Let $\mathscr{F}_{n,d}$ be the class of all functions $f:\{-1,1\}^n\to[-1,1]$ on the $n$-dimensional discrete hypercube of degree at most $d$. In the first part of this paper, we prove that any (deterministic or randomized) algorithm which learns $\mathscr{F}_{n,d}$ with $L_2$-accuracy $\varepsilon$ requires at least $\Omega((1-\sqrt{\varepsilon})2^d\log n)$ queries for large enough $n$, thus establishing the sharpness as $n\to\infty$ of a recent upper bound of Eskenazis and Ivanisvili (2021). To do this, we show that the $L_2$-packing numbers $\mathsf{M}(\mathscr{F}_{n,d},\|\cdot\|_{L_2},\varepsilon)$ of the concept class $\mathscr{F}_{n,d}$ satisfy the two-sided estimate $$c(1-\varepsilon)2^d\log n \leq \log \mathsf{M}(\mathscr{F}_{n,d},\|\cdot\|_{L_2},\varepsilon) \leq \frac{2^{Cd}\log n}{\varepsilon^4}$$ for large enough $n$, where $c, C>0$ are universal constants. In the second part of the paper, we present a logarithmic upper bound for the randomized query complexity of classes of bounded approximate polynomials whose Fourier spectra are concentrated on few subsets. As an application, we prove new estimates for the number of random queries required to learn approximate juntas of a given degree, functions with rapidly decaying Fourier tails and constant depth circuits of given size. Finally, we obtain bounds for the number of queries required to learn the polynomial class $\mathscr{F}_{n,d}$ without error in the query and random example models.

翻译：设 $\mathscr{F}_{n,d}$ 为所有定义在 $n$ 维离散超立方体 $\{-1,1\}^n$ 上、取值于 $[-1,1]$ 且度数至多为 $d$ 的函数 $f$ 构成的函数类。在本文第一部分，我们证明任何（确定性或随机化）以 $L_2$ 精度 $\varepsilon$ 学习 $\mathscr{F}_{n,d}$ 的算法，在 $n$ 足够大时至少需要 $\Omega((1-\sqrt{\varepsilon})2^d\log n)$ 次查询，从而当 $n\to\infty$ 时，确立了 Eskenazis 与 Ivanisvili (2021) 近期上界的最优性。为此，我们证明了概念类 $\mathscr{F}_{n,d}$ 的 $L_2$ 填充数 $\mathsf{M}(\mathscr{F}_{n,d},\|\cdot\|_{L_2},\varepsilon)$ 满足双边估计：当 $n$ 足够大时，有 $$c(1-\varepsilon)2^d\log n \leq \log \mathsf{M}(\mathscr{F}_{n,d},\|\cdot\|_{L_2},\varepsilon) \leq \frac{2^{Cd}\log n}{\varepsilon^4}$$ 其中 $c, C>0$ 为普适常数。在本文第二部分，我们给出了傅里叶谱集中于少数子集的有界近似多项式类随机查询复杂度的对数上界。作为应用，我们证明了学习给定度数的近似junta函数、具有快速衰减傅里叶尾部的函数以及给定大小的常深度电路所需随机查询次数的新估计。最后，我们获得了在查询模型与随机样本模型中无误差学习多项式类 $\mathscr{F}_{n,d}$ 所需查询次数的界。