Quantile Optimization via Multiple Timescale Local Search for Black-box Functions

TL;DR

This paper introduces two novel three-timescale local search algorithms for quantile optimization of noisy black-box functions, demonstrating convergence and competitive performance through theoretical analysis and simulations.

Contribution

It proposes two new iterative algorithms with different gradient estimators for quantile optimization, including convergence proofs and finite-time rate analysis.

Findings

Both algorithms converge almost surely under certain conditions.

The second algorithm is more practical for high-dimensional problems.

Simulation results show competitive performance with existing methods.

Abstract

We consider quantile optimization of black-box functions that are estimated with noise. We propose two new iterative three-timescale local search algorithms. The first algorithm uses an appropriately modified finite-difference-based gradient estimator that requires $2d$ + 1 samples of the black-box function per iteration of the algorithm, where $d$ is the number of decision variables (dimension of the input vector). For higher-dimensional problems, this algorithm may not be practical if the black-box function estimates are expensive. The second algorithm employs a simultaneous-perturbation-based gradient estimator that uses only three samples for each iteration regardless of problem dimension. Under appropriate conditions, we show the almost sure convergence of both algorithms. In addition, for the class of strongly convex functions, we further establish their (finite-time) convergence rate through a novel fixed-point argument. Simulation experiments indicate that the algorithms work well on a variety of test problems and compare well with recently proposed alternative methods.