Unified Optimal Analysis of the (Stochastic) Gradient Method

In this note we give a simple proof for the convergence of stochastic gradient (SGD) methods on $\mu$-strongly convex functions under a (milder than standard) $L$-smoothness assumption. We show that SGD converges after $T$ iterations as $O\left( L \|x_0-x^\star\|^2 \exp \bigl[-\frac{\mu}{4L}T \bigr] + \frac{\sigma^2}{\mu T} \right)$ where $\sigma^2$ measures the variance. For deterministic gradient descent (GD) and SGD in the interpolation setting we have $\sigma^2 =0$ and we recover the exponential convergence rate. The bound matches with the best known iteration complexity of GD and SGD, up to constants.