  We lower bound the complexity of finding $\epsilon$-stationary points (with
gradient norm at most $\epsilon$) using stochastic first-order methods. In a
well-studied model where algorithms access smooth, potentially non-convex
functions through queries to an unbiased stochastic gradient oracle with
bounded variance, we prove that (in the worst case) any algorithm requires at
least $\epsilon^{-4}$ queries to find an $\epsilon$ stationary point. The lower
bound is tight, and establishes that stochastic gradient descent is minimax
optimal in this model. In a more restrictive model where the noisy gradient
estimates satisfy a mean-squared smoothness property, we prove a lower bound of
$\epsilon^{-3}$ queries, establishing the optimality of recently proposed
variance reduction techniques.
