We analyze the behavior of stochastic approximation algorithms where iterates, in expectation, make progress towards an objective at each step. When progress is proportional to the step size of the algorithm, we prove exponential concentration bounds. These tail-bounds contrast asymptotic normality results which are more frequently associated with stochastic approximation. The methods that we develop rely on a geometric ergodicity proof. This extends a result on Markov chains due to Hajek (1982) to the area of stochastic approximation algorithms. For Projected Stochastic Gradient Descent with a non-vanishing gradient, our results can be used to prove $O(1/t)$ and linear convergence rates.

翻译：我们分析了随机逼近算法的行为，其中迭代在期望意义上每一步都朝着目标进步。当进步幅度与算法步长成正比时，我们证明了指数集中界。这些尾部界与通常关联于随机逼近的渐近正态性结果形成对比。我们开发的方法依赖于几何遍历性证明。这将Hajek（1982）关于马尔可夫链的结果推广到了随机逼近算法领域。对于具有非消失梯度的投影随机梯度下降法，我们的结果可用于证明$O(1/t)$和线性收敛速率。