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Worst-Case Learning under a Multi-fidelity Model

Simon Foucart, Nicolas Hengartner

TL;DR

This work reframes multi-fidelity surrogate design as an Optimal Recovery problem to obtain deterministic, worst-case error bounds. By modeling high- and low-fidelity codes with sets $\mathcal{K}_0$ and $\mathcal{K}_1$ and data constraints, it derives both global and local recovery strategies: (i) globally optimal estimation of linear functionals via linear, efficiently computable coefficient maps; (ii) globally optimal recovery in Hilbert spaces using a constrained regularization parameter $\tau^{\sharp}$ solved by SDP; and (iii) a Chebyshev-center analysis for the intersection of two hyperellipsoids to bound and certify local radius of information. It further introduces locally optimal recovery in Hilbert spaces with data-dependent regularization parameters, and provides SDP formulations to compute optimal arms, including cases where the two-fidelity setting suffices or where S-procedure limitations arise. The approach yields explicit, verifiable, deterministic uncertainty bounds for multi-fidelity surrogates and offers scalable, sparse recovery procedures with potential extensions to time-series and transfer-learning contexts.

Abstract

Inspired by multi-fidelity methods in computer simulations, this article introduces procedures to design surrogates for the input/output relationship of a high-fidelity code. These surrogates should be learned from runs of both the high-fidelity and low-fidelity codes and be accompanied by error guarantees that are deterministic rather than stochastic. For this purpose, the article advocates a framework tied to a theory focusing on worst-case guarantees, namely Optimal Recovery. The multi-fidelity considerations triggered new theoretical results in three scenarios: the globally optimal estimation of linear functionals, the globally optimal approximation of arbitrary quantities of interest in Hilbert spaces, and their locally optimal approximation, still within Hilbert spaces. The latter scenario boils down to the determination of the Chebyshev center for the intersection of two hyperellipsoids. It is worth noting that the mathematical framework presented here, together with its possible extension, seems to be relevant in several other contexts briefly discussed.

Worst-Case Learning under a Multi-fidelity Model

TL;DR

This work reframes multi-fidelity surrogate design as an Optimal Recovery problem to obtain deterministic, worst-case error bounds. By modeling high- and low-fidelity codes with sets and and data constraints, it derives both global and local recovery strategies: (i) globally optimal estimation of linear functionals via linear, efficiently computable coefficient maps; (ii) globally optimal recovery in Hilbert spaces using a constrained regularization parameter solved by SDP; and (iii) a Chebyshev-center analysis for the intersection of two hyperellipsoids to bound and certify local radius of information. It further introduces locally optimal recovery in Hilbert spaces with data-dependent regularization parameters, and provides SDP formulations to compute optimal arms, including cases where the two-fidelity setting suffices or where S-procedure limitations arise. The approach yields explicit, verifiable, deterministic uncertainty bounds for multi-fidelity surrogates and offers scalable, sparse recovery procedures with potential extensions to time-series and transfer-learning contexts.

Abstract

Inspired by multi-fidelity methods in computer simulations, this article introduces procedures to design surrogates for the input/output relationship of a high-fidelity code. These surrogates should be learned from runs of both the high-fidelity and low-fidelity codes and be accompanied by error guarantees that are deterministic rather than stochastic. For this purpose, the article advocates a framework tied to a theory focusing on worst-case guarantees, namely Optimal Recovery. The multi-fidelity considerations triggered new theoretical results in three scenarios: the globally optimal estimation of linear functionals, the globally optimal approximation of arbitrary quantities of interest in Hilbert spaces, and their locally optimal approximation, still within Hilbert spaces. The latter scenario boils down to the determination of the Chebyshev center for the intersection of two hyperellipsoids. It is worth noting that the mathematical framework presented here, together with its possible extension, seems to be relevant in several other contexts briefly discussed.
Paper Structure (6 sections, 9 theorems, 77 equations)

This paper contains 6 sections, 9 theorems, 77 equations.

Table of Contents

  1. Introduction
  2. Preliminaries, Problem Formulation, and Summary of Results
  3. Globally Optimal Estimation of Linear Functionals
  4. Globally Optimal Recovery in Hilbert Spaces
  5. Chebyshev Center for the Intersection of Two Hyperellipsoids
  6. Locally Optimal Recovery in Hilbert Spaces

Key Result

Theorem 1

Suppose that the model sets $\mathcal{K}_0,\mathcal{K}_1,\ldots,\mathcal{K}_T$ are symmetric and convex. If the quantity of interest $Q$ is a linear functional, then a globally optimal recovery map is given by where the vector $a^{\rm opt} = [a^{\rm opt}_0; a^{\rm opt}_1; \ldots; a^{\rm opt}_T] \in \mathbb{R}^{m}$ is a solution to

Theorems & Definitions (21)

  • Theorem 1
  • proof
  • Theorem 2
  • proof
  • Corollary 3
  • Theorem 4
  • proof
  • Remark
  • Theorem 5
  • Lemma 6
  • ...and 11 more