光滑函数及其导数的序贯无梯度最小化的极小极大最优性
Minimax optimality for sequential gradient-free minimization of smooth functions and their derivatives
- ENSAE, CREST（法国国立经济统计行政学院，中央研究经济学研究所）
- Univ Rennes, Ensai, CNRS, CREST—UMR 9194（雷恩大学，ENSAI，法国国家科学研究中心，中央研究经济学研究所）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
针对β-Hölder函数及其导数的带噪无梯度最小化，提出非渐近极小极大速率，证明局部多项式估计量速率最优，并给出多项式时间算法匹配下界。
AI中文摘要：
我们考虑在d维立方体上支撑的β-Hölder函数的k阶偏导数的带噪无梯度最小化问题。我们证明T^{(β+d+k)/(2β+d)} log(T)^{(β-k)/(2β+d)}是所有β≥0的T步累积遗憾的非渐近极小极大速率。在特殊情况k=0下，我们的结果涵盖了β-Hölder函数的带噪无梯度最小化问题，弥合了已知上界和下界之间的现有差距。我们证明，适当选择的局部多项式估计量的极小化器是速率最优的。极小极大最优上界在被动设计下实现，即当查询点是独立同分布时。因此，当仅知道f是β-Hölder函数且没有额外性质时，考虑序贯设计没有优势。我们提出了一种在多项式时间内可行的算法，该算法构造局部多项式估计量的极小化器的代理。该过程需要在辅助随机点上计算估计量。所得到的多项式时间算法与下界匹配。
英文摘要：
We consider the problem of noisy gradient-free minimization of the k-th order partial derivative of a $β$-H{ö}lder function supported on a d-dimensional cube. We show that T ^{($β$+d+k)/(2$β$+d)} log(T )^{(β-k)/(2β+d)} is a non-asymptotic minimax rate of the T step cumulative regret for all $β$ \ge 0. In the special case k = 0, our results cover the problem of noisy gradient-free minimization of $β$-H{ö}lder functions, closing the existing gap between the known upper and lower bounds. We show that a minimizer of a suitably chosen local polynomial estimator is rate-optimal. The minimax optimal upper bound is achieved under the passive design, that is, when the query points are i.i.d. Thus, there is no advantage in considering sequential designs when it is only known that f is a $β$-H{ö}lder function with no additional property. We propose an algorithm feasible in polynomial time that constructs a proxy of the minimizer of the local polynomial estimator. The procedure requires computing the estimator on auxiliary random points. The resulting polynomial time algorithm matches the lower bound.
