  In this work, we compare the direct and indirect approaches to data-driven
predictive control of stochastic linear time-invariant systems. The distinction
between the two approaches lies in the fact that the indirect approach involves
identifying a lower dimensional model from data which is then used in a
certainty-equivalent control design, while the direct approach avoids this
intermediate step altogether. Working within an optimization-based framework,
we find that the suboptimality gap measuring the control performance w.r.t. the
optimal model-based control design vanishes with the size of the dataset only
with the direct approach. The indirect approach has a higher rate of
convergence, but its suboptimality gap does not vanish as the size of the
dataset increases. This reveals the existence of two distinct regimes of
performance as the size of the dataset of input-output behaviors is increased.
We show that the indirect approach, by relying on the identification of a lower
dimensional model, has lower variance and outperforms the direct approach for
smaller datasets, while it incurs an asymptotic bias as a result of the process
noise and a (possibly) incorrect assumption on the order of the identified
model. The direct approach, however, does not incur an asymptotic bias, and
outperforms the indirect approach for larger datasets. Ultimately, by revealing
the existence of two non-asymptotic regimes for the performance of direct and
indirect data-driven predictive control designs, our study suggests that
neither approach is invariably superior and that the choice of design must, in
practice, be informed by the available dataset.
