We address the regression problem for a general function $f:[-1,1]^d\to \mathbb R$ when the learner selects the training points $\{x_i\}_{i=1}^n$ to achieve a uniform error bound across the entire domain. In this setting, known historically as nonparametric regression, we aim to establish a sample complexity bound that depends solely on the function's degree of smoothness. Assuming periodicity at the domain boundaries, we introduce PADUA, an algorithm that, with high probability, provides performance guarantees optimal up to constant or logarithmic factors across all problem parameters. Notably, PADUA is the first parametric algorithm with optimal sample complexity for this setting. Due to this feature, we prove that, differently from the non-parametric state of the art, PADUA enjoys optimal space complexity in the prediction phase. To validate these results, we perform numerical experiments over functions coming from real audio data, where PADUA shows comparable performance to state-of-the-art methods, while requiring only a fraction of the computational time.

翻译：本文研究了一般函数$f:[-1,1]^d\to \mathbb R$的回归问题，其中学习者通过选择训练点$\{x_i\}_{i=1}^n$来实现整个定义域上的均匀误差界。在此历史称为非参数回归的设置中，我们旨在建立仅依赖于函数光滑度的样本复杂度界。假设函数在定义域边界具有周期性，我们提出PADUA算法——该算法以高概率提供在所有问题参数上达到常数或对数因子最优的性能保证。值得注意的是，PADUA是该设置中首个具有最优样本复杂度的参数化算法。基于这一特性，我们证明与现有非参数方法不同，PADUA在预测阶段具有最优空间复杂度。为验证这些结果，我们对真实音频数据生成的函数进行了数值实验，结果表明PADUA在仅需部分计算时间的情况下，取得了与前沿方法相当的性能。