In this paper, we present a stochastic gradient algorithm for minimizing a smooth objective function that is an expectation over noisy cost samples, and only the latter are observed for any given parameter. Our algorithm employs a gradient estimation scheme with random perturbations, which are formed using the truncated Cauchy distribution from the delta sphere. We analyze the bias and variance of the proposed gradient estimator. Our algorithm is found to be particularly useful in the case when the objective function is non-convex, and the parameter dimension is high. From an asymptotic convergence analysis, we establish that our algorithm converges almost surely to the set of stationary points of the objective function and obtains the asymptotic convergence rate. We also show that our algorithm avoids unstable equilibria, implying convergence to local minima. Further, we perform a non-asymptotic convergence analysis of our algorithm. In particular, we establish here a non-asymptotic bound for finding an epsilon-stationary point of the non-convex objective function. Finally, we demonstrate numerically through simulations that the performance of our algorithm outperforms GSF, SPSA, and RDSA by a significant margin over a few non-convex settings and further validate its performance over convex (noisy) objectives.

翻译：本文提出一种随机梯度算法，用于最小化光滑目标函数（该函数为含噪声代价样本的期望值，且对于任意给定参数仅能观测到噪声样本）。该算法采用基于截断德尔塔球面柯西分布的随机扰动梯度估计方案。我们分析了所提梯度估计量的偏差与方差。实验表明，该算法在目标函数非凸且参数维度较高时尤为有效。通过渐近收敛性分析，我们证明了算法几乎必然收敛至目标函数的驻点集，并获得了渐近收敛速率。同时证明该算法可规避不稳定平衡点，即收敛至局部极小值。进一步地，我们开展了算法的非渐近收敛性分析，特别建立了寻找非凸目标函数ε-驻点的非渐近界。最后，通过数值仿真验证：在多个非凸场景下，该算法性能显著优于GSF、SPSA及RDSA，并在含噪凸目标函数上进一步验证了其有效性。