  A framework is introduced for sequentially solving convex stochastic
minimization problems, where the objective functions change slowly, in the
sense that the distance between successive minimizers is bounded. The
minimization problems are solved by sequentially applying a selected
optimization algorithm, such as stochastic gradient descent (SGD), based on
drawing a number of samples in order to carry the iterations. Two tracking
criteria are introduced to evaluate approximate minimizer quality: one based on
being accurate with respect to the mean trajectory, and the other based on
being accurate in high probability (IHP). An estimate of a bound on the
minimizers' change, combined with properties of the chosen optimization
algorithm, is used to select the number of samples needed to meet the desired
tracking criterion. A technique to estimate the change in minimizers is
provided along with analysis to show that eventually the estimate upper bounds
the change in minimizers. This estimate of the change in minimizers provides
sample size selection rules that guarantee that the tracking criterion is met
for sufficiently large number of time steps. Simulations are used to confirm
that the estimation approach provides the desired tracking accuracy in
practice, while being efficient in terms of number of samples used in each time
step.
