Interpolation-Based Gradient-Error Bounds for Use in Derivative-Free Optimization of Noisy Functions

TL;DR

This paper analyzes the accuracy of gradient estimates from linear interpolation in noisy functions, proposing less conservative error bounds and a derivative-free optimization scheme that incorporates these bounds as constraints.

Contribution

It introduces approximate gradient error bounds for noisy functions and integrates them into a sequential programming DFO scheme, improving over traditional conservative bounds.

Findings

Derived upper bounds for gradient errors due to noise and interpolation

Compared conservative and approximate bounds through examples

Proposed a DFO method enforcing error bounds as constraints

Abstract

In this paper, we analyze the accuracy of gradient estimates obtained by linear interpolation when the underlying function is subject to bounded measurement noise. The total gradient error is decomposed into a deterministic component arising from the interpolation (finite-difference) approximation, and a stochastic component due to noise. Various upper bounds for both error components are derived and compared through several illustrative examples. Our comparative study reveals that strict deterministic bounds, including those commonly used in derivative-free optimization (DFO), tend to be overly conservative. To address this, we propose approximate gradient error bounds that aim to upper bound the gradient error norm more realistically, without the excessive conservatism of classical bounds. Finally, drawing inspiration from dual real-time optimization strategies, we present a DFO scheme based on sequential programming, where the approximate gradient error bounds are enforced as constraints within the optimization problem.