  We study online linear regression problems in a distributed setting, where
the data is spread over a network. In each round, each network node proposes a
linear predictor, with the objective of fitting the \emph{network-wide} data.
It then updates its predictor for the next round according to the received
local feedback and information received from neighboring nodes. The predictions
made at a given node are assessed through the notion of regret, defined as the
difference between their cumulative network-wide square errors and those of the
best off-line network-wide linear predictor. Various scenarios are
investigated, depending on the nature of the local feedback (full information
or bandit feedback), on the set of available predictors (the decision set), and
the way data is generated (by an oblivious or adaptive adversary). We propose
simple and natural distributed regression algorithms, involving, at each node
and in each round, a local gradient descent step and a communication and
averaging step where nodes aim at aligning their predictors to those of their
neighbors. We establish regret upper bounds typically in ${\cal O}(T^{3/4})$
when the decision set is unbounded and in ${\cal O}(\sqrt{T})$ in case of
bounded decision set.
