We study the classical problem of approximating a non-decreasing function $f: \mathcal{X} \to \mathcal{Y}$ in $L^p(\mu)$ norm by sequentially querying its values, for known compact real intervals $\mathcal{X}$, $\mathcal{Y}$ and a known probability measure $\mu$ on $\cX$. For any function~$f$ we characterize the minimum number of evaluations of $f$ that algorithms need to guarantee an approximation $\hat{f}$ with an $L^p(\mu)$ error below $\epsilon$ after stopping. Unlike worst-case results that hold uniformly over all $f$, our complexity measure is dependent on each specific function $f$. To address this problem, we introduce GreedyBox, a generalization of an algorithm originally proposed by Novak (1992) for numerical integration. We prove that GreedyBox achieves an optimal sample complexity for any function $f$, up to logarithmic factors. Additionally, we uncover results regarding piecewise-smooth functions. Perhaps as expected, the $L^p(\mu)$ error of GreedyBox decreases much faster for piecewise-$C^2$ functions than predicted by the algorithm (without any knowledge on the smoothness of $f$). A simple modification even achieves optimal minimax approximation rates for such functions, which we compute explicitly. In particular, our findings highlight multiple performance gaps between adaptive and non-adaptive algorithms, smooth and piecewise-smooth functions, as well as monotone or non-monotone functions. Finally, we provide numerical experiments to support our theoretical results.

翻译：我们研究通过顺序查询函数值来在$L^p(\mu)$范数下逼近非递减函数$f: \mathcal{X} \to \mathcal{Y}$的经典问题，其中$\mathcal{X}$、$\mathcal{Y}$为已知紧实数区间，且$\mu$为$\cX$上的已知概率测度。对于任意函数$f$，我们刻画了保证逼近误差$\hat{f}$在停止后$L^p(\mu)$误差低于$\epsilon$所需的最小函数$f$评估次数。与对所有$f$一致成立的悲观情形结果不同，我们的复杂度度量依赖于每个具体函数$f$。为解决此问题，我们引入GreedyBox算法——该算法最初由Novak（1992）为数值积分提出的算法的泛化形式。我们证明GreedyBox对任意函数$f$（对数因子范围内）均能达到最优样本复杂度。此外，我们揭示了关于分段光滑函数的新结果。可能正如预期，对于分段$C^2$函数，GreedyBox的$L^p(\mu)$误差下降速度远快于算法（在无任何$f$光滑性先验知识下）的理论预测值。通过简单改进，我们甚至显式计算出了此类函数的最优极小极大逼近率。特别地，我们的发现凸显了自适应与非自适应算法、光滑与分段光滑函数、以及单调与非单调函数之间的多重性能差距。最后，我们通过数值实验验证了理论结果。