  We consider a partial-feedback variant of the well-studied online PCA problem
where a learner attempts to predict a sequence of $d$-dimensional vectors in
terms of a quadratic loss, while only having limited feedback about the
environment's choices. We focus on a natural notion of bandit feedback where
the learner only observes the loss associated with its own prediction. Based on
the classical observation that this decision-making problem can be lifted to
the space of density matrices, we propose an algorithm that is shown to achieve
a regret of $O(d^{3/2}\sqrt{T})$ after $T$ rounds in the worst case. We also
prove data-dependent bounds that improve on the basic result when the loss
matrices of the environment have bounded rank or the loss of the best action is
bounded. One version of our algorithm runs in $O(d)$ time per trial which
massively improves over every previously known online PCA method. We complement
these results by a lower bound of $\Omega(d\sqrt{T})$.
