AI 中文总结
研究零阶凸优化中精确函数值访问的预言机限制，针对在\(d\)维欧几里得单位球上最小化非光滑凸函数的自适应随机算法，给出查询次数下限，在不同精度 regime 与已有上界匹配，采用新方法证明。
AI 中文摘要
在零阶优化中，精确函数值访问的基本预言机限制仍未得到很好的理解：即使对于典型的凸问题，对维度\(d\)和精度\(\epsilon\)的最优联合依赖仍未解决。我们针对在\(d\)维欧几里得单位球上最小化非光滑\(L_0\)-Lipschitz凸函数的任意自适应随机算法解决了这个问题，其中每个查询仅返回固定目标的精确标量值\(f(\mathbf x)\)。对于通用的\(L_0>0\)，令\(T_\epsilon\)表示以至少\(1/2\)的概率返回\(\epsilon\)-次优解所需的最小查询次数，在函数类上均匀分布。我们证明，对于所有\(d\geq d_0\)和\(0<\epsilon\leq\epsilon_0\)，对于通用常数\(c,\epsilon_0>0\)和\(d_0\in\mathbb N\)，有\(T_\epsilon \geq c\, \frac{ d\min\{d,\epsilon^{-2}\} }{ \log\!\bigl(e\min\{d,\epsilon^{-2}\}\bigr) }\)。在低精度 regime\(\epsilon\geq d^{-1/2}\)中，这与\(O(d\epsilon^{-2})\)两点精确值上界在对数因子上匹配。在高精度 regime\(\epsilon\leq d^{-1/2}\)中，下界饱和为\(\Omega(d^2/\log(ed))\)，与\(\epsilon\)无关，与\(\widetilde O(d^2)\)评估预言机上界在多对数因子上匹配。证明使用了随机支持函数硬族，并为自适应精确最大观测开发了后验平均能量方法，代替了一阶零链构造和基于噪声的转录不等式。
英文摘要
Whether exact scalar feedback intrinsically incurs the additional dimension $d$ paid by known zeroth-order methods remains open even for Lipschitz convex optimization. For a universal Lipschitz scale, the value only bound $O(d^2\log(d+1)\log(1/ε))$ and two-point bound $O(dε^{-2})$ yield the upper bound $\widetilde O\left(d\min\{d,ε^{-2}\}\right)$. By contrast, prior lower bounds for arbitrary randomized algorithms give only $Ω(\min\{d,ε^{-2}\})$, leaving a factor $d$ unexplained. We close this gap, up to logarithmic factors, for arbitrary adaptive randomized algorithms minimizing a convex objective with a universal Lipschitz scale over the $d$-dimensional Euclidean unit ball, where each query returns only the exact scalar value. Let $T_ε$ denote the minimum number of queries required to return an $ε$-suboptimal point with probability at least $1/2$, uniformly over the function class. We prove that \[T_ε\ge c\,\frac{d\min\{d,ε^{-2}\}}{\log\!\bigl(\min\{d,ε^{-2}\}\bigr)},\] for $d\ge d_0$ and $0<ε\leε_0$, where $c,ε_0>0$ and $d_0\in\mathbb N$ are universal constants. This gives $Ω\left(\frac{d}{ε^2\log(1/ε)}\right)$ in the low-accuracy regime $ε\ge d^{-1/2}$ and $Ω\left(\frac{d^2}{\log d}\right)$ in the high-accuracy regime $ε\le d^{-1/2}$ with the latter independent of $ε$. These bounds match the corresponding upper bound up to logarithmic factors. To our knowledge, this is the first near-optimal lower bound for arbitrary adaptive randomized algorithms throughout both accuracy regimes of exact value Lipschitz convex optimization. The proof uses a random support function hard family and develops a posterior mean energy method for adaptive exact max observations, in place of first-order zero chain constructions and noise based transcript inequalities.
