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：我们给出了凸优化及其相关可行性问题的查询复杂度下界。研究表明，对于一阶凸优化，要达到最优的预言机复杂度，必须使用二次内存。特别地，这表明维度为$d$、使用了$\tilde O(d^2)$内存和$\tilde O(d)$次查询的质心切割平面算法，在凸优化和可行性问题上均为帕累托最优（在对数因子意义下）。具体而言，我们证明：对于在单位球面上最小化$1$-Lipschitz凸函数至$1/d^4$精度的问题，任何使用至多$d^{2-\delta}$位内存的确定性一阶算法，都需进行$\tilde\Omega(d^{1+\delta/3})$次查询，其中$\delta\in[0,1]$。对于仅能访问分隔预言机的可行性问题，我们给出了更强的权衡关系：当内存不超过$d^{2-\delta}$时，所需查询次数为$\tilde\Omega(d^{1+\delta})$。这一结果解决了Woodworth和Srebro在COLT 2019上提出的公开问题。