In this paper, we address the trade-off between radiated power and achieved throughput in wireless multiple-input and multiple-output (MIMO) systems that evolve over time in an unpredictable fashion (e.g. due to changes in the wireless medium or the users' QoS requirements). Contrary to the static/stationary channel regime, there is no optimal power allocation profile to converge to (either static or in the mean), so the system's users must adapt to changes in the environment “on the fly”, without being able to predict the system's evolution ahead of time. In this dynamic context, we formulate the users' power/throughput trade-off as an online optimization problem and we provide a matrix exponential learning algorithm that leads to no regret - i.e. the proposed transmit policy is asymptotically optimal in hindsight, irrespective of how the system varies with time. As a result, users are able to track the evolution of their individually optimum transmit profiles remarkably well, even in arbitrarily changing wireless environments.
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