The adaptive round complexity of finding an epsilon-stationary point is Omega(epsilon^{-(p+1)/p}) in high dimension even with poly(d) parallel queries, and near-matching per-round query bounds are given in constant dimension.
Lower Bound for Randomized First Order Convex Optimization
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abstract
We provide an explicit construction and direct proof for the lower bound on the number of first order oracle accesses required for a randomized algorithm to minimize a convex Lipschitz function.
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The Adaptive Complexity of Finding a Stationary Point
The adaptive round complexity of finding an epsilon-stationary point is Omega(epsilon^{-(p+1)/p}) in high dimension even with poly(d) parallel queries, and near-matching per-round query bounds are given in constant dimension.