How To Make the Gradients Small Stochastically: Even Faster Convex and Nonconvex SGD

Zeyuan Allen-Zhu

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arXiv · 1801.02982

How To Make the Gradients Small Stochastically: Even Faster Convex and Nonconvex SGD

Zeyuan Allen-Zhu

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Abstract

Stochastic gradient descent (SGD) gives an optimal convergence rate when minimizing convex stochastic objectives $f(x)$. However, in terms of making the gradients small, the original SGD does not give an optimal rate, even when $f(x)$ is convex. If $f(x)$ is convex, to find a point with gradient norm $\varepsilon$, we design an algorithm SGD3 with a near-optimal rate $\tilde{O}(\varepsilon^{-2})$, improving the best known rate $O(\varepsilon^{-8/3})$ of [18]. If $f(x)$ is nonconvex, to find its $\varepsilon$-approximate local minimum, we design an algorithm SGD5 with rate $\tilde{O}(\varepsilon^{-3.5})$, where previously SGD variants only achieve $\tilde{O}(\varepsilon^{-4})$ [6, 15, 33]. This is no slower than the best known stochastic version of Newton's method in all parameter regimes [30].

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Zeyuan Allen-Zhu. 2018-01-08. How To Make the Gradients Small Stochastically: Even Faster Convex and Nonconvex SGD. https://arxiv.org/abs/1801.02982

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