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.<\/p>\n","protected":false},"excerpt":{"rendered":"
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 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