In this paper, we introduce a distributed algorithm that optimizes the Gaussian signal covariance matrices of multi-antenna users transmitting to a common multi-antenna receiver under imperfect and possibly delayed channel state information. The algorithm is based on an extension of exponential learning techniques to a semidefinite setting and it requires the same information as distributed water-filling methods. Unlike water-filling however, the proposed matrix exponential learning (MXL) algorithm converges to the system's optimum signal covariance profile under very mild conditions on the channel uncertainty statistics; moreover, the algorithm retains its convergence properties even in the presence of user update asynchronicities, random delays and/or ergodically changing channel conditions. In particular, by properly tuning the algorithm's learning rate (or step size), the algorithm converges within a few iterations, even for large numbers of users and/or antennas per user. Our theoretical analysis is complemented by numerical simulations which illustrate the algorithm's robustness and scalability in realistic network conditions.