  We show that any memory-constrained, first-order algorithm which minimizes
$d$-dimensional, $1$-Lipschitz convex functions over the unit ball to
$1/\mathrm{poly}(d)$ 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.
