Least Squares Regression with Markovian Data: Fundamental Limits and Algorithms

We study the problem of least squares linear regression where the datapoints are dependent and are sampled from a Markov chain. We establish sharp information theoretic minimax lower bounds for this problem in terms of \tmix, the mixing time of the underlying Markov chain, under different noise settings. Our results establish that in general, optimization with Markovian data is strictly harder than optimization with independent data and a trivial algorithm (SGD-DD) that works with only one in every \tmix samples, which are approximately independent, is minimax optimal. In fact, it is strictly better than the popular Stochastic Gradient Descent (SGD) method with constant step-size which is otherwise minimax optimal in the regression with independent data setting. Beyond a worst case analysis, we investigate whether structured datasets seen in practice such as Gaussian auto-regressive dynamics can admit more efficient optimization schemes.