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.

翻译：我们考虑存在噪声估计的黑箱函数的分位数优化问题。本文提出两种新的迭代式三时间尺度局部搜索算法。第一种算法采用经适当改进的有限差分梯度估计器，每次迭代需对黑箱函数进行$2d$+1次采样，其中$d$为决策变量数（输入向量维度）。对于高维问题，当黑箱函数估值代价较高时该算法可能缺乏实用性。第二种算法采用同步扰动梯度估计器，无论问题维度高低，每次迭代仅需三个样本。在适当条件下，我们证明了两类算法的几乎必然收敛性。此外，针对强凸函数类，通过新颖的不动点论证方法进一步建立了算法的有限时间收敛速率。仿真实验表明，该算法在多种测试问题中表现良好，并与近期提出的替代方法相比具有竞争力。