  Large-scale nonconvex optimization problems are ubiquitous in modern machine
learning, and among practitioners interested in solving them, Stochastic
Gradient Descent (SGD) reigns supreme. We revisit the analysis of SGD in the
nonconvex setting and propose a new variant of the recently introduced expected
smoothness assumption which governs the behaviour of the second moment of the
stochastic gradient. We show that our assumption is both more general and more
reasonable than assumptions made in all prior work. Moreover, our results yield
the optimal $\mathcal{O}(\varepsilon^{-4})$ rate for finding a stationary point
of nonconvex smooth functions, and recover the optimal
$\mathcal{O}(\varepsilon^{-1})$ rate for finding a global solution if the
Polyak-{\L}ojasiewicz condition is satisfied. We compare against convergence
rates under convexity and prove a theorem on the convergence of SGD under
Quadratic Functional Growth and convexity, which might be of independent
interest. Moreover, we perform our analysis in a framework which allows for a
detailed study of the effects of a wide array of sampling strategies and
minibatch sizes for finite-sum optimization problems. We corroborate our
theoretical results with experiments on real and synthetic data.
