Approximating Higher-Order Derivative Tensors Using Secant Updates

TL;DR

This paper introduces a method to approximate higher-order derivative tensors, such as third derivatives, using secant updates inspired by quasi-Newton methods, with proven convergence and numerical validation.

Contribution

It generalizes quasi-Newton secant updates to higher-order derivatives, providing a full characterization and convergence analysis for these tensor approximations.

Findings

Convergence of higher-order derivative approximations established.

Characterization of least-change updates in weighted Frobenius norms.

Numerical experiments demonstrate approximation quality.

Abstract

Quasi-Newton methods employ an update rule that gradually improves the Hessian approximation using the already available gradient evaluations. We propose higher-order secant updates which generalize this idea to higher-order derivatives, approximating for example third derivatives (which are tensors) from given Hessian evaluations. Our generalization is based on the observation that quasi-Newton updates are least-change updates satisfying the secant equation, with different methods using different norms to measure the size of the change. We present a full characterization for least-change updates in weighted Frobenius norms (satisfying an analogue of the secant equation) for derivatives of arbitrary order. Moreover, we establish convergence of the approximations to the true derivative under standard assumptions and explore the quality of the generated approximations in numerical experiments.