# Adaptive approximation of monotone functions

arXiv: https://arxiv.org/abs/2309.07530 · HTML: https://mlsanity.com/paper/2309.07530/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 3 of its 32 references that are in the map.
- First posted on arXiv 14 Sep 2023.

## Builds on (most cited first, 3 of 3)

- [Multivariate extensions of isotonic regression and total variation denoising via entire monotonicity and Hardy-Krause variation](https://mlsanity.com/paper/1903.01395/) (2019)
- [Instance-Dependent Bounds for Zeroth-order Lipschitz Optimization with Error Certificates](https://mlsanity.com/paper/2102.01977/) (2021)
- [Adaptive reconstruction of imperfectly-observed monotone functions, with applications to uncertainty quantification](https://mlsanity.com/paper/2007.05236/) (2020)

## Abstract (arXiv)

We study the classical problem of approximating a non-decreasing function f: X → Y in Lp(μ) norm by sequentially querying its values, for known compact real intervals X, 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 hatf with an Lp(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 Lp(μ) error of GreedyBox decreases much faster for piecewise-C2 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.