Motivated by the optimization of bounded binary black-box functions, we study the problem of learning polynomial surrogates over the Boolean hypercube. To ensure that optimizing the surrogate yields good solutions for the underlying objective, we require uniform $L_\infty$-error guarantees rather than the usual $L_2$-type guarantees. We characterize the minimax sample complexity of uniform estimation under subgaussian noise for two classes of bounded polynomials. First, for polynomials of degree at most $d$ on $n$ variables, the sample complexity scales as $n^{d+1}$. Second, for $s$-sparse Fourier-Walsh polynomials with $s \leq n$, it scales as $ns^2$. These rates differ structurally from the noiseless setting, where uniform exact recovery scales as $n^d$ and $ns$, respectively. Our lower bounds hold even for arbitrary adaptive learners, showing that the additional factors are intrinsic to the noisy cases. Standard Fourier-analysis tools for the $L_2$-norm do not naturally extend to the $L_\infty$-setting in a way that yields uniform guarantees. Our proofs overcome this difficulty by relying on suitably chosen auxiliary norms that serve as proxies for controlling the $L_\infty$-error. Together, our results provide a tight characterization of the sample complexity of learning optimization-safe polynomial surrogates.

翻译：受有界二元黑箱函数优化的驱动，我们研究在布尔超立方体上学习多项式替代模型的问题。为确保优化替代模型能获得目标函数的优质解，我们要求均匀 $L_\infty$ 误差保证，而非通常的 $L_2$ 型保证。针对两类有界多项式，我们刻画了子高斯噪声下均匀估计的极小极大样本复杂度。首先，对于 $n$ 变量上次数不超过 $d$ 的多项式，样本复杂度为 $n^{d+1}$。其次，对于 $s$-稀疏傅里叶-沃尔什多项式且 $s \leq n$，样本复杂度为 $ns^2$。这些速率在结构上区别于无噪声情形，后者的均匀精确恢复复杂度分别为 $n^d$ 和 $ns$。我们的下界对任意自适应学习器均成立，表明额外因子是噪声情形固有的。用于 $L_2$ 范数的标准傅里叶分析工具无法自然扩展至 $L_\infty$ 情形以提供均匀保证。本文的证明通过依赖适当选取的辅助范数克服了这一困难，这些范数作为控制 $L_\infty$ 误差的代理。综合而言，我们的结果给出了学习优化安全多项式替代模型样本复杂度的紧致刻画。