Recently the adaption problem of Information-Based Complexity (IBC) for linear problems in the randomized setting was solved in Heinrich (J. Complexity 82, 2024, 101821). Several papers treating further aspects of this problem followed. However, all examples obtained so far were vector-valued. In this paper we settle the scalar-valued case. We study the complexity of mean computation in finite dimensional sequence spaces with mixed $L_p^N$ norms. We determine the $n$-th minimal errors in the randomized adaptive and non-adaptive setting. It turns out that among the problems considered there are examples where adaptive and non-adaptive $n$-th minimal errors deviate by a power of $n$. The gap can be (up to log factors) of the order $n^{1/4}$. We also show how to turn such results into infinite dimensional examples with suitable deviation for all $n$ simultaneously.

翻译：近期，Heinrich (J. Complexity 82, 2024, 101821) 解决了随机环境下线性问题的信息复杂度自适应问题。随后出现了多篇探讨该问题其他方面的论文，但迄今所有实例均为向量值情形。本文解决了标量值情形。我们研究了具有混合 $L_p^N$ 范数的有限维序列空间中均值计算的复杂度，确定了随机自适应与非自适应设置下的 $n$ 阶最小误差。结果表明，在所考虑的问题中，存在自适应与$n$阶最小误差相差 $n$ 的幂次的实例。该差距（忽略对数因子）可达 $n^{1/4}$ 量级。本文还展示了如何将此类结果转化为对所有 $n$ 同时具有适当偏差的无穷维实例。