, whose noise amplitude is governed by the ''temperature'' $T\equiv \eta/B$. Yet this description is observed to break down for sufficiently large batches $B\geq B^*$, 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 $B$-$\eta$ 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 $B^*$ separating regimes (i) and (ii) scale with the size $P$ of the training set, with an exponent that characterizes the hardness of the classification problem.