发表机构
University of Toronto; Vector Institute; Weizmann Institute of Science（多伦多大学; 向量研究所; 魏茨曼科学研究所）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
研究在单位球上最小化d维1-利普希茨凸函数且有m比特内存时的一阶预言机复杂度下界，通过证明给出随机和确定性算法的新下界，改进了之前结果，还揭示了确定性算法在$m\approx d^2$附近的复杂度相变及随机算法内存需求。
AI 中文摘要
我们证明了在单位球上最小化一个d维1-利普希茨凸函数且有m比特内存时一阶预言机复杂度的两个下界。首先表明任何这样的（可能是随机的）算法必须进行$\tilde{\Omega}(\frac{d^2}{\sqrt{m}})$次预言机查询。对于确定性优化算法，表明需要$\tilde{\Omega}(\min\{d^{1.6},\frac{d^{8/3}}{m^{2/3}}\})$次查询。在所有感兴趣的内存范围中，这些改进了之前随机和确定性算法分别的最佳已知下界$\tilde{\Omega}(\max\{\frac{d^{8/3}}{m^{4/3}},\frac{d^{4/3}}{m^{1/6}}\})$和$\tilde{\Omega}(\frac{d^{5/3}}{m^{1/3}})$。值得注意的是，由于现有上界，我们关于确定性算法的下界首次表明在$m\approx d^2$附近有一个尖锐的预言机复杂度相变，内存中的多对数变化会导致所需预言机调用次数的多项式变化。此外，当次优性在d中是多项式小的时候，但之前这种结果仅在次优性在d中是拟多项式小的情况下已知。我们关于随机算法的下界首次表明，对于无内存限制的算法中几乎匹配最优查询复杂度，需要$\tilde{\Omega}(d^2)$内存。
英文摘要
We prove two lower bounds for the first order oracle complexity of minimizing a $d$-dimensional $1$-Lipschitz convex function over the unit ball with $m$ bits of memory. We first show that any such (possibly randomized) algorithm must make $\tildeΩ(\frac{d^2}{\sqrt{m}})$ oracle queries. For deterministic optimization algorithms, we show that $\tildeΩ(\min\{d^{1.6},\frac{d^{8/3}}{m^{2/3}}\})$ queries are required. For all memory regimes of interest, these improves upon the previous best known lower bounds of $\tildeΩ(\max\{\frac{d^{8/3}}{m^{4/3}},\frac{d^{4/3}}{m^{1/6}}\})$ and $\tildeΩ(\frac{d^{5/3}}{m^{1/3}})$ for randomized and deterministic algorithms respectively. Notably, due to existing upper bounds, our lower bound for deterministic algorithms is the first to show a sharp oracle complexity phase transition around $m\approx d^2$, where a polylogarithmic change in memory leads to a $\mathsf{poly}(d)$ change in the number of required oracle calls. Further, when the suboptimality is polynomially small in $d$, our lower bound randomized algorithms is the first to show that $\tildeΩ(d^2)$ memory is necessary to nearly match the optimal query complexity among algorithms without memory constraints. Previously, such a result was only known for the regime where the suboptimality is quasipolynomially small in $d$.
