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.

介绍了一种框架，用于解决凸随机极小化问题，其中目标函数变化缓慢，通过应用选择的优化算法，例如随机梯度下降（SGD），来顺序地解决极小化问题。有两种跟踪标准来评估近似极小值的质量，一种是基于对均值轨迹的准确性，另一种是基于高概率的准确性。提供了估计极小值变化的技术，以及分析结果表明最终估计会上限极小值的变化。这个估计量提供了样本量选择规则，保证了足够的时间步长使跟踪标准得到满足。实验表明，该估计方法在实践中提供了所需的跟踪精度，同时在每个时间步中使用的样本数量方面是高效的。