  Minimum variance controllers have been employed in a wide-range of industrial
applications. A key challenge experienced by many adaptive controllers is their
poor empirical performance in the initial stages of learning. In this paper, we
address the problem of initializing them so that they provide acceptable
transients, and also provide an accompanying finite-time regret analysis, for
adaptive minimum variance control of an auto-regressive system with exogenous
inputs (ARX). Following [3], we consider a modified version of the Certainty
Equivalence (CE) adaptive controller, which we call PIECE, that utilizes
probing inputs for exploration. We show that it has a $C \log T$ bound on the
regret after $T$ time-steps for bounded noise, and $C\log^2 T$ in the case of
sub-Gaussian noise. The simulation results demonstrate the advantage of PIECE
over the algorithm proposed in [3] as well as the standard Certainty
Equivalence controller especially in the initial learning phase. To the best of
our knowledge, this is the first work that provides finite-time regret bounds
for an adaptive minimum variance controller.
