We study the randomized $n$-th minimal errors (and hence the complexity) of vector valued approximation. In a recent paper by the author [Randomized complexity of parametric integration and the role of adaption I. Finite dimensional case (preprint)] a long-standing problem of Information-Based Complexity was solved: Is there a constant $c>0$ such that for all linear problems $\mathcal{P}$ the randomized non-adaptive and adaptive $n$-th minimal errors can deviate at most by a factor of $c$? That is, does the following hold for all linear $\mathcal{P}$ and $n\in {\mathbb N}$ \begin{equation*} e_n^{\rm ran-non} (\mathcal{P})\le ce_n^{\rm ran} (\mathcal{P}) \, {\bf ?} \end{equation*} The analysis of vector-valued mean computation showed that the answer is negative. More precisely, there are instances of this problem where the gap between non-adaptive and adaptive randomized minimal errors can be (up to log factors) of the order $n^{1/8}$. This raises the question about the maximal possible deviation. In this paper we show that for certain instances of vector valued approximation the gap is $n^{1/2}$ (again, up to log factors).

翻译：我们研究向量值逼近的随机化$n$阶最小误差（进而研究其复杂度）。作者近期论文[随机化参数积分复杂度与自适应性的作用 I：有限维情形（预印本）]解决了信息复杂度领域一个长期未决问题：是否存在常数$c>0$，使得对所有线性问题$\mathcal{P}$，其随机化非自适应与自适应$n$阶最小误差的偏差不超过因子$c$？即，是否对所有线性$\mathcal{P}$及$n\in {\mathbb N}$满足 \begin{equation*} e_n^{\rm ran-non} (\mathcal{P})\le ce_n^{\rm ran} (\mathcal{P}) \, {\bf ?} \end{equation*} 向量值均值计算的分析表明该答案为否定。更精确地说，该问题存在实例使得非自适应与自适应随机化最小误差间的差距（在对数因子意义下）可达$n^{1/8}$量级。这引出了最大可能偏差的探究。本文证明，对向量值逼近的某些实例，该差距为$n^{1/2}$（同样在对数因子意义下）。