The Impact of the Mini-batch Size on the Dynamics of SGD: Variance and Beyond

Xin Qian, Diego Klabjan

Abstract: We study mini-batch stochastic gradient descent (SGD) dynamics under linear regression and deep linear networks by focusing on the variance of the gradients only given the initial weights and mini-batch size, which is the first study of this nature. In the linear regression case, we show that in each iteration the norm of the gradient is a decreasing function of the mini-batch size $b$ and thus the variance of the stochastic gradient estimator is a decreasing function of $b$. For deep neural networks with $L_2$ loss we show that the variance of the gradient is a polynomial in $1/b$. The results theoretically back the important intuition that smaller batch sizes yield larger variance of the stochastic gradients and lower loss function values which is a common believe among the researchers. The proof techniques exhibit a relationship between stochastic gradient estimators and initial weights, which is useful for further research on the dynamics of SGD. We empirically provide insights to our results on various datasets and commonly used deep network structures. We further discuss possible extensions of the approaches we build in studying the generalization ability of the deep learning models.

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