Randomized Complexity of Mean Computation and the Adaption Problem
Stefan Heinrich
TL;DR
This work resolves the scalar-valued adaption problem in information-based complexity for mean computation in mixed-norm spaces. It demonstrates that, for $1\le p<2<u\le\infty$, adaptive and non-adaptive randomized $n$-th minimal errors can differ by a power of $n$, up to $n^{1/4}$ (modulo logarithmic factors), by constructing finite-dimensional examples and extending them to infinite dimensions via direct sums. The authors develop concrete adaptive and non-adaptive algorithms with explicit error bounds, and they establish sharp lower bounds that quantify the adaptivity gap. A key contribution is a general framework to transfer finite-dimensional gaps to infinite-dimensional problems, yielding gaps that persist across all $n$ in both sum and diagonal models. These results provide a negative answer to the universality of adaptivity advantages in the scalar case and offer a principled pathway to infinite-dimensional instantiations, illuminating the limits of adaptive information in randomized settings.
Abstract
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.