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-planes algorithms in dimension $d$ which use $\tilde O(d^2)$ memory and $\tilde O(d)$ queries are Pareto-optimal for both convex optimization and the feasibility problem, up to logarithmic factors. Precisely, we prove that to minimize $1$-Lipschitz convex functions over the unit ball to $1/d^4$ accuracy, any deterministic first-order algorithms using at most $d^{2-\delta}$ bits of memory must make $\tilde\Omega(d^{1+\delta/3})$ queries, for any $\delta\in[0,1]$. For the feasibility problem, in which an algorithm only has access to a separation oracle, we show a stronger trade-off: for at most $d^{2-\delta}$ memory, the number of queries required is $\tilde\Omega(d^{1+\delta})$. This resolves a COLT 2019 open problem of Woodworth and Srebro.
翻译：我们为 convex 优化及相关的可行性问题给出了更低的复杂度。 我们显示, 二次内存对于实现一级 convex 优化最优化的极复杂度是必要的。 特别是, 这显示, 使用 $tilde O (d ⁇ 2) 的存储量和 $tilde O (d) $tilde O (d) $d) 的询问, 使用 $treto- 最优的 convex 优化和可行性问题查询。 确切地说, 我们证明, 要将单位球上的 $- Lipschitz convex 函数最小化到 $/ d ⁇ 4 的精确度, 任何使用 $d ⁇ 2\\\\ delta} 美元 的确定性第一序列算法的算法, 使用 $tilde\ Omega (d ⁇ 1\ delta/3} $ 美元查询, 任何 $delto- pin, 和 opplements, 我们展示了更强烈的交易交易: $d\\\\\\\\\\\\\ delevleas leas leas demodeal demissional lement lementlement 20 lementlementlemental lemental declemental $.
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