  We study the problem of regret minimization in a multi-armed bandit setup
where the agent is allowed to play multiple arms at each round by spreading the
resources usually allocated to only one arm. At each iteration the agent
selects a normalized power profile and receives a Gaussian vector as outcome,
where the unknown variance of each sample is inversely proportional to the
power allocated to that arm. The reward corresponds to a linear combination of
the power profile and the outcomes, resembling a linear bandit. By spreading
the power, the agent can choose to collect information much faster than in a
traditional multi-armed bandit at the price of reducing the accuracy of the
samples. This setup is fundamentally different from that of a linear bandit --
the regret is known to scale as $\Theta(\sqrt{T})$ for linear bandits, while in
this setup the agent receives a much more detailed feedback, for which we
derive a tight $\log(T)$ problem-dependent lower-bound. We propose a
Thompson-Sampling-based strategy, called Weighted Thompson Sampling (\WTS),
that designs the power profile as its posterior belief of each arm being the
best arm, and show that its upper bound matches the derived logarithmic lower
bound. Finally, we apply this strategy to a problem of control and system
identification, where the goal is to estimate the maximum gain (also called
$\mathcal{H}_\infty$-norm) of a linear dynamical system based on batches of
input-output samples.
