Efficient Convex Optimization Requires Superlinear Memory
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We show that any memory-constrained, first-order algorithm which minimizes d-dimensional, 1-Lipschitz convex functions over the unit ball to 1/poly(d) accuracy using at most d^1.25 - δ bits of memory must make at least Ω̃(d^1 + (4/3)δ) first-order queries (for any constant δ∈ [0, 1/4]). Consequently, the performance of such memory-constrained algorithms are a polynomial factor worse than the optimal Õ(d) query bound for this problem obtained by cutting plane methods that use Õ(d^2) memory. This resolves a COLT 2019 open problem of Woodworth and Srebro.
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