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：本文提出了一种随机梯度算法,用于最小化一个光滑的目标函数,该函数是对噪声代价样本的期望,且对于任意给定参数,仅能观测到这些噪声样本。我们的算法采用基于随机扰动的梯度估计方案,该扰动由δ球面上的截断柯西分布生成。我们分析了所提梯度估计器的偏差与方差。当目标函数非凸且参数维度较高时,该算法尤为有效。通过渐近收敛性分析,我们证明算法几乎必然收敛至目标函数的驻点集,并得到了渐近收敛速率。同时,我们证明该算法能够避免不稳定平衡点,从而收敛至局部极小值。进一步地,我们对算法进行了非渐近收敛性分析,特别地,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,我们证明所提算法在多个非凸场景下的性能显著优于GSF、SPSA和RDSA,并在凸(含噪)目标函数上进一步验证其有效性。