  We propose robust sparse reduced rank regression for analyzing large and
complex high-dimensional data with heavy-tailed random noise. The proposed
method is based on a convex relaxation of a rank- and sparsity-constrained
non-convex optimization problem, which is then solved using the alternating
direction method of multipliers algorithm. We establish non-asymptotic
estimation error bounds under both Frobenius and nuclear norms in the
high-dimensional setting. This is a major contribution over existing results in
reduced rank regression, which mainly focus on rank selection and prediction
consistency. Our theoretical results quantify the tradeoff between
heavy-tailedness of the random noise and statistical bias. For random noise
with bounded $(1+\delta)$th moment with $\delta \in (0,1)$, the rate of
convergence is a function of $\delta$, and is slower than the sub-Gaussian-type
deviation bounds; for random noise with bounded second moment, we obtain a rate
of convergence as if sub-Gaussian noise were assumed. Furthermore, the
transition between the two regimes is smooth. We illustrate the performance of
the proposed method via extensive numerical studies and a data application.
