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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 $$d^{1.25-\delta}$$ bits of memory must make at least $$\tilde{Omega}(d^{1+(4/3)\delta})$$ first-order queries (for any constant $$\delta in [0,1/4]$$). Consequently, the performance of such memory-constrained algorithms are a polynomial factor worse than the optimal $$\tilde{O}(d)$$ query bound for this problem obtained by cutting plane methods that use $$\tilde{O}(d^2)$$ memory. This resolves a COLT 2019 open problem of Woodworth and Srebro.more » « less
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