  Conventional machine learning applications typically assume that data samples
are independently and identically distributed (i.i.d.). However, practical
scenarios often involve a data-generating process that produces highly
dependent data samples, which are known to heavily bias the stochastic
optimization process and slow down the convergence of learning. In this paper,
we conduct a fundamental study on how different stochastic data sampling
schemes affect the sample complexity of online stochastic gradient descent
(SGD) over highly dependent data. Specifically, with a $\phi$-mixing model of
data dependence, we show that online SGD with proper periodic data-subsampling
achieves an improved sample complexity over the standard online SGD in the full
spectrum of the data dependence level. Interestingly, even subsampling a subset
of data samples can accelerate the convergence of online SGD over highly
dependent data. Moreover, we show that online SGD with mini-batch sampling can
further substantially improve the sample complexity over online SGD with
periodic data-subsampling over highly dependent data. Numerical experiments
validate our theoretical results.
