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Learning the Maximum of a Hölder Function from Inexact Data

Simon Foucart

TL;DR

The paper addresses learning the maximum of a Hölder function from inexact point evaluations at prescribed datasites within the Optimal Recovery framework. It develops locally optimal (Chebyshev-center) and globally optimal procedures for general monotone quantities, and specializes to the maximum, deriving explicit formulas for both the local estimator $\Delta^{\rm loc}$ and the global estimator $\Delta^{\rm glo}$, including a simple correction term dependent on data locations. In the nonlinear maximum case, it shows that the globally optimal estimator is $\Delta^{\rm glo}(\mathbf{y}) = \max_m(y_m) + \tfrac{1}{2}\max[U]$ with $U(x)=\min_m \operatorname{dist}(x,x^{(m)})^\alpha$ (under equal observation error), and contrasts this with the locally optimal rule, also providing bounds and extensions to jittered data. Finally, it quantifies the minimal global worst-case error, proving bounds that scale with the observation error $\varepsilon$ and the grid resolution $M^{-{\alpha}/d}$, highlighting the curse of dimensionality and delivering exact results in the cube-grid setting.

Abstract

Within the theoretical framework of Optimal Recovery, one determines in this article the {\em best} procedures to learn a quantity of interest depending on a Hölder function acquired via inexact point evaluations at fixed datasites. {\em Best} here refers to procedures minimizing worst-case errors. The elementary arguments hint at the possibility of tackling nonlinear quantities of interest, with a particular focus on the function maximum. In a local setting, i.e., for a fixed data vector, the optimal procedure (outputting the so-called Chebyshev center) is precisely described relatively to a general model of inexact evaluations. Relatively to a slightly more restricted model and in a global setting, i.e., uniformly over all data vectors, another optimal procedure is put forward, showing how to correct the natural underestimate that simply returns the data vector maximum. Jitterred data are also briefly discussed as a side product of evaluating the minimal worst-case error optimized over the datasites.

Learning the Maximum of a Hölder Function from Inexact Data

TL;DR

The paper addresses learning the maximum of a Hölder function from inexact point evaluations at prescribed datasites within the Optimal Recovery framework. It develops locally optimal (Chebyshev-center) and globally optimal procedures for general monotone quantities, and specializes to the maximum, deriving explicit formulas for both the local estimator and the global estimator , including a simple correction term dependent on data locations. In the nonlinear maximum case, it shows that the globally optimal estimator is with (under equal observation error), and contrasts this with the locally optimal rule, also providing bounds and extensions to jittered data. Finally, it quantifies the minimal global worst-case error, proving bounds that scale with the observation error and the grid resolution , highlighting the curse of dimensionality and delivering exact results in the cube-grid setting.

Abstract

Within the theoretical framework of Optimal Recovery, one determines in this article the {\em best} procedures to learn a quantity of interest depending on a Hölder function acquired via inexact point evaluations at fixed datasites. {\em Best} here refers to procedures minimizing worst-case errors. The elementary arguments hint at the possibility of tackling nonlinear quantities of interest, with a particular focus on the function maximum. In a local setting, i.e., for a fixed data vector, the optimal procedure (outputting the so-called Chebyshev center) is precisely described relatively to a general model of inexact evaluations. Relatively to a slightly more restricted model and in a global setting, i.e., uniformly over all data vectors, another optimal procedure is put forward, showing how to correct the natural underestimate that simply returns the data vector maximum. Jitterred data are also briefly discussed as a side product of evaluating the minimal worst-case error optimized over the datasites.

Paper Structure

This paper contains 6 sections, 7 theorems, 51 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Locally Optimal Solutions
  3. Globally Optimal Solutions
  4. The case of linear quantities of interest
  5. The case of the maximum
  6. Estimations of the Minimal Global Worst-Case Errors

Key Result

Theorem 1

For the recovery of a monotone quantity of interest $\Gamma$ mapping into a Banach lattice $Z$, given the data DefY and the model set DefH, a minimizer of ${\rm lwce}_{{\bf y},{\bm \varepsilon}}(z)$ over all $z \in Z$ is given by while the value of the minimum is

Figures (1)

  • Figure 1: An example of a univariate Lipschitz function $f$ (solid blue curve) acquired via inexact point values (black circles). The left panel displays the lower function $\ell_{{\bf y},{\bm \varepsilon}}$ and the upper function $u_{{\bf y},{\bm \varepsilon}}$ from \ref{['DefLy']}-\ref{['DefUy']} (dotted curves), as well as the locally optimal estimation of $f$ from Theorem \ref{['ThmLoc']} (dashed red curve). The right panel shows, as horizontal lines, the locally optimal estimation of $\max[f]$ from Theorem \ref{['ThmLoc']} (dashed red line) and the globally optimal estimation of $\max[f]$ from Theorem \ref{['ThmGloMax']} (dash-dotted green line).

Theorems & Definitions (14)

  • Theorem 1
  • proof
  • Proposition 2
  • proof
  • Theorem 3
  • proof
  • Theorem 4
  • Theorem 5
  • proof
  • Remark
  • ...and 4 more