Adaptive Finite-Difference Interval Estimation for Noisy Derivative-Free Optimization

A common approach for minimizing a smooth nonlinear function is to employ finite-difference approximations to the gradient. While this can be easily performed when no error is present within the function evaluations, when the function is noisy, the optimal choice requires information about the noise level and higher-order derivatives of the function, which is often unavailable. Given the noise level of the function, we propose a bisection search for finding a finite-difference interval for any finite-difference scheme that balances the truncation error, which arises from the error in the Taylor series approximation, and the measurement error, which results from noise in the function evaluation. Our procedure produces reliable estimates of the finite-difference interval at low cost without explicitly approximating higher-order derivatives. We show its numerical reliability and accuracy on a set of test problems. When combined with L-BFGS, we obtain a robust method for minimizing noisy black-box functions, as illustrated on a subset of unconstrained CUTEst problems with synthetically added noise.