光滑强凸优化的近最优精确值零阶复杂度

中文摘要 AI 辅助

我们研究使用精确标量函数值对全局$\beta$-光滑、$\mu$-强凸函数进行确定性自适应优化。查询和输出位于$B_2^d(R)$中，最小化器位于$B_2^d(R/2)$中。设$\kappa=\beta/\mu$，$Q=\beta R^2/\epsilon$，$D_d=(d/\log(ed))^{1/3}$。对于足够大的$d$和$0<\epsilon\le c_\epsilon\beta R^2$，极小极大值复杂度$N_\epsilon$满足\\[ \begin{aligned} N_\epsilon &\ge c d\min\{\sqrt Q,\sqrt\kappa,D_d\},\\\\ N_\epsilon &\le C d\min\left\{ \sqrt Q,\sqrt\kappa[1+\log_+(Q/\kappa)] \right\}, \end{aligned} \\]其中$c,C,c_\epsilon>0$是通用常数，$\log_+(t)=\max\{0,\log t\}$。下界使用精确屏蔽的光滑链和批量延迟旋转；上界结合有限差分、加速和重启。当$\min\{Q,\kappa\}\le D_d^2$时，这些界在精度主导的区间$Q\le\kappa$和恒定相对精度$\epsilon=\Theta(\mu R^2)$下与常数匹配。对于$\kappa\le D_d^2$范围内任意更高的精度，这些界最多相差$1+\log(\mu R^2/\epsilon)$；最优精度依赖性在一般情况下仍未解决。

英文摘要

We study deterministic adaptive optimization of globally $β$-smooth, $μ$-strongly convex functions using exact scalar function values. Queries and outputs lie in $B_2^d(R)$, and the minimizer lies in $B_2^d(R/2)$. Set $κ=β/μ$, $Q=βR^2/ε$, and $D_d=(d/\log(ed))^{1/3}$. For sufficiently large $d$ and $0<ε\le c_εβR^2$, the minimax value complexity $N_ε$ satisfies \[ \begin{aligned} N_ε&\ge c d\min\{\sqrt Q,\sqrtκ,D_d\},\\ N_ε&\le C d\min\left\{ \sqrt Q,\sqrtκ[1+\log_+(Q/κ)] \right\}, \end{aligned} \] where $c,C,c_ε>0$ are universal constants and $\log_+(t)=\max\{0,\log t\}$. The lower bound uses an exactly shielded smooth chain and batched delayed rotations; the upper bound combines finite differences, acceleration, and restart. When $\min\{Q,κ\}\le D_d^2$, these bounds match up to constants in the accuracy-dominated regime $Q\leκ$ and at constant relative accuracy $ε=Θ(μR^2)$. For arbitrarily higher accuracy in the range $κ\le D_d^2$, the bounds differ by at most $1+\log(μR^2/ε)$; the optimal accuracy dependence remains unresolved in general.

发表机构

• Peking University（北京大学）

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