Finite-sample analysis of iterate averaging method for stochastic approximation with quadratic loss function

Consider the stochastic approximation algorithms with quadratic loss functions, where we have noisy measurements of gradients at each iteration. A widely-used technique of stochastic approximation is to average some or all of the iterates in order to reduce the variance of the resulting estimate. Under proper settings for coefficients, the averaged sequence converges to its limit at an optimum rate. In practice, however, the results on iterate averaging are more mixed than the above suggests. Under proper assumptions, we provide a formal analysis under finite iterations and derive conditions under which iterate averaging benefits the algorithm output, in terms of reducing mean squared errors. Simulations and examples support the practical importance of the given conditions.