  We consider a sequential learning problem with Gaussian payoffs and side
information: after selecting an action $i$, the learner receives information
about the payoff of every action $j$ in the form of Gaussian observations whose
mean is the same as the mean payoff, but the variance depends on the pair
$(i,j)$ (and may be infinite). The setup allows a more refined information
transfer from one action to another than previous partial monitoring setups,
including the recently introduced graph-structured feedback case. For the first
time in the literature, we provide non-asymptotic problem-dependent lower
bounds on the regret of any algorithm, which recover existing asymptotic
problem-dependent lower bounds and finite-time minimax lower bounds available
in the literature. We also provide algorithms that achieve the
problem-dependent lower bound (up to some universal constant factor) or the
minimax lower bounds (up to logarithmic factors).
