Nesterov's method of dichotomy via Order Oracle: The problem of optimizing a two-variable function on a square

TL;DR

This paper introduces new optimization algorithms using an Order Oracle with limited information, providing convergence guarantees and practical applications for noisy, black-box functions in two variables.

Contribution

It develops convergence rate estimates for a one-dimensional search and a two-variable algorithm under oracle inaccuracies, outperforming existing methods.

Findings

Convergence rates established for the golden ratio method with noisy oracles.

Algorithm demonstrated effective in maximizing a preference function with noisy parameters.

Results comparable to methods with more detailed function information.

Abstract

The challenges of black box optimization arise due to imprecise responses and limited output information. This article describes new results on optimizing multivariable functions using an Order Oracle, which provides access only to the order between function values and with some small errors. We obtained convergence rate estimates for the one-dimensional search method (golden ratio method) under the condition of oracle inaccuracy, as well as convergence results for the algorithm on a "square" (also with noise), which outperforms its alternatives. The results obtained are similar to those in problems with oracles providing significantly more information about the optimized function. Additionally, the practical application of the algorithm has been demonstrated in maximizing a preference function, where the parameters are the acidity and sweetness of the drink. This function is expected to be convex or at least quasi-convex.