Convergence Rates for Stochastic Approximation: Biased Noise with Unbounded Variance, and Applications

The Stochastic Approximation (SA) algorithm introduced by Robbins and Monro in 1951 has been a standard method for solving equations of the form $\mathbf{f}({\boldsymbol {\theta}}) = \mathbf{0}$, when only noisy measurements of $\mathbf{f}(\cdot)$ are available. If $\mathbf{f}({\boldsymbol {\theta}}) = \nabla J({\boldsymbol {\theta}})$ for some function $J(\cdot)$, then SA can also be used to find a stationary point of $J(\cdot)$. At each time $t$, the current guess ${\boldsymbol {\theta}}_t$ is updated to ${\boldsymbol {\theta}}_{t+1}$ using a noisy measurement of the form $\mathbf{f}({\boldsymbol {\theta}}_t) + {\boldsymbol {\xi}}_{t+1}$. In much of the literature, it is assumed that the error term ${\boldsymbol {\xi}}_{t+1}$ has zero conditional mean, and/or that its conditional variance is bounded as a function of $t$ (though not necessarily with respect to ${\boldsymbol {\theta}}_t$). Over the years, SA has been applied to a variety of areas, out of which the focus in this paper is on convex and nonconvex optimization. As it turns out, in these applications, the above-mentioned assumptions on the measurement error do not always hold. In zero-order methods, the error neither has zero mean nor bounded conditional variance. In the present paper, we extend SA theory to encompass errors with nonzero conditional mean and/or unbounded conditional variance. In addition, we derive estimates for the rate of convergence of the algorithm, and compute the ``optimal step size sequences'' to maximize the estimated rate of convergence.