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The Stochastic Approximation (SA) algorithm introduced by Robbins and Monroin 1951 has been a standard method for solving equations of the form$\mathbf{f}({\boldsymbol {\theta}}) = \mathbf{0}$, when only noisy measurementsof $\mathbf{f}(\cdot)$ are available. If $\mathbf{f}({\boldsymbol {\theta}}) =\nabla J({\boldsymbol {\theta}})$ for some function $J(\cdot)$, then SA canalso be used to find a stationary point of $J(\cdot)$. At each time $t$, thecurrent 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 isassumed that the error term ${\boldsymbol {\xi}}_{t+1}$ has zero conditionalmean, and/or that its conditional variance is bounded as a function of $t$(though not necessarily with respect to ${\boldsymbol {\theta}}_t$). Over theyears, SA has been applied to a variety of areas, out of which the focus inthis paper is on convex and nonconvex optimization. As it turns out, in theseapplications, the above-mentioned assumptions on the measurement error do notalways hold. In zero-order methods, the error neither has zero mean nor boundedconditional variance. In the present paper, we extend SA theory to encompasserrors with nonzero conditional mean and/or unbounded conditional variance. Inaddition, we derive estimates for the rate of convergence of the algorithm, andcompute the ``optimal step size sequences'' to maximize the estimated rate ofconvergence.

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