Abstract
We give query complexity lower bounds for convex optimization and the related feasibility problem. We show that quadratic memory is necessary to achieve the optimal oracle complexity for first-order convex optimization. In particular, this shows that center-of-mass cutting-plane algorithms in dimension d, which use [Formula: see text] memory and [Formula: see text] queries, are Pareto optimal for both convex optimization and the feasibility problem, up to logarithmic factors. Precisely, building upon techniques introduced in previous works, we prove that to minimize 1-Lipschitz convex functions over the unit ball to [Formula: see text] accuracy, any deterministic first-order algorithms using at most [Formula: see text] bits of memory must make [Formula: see text] queries for any [Formula: see text]. For the feasibility problem, in which an algorithm only has access to a separation oracle, we show a stronger trade-off; for at most [Formula: see text] memory, the number of queries required is [Formula: see text]. This resolves a Conference on Learning Theory 2019 open problem. Funding: This work was partly supported by the Air Force Office of Scientific Research [Grant FA9550-19-1-0263] and the Office of Naval Research [Grant N00014-18-1-2122].
Published Version
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