多重非孤立极值情形下随机梯度搜索的收敛性和收敛速度
Convergence and Convergence Rate of Stochastic Gradient Search in the Case of Multiple and Non-Isolated Extrema
摘要：
研究了随机梯度算法的渐近性态。借助微分几何的结果（Lojasiewicz梯度不等式），证明了算法迭代的单极限点收敛性，并给出了相对严格的收敛速度界.与现有的渐近结果形成鲜明对比，这里提出的新结果允许目标函数有多个和非孤立的最小值。新的结果也提供了新的见解的渐近性质的几类递归算法，经常用于工程，统计，机器学习和运筹学。
英文摘要：
The asymptotic behavior of stochastic gradient algorithms is studied. Relying on results from differential geometry (Lojasiewicz gradient inequality), the single limit-point convergence of the algorithm iterates is demonstrated and relatively tight bounds on the convergence rate are derived. In sharp contrast to the existing asymptotic results, the new results presented here allow the objective function to have multiple and non-isolated minima. The new results also offer new insights into the asymptotic properties of several classes of recursive algorithms which are routinely used in engineering, statistics, machine learning and operations research.
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