Adaptive approximation of monotone functions
This work addresses a classical approximation problem for researchers in numerical analysis and machine learning, offering theoretical insights and an optimal algorithm, though it is incremental as it builds on prior methods like Novak's.
The paper tackles the problem of approximating monotone functions in L^p norm with adaptive queries, characterizing the minimum evaluations needed for a given error and introducing GreedyBox, which achieves optimal sample complexity up to logarithmic factors. It also shows faster error decay for piecewise-smooth functions and highlights performance gaps between adaptive and non-adaptive algorithms.
We study the classical problem of approximating a non-decreasing function $f: \mathcal{X} \to \mathcal{Y}$ in $L^p(μ)$ norm by sequentially querying its values, for known compact real intervals $\mathcal{X}$, $\mathcal{Y}$ and a known probability measure $μ$ 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 $L^p(μ)$ error below $ε$ 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 $L^p(μ)$ 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.