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Stochastic gradient descent (SGD) gives an optimal convergence rate whenminimizing convex stochastic objectives $f(x)$. However, in terms of making thegradients 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 $\varepsilon$, wedesign 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 $\tilde{O}(\varepsilon^{-3.5})$, wherepreviously 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 inall parameter regimes [30].

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