Equivalent necessary and sufficient conditions on noise sequences for stochastic approximation algorithms

The authors consider stochastic approximation algorithms on a general Hilbert space, and study four conditions on noise sequences for their analysis: Kushner and Clark’s condition, Chen’s condition, a decomposition condition, and Kulkarni and Horn’s condition. They discuss various properties of these conditions. In the present main result the authors show that the four conditions are all equivalent, and are both necessary and sufficient for convergence of stochastic approximation algorithms under appropriate assumptions.