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arXiv · 1712.01906

On the linear convergence of the stochastic gradient method with constant step-size

Volkan Cevher·Bang Cong Vu
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Abstract

The strong growth condition (SGC) is known to be a sufficient condition for linear convergence of the stochastic gradient method using a constant step-size $\gamma$ (SGM-CS). In this paper, we provide a necessary condition, for the linear convergence of SGM-CS, that is weaker than SGC. Moreover, when this necessary is violated up to a additive perturbation $\sigma$, we show that both the projected stochastic gradient method using a constant step-size (PSGM-CS) and the proximal stochastic gradient method exhibit linear convergence to a noise dominated region, whose distance to the optimal solution is proportional to $\gamma \sigma$.

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Volkan Cevher, Bang Cong Vu. 2017-12-05. On the linear convergence of the stochastic gradient method with constant step-size. https://arxiv.org/abs/1712.01906

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