In this work, we analyze the learnability of reproducing kernel Hilbert spaces (RKHS) under the $L^\infty$ norm, which is critical for understanding the performance of kernel methods and random feature models in safety- and security-critical applications. Specifically, we relate the $L^\infty$ learnability of a RKHS to the spectrum decay of the associate kernel and both lower bounds and upper bounds of the sample complexity are established. In particular, for dot-product kernels on the sphere, we identify conditions when the $L^\infty$ learning can be achieved with polynomial samples. Let $d$ denote the input dimension and assume the kernel spectrum roughly decays as $\lambda_k\sim k^{-1-\beta}$ with $\beta>0$. We prove that if $\beta$ is independent of the input dimension $d$, then functions in the RKHS can be learned efficiently under the $L^\infty$ norm, i.e., the sample complexity depends polynomially on $d$. In contrast, if $\beta=1/\mathrm{poly}(d)$, then the $L^\infty$ learning requires exponentially many samples.

翻译：本文分析了再生核希尔伯特空间（RKHS）在$L^\infty$范数下的可学习性，这对于理解核方法和随机特征模型在安全关键型应用中的性能至关重要。具体而言，我们将RKHS的$L^\infty$可学习性与关联核的谱衰减联系起来，并建立了样本复杂度的下界和上界。特别地，对于球面上的点积核，我们确定了能够以多项式样本实现$L^\infty$学习的条件。设$d$为输入维度，假设核谱大致以$\lambda_k\sim k^{-1-\beta}$（其中$\beta>0$）的速率衰减。我们证明：若$\beta$与输入维度$d$无关，则RKHS中的函数可在$L^\infty$范数下高效学习（即样本复杂度关于$d$呈多项式依赖）；反之，若$\beta=1/\mathrm{poly}(d)$，则$L^\infty$学习需要指数级样本量。