  We propose fully distributed multi-group multicast precoding designs for
cell-free massive multiple-input multiple-output (MIMO) systems with modest
training overhead. We target the minimization of the sum of the maximum mean
squared errors (MSEs) over the multicast groups, which is then approximated
with a weighted sum MSE minimization to simplify the computation and signaling.
To design the joint network-wide multi-group multicast precoders at the base
stations (BSs) and the combiners at the user equipments (UEs) in a fully
distributed fashion, we adopt an iterative bi-directional training scheme with
UE-specific or group-specific precoded uplink pilots and group-specific
precoded downlink pilots. To this end, we introduce a new group-specific uplink
training resource that entirely eliminates the need for backhaul signaling for
the channel state information (CSI) exchange. The precoders are optimized
locally at each BS by means of either best-response or gradient-based updates,
and the convergence of the two approaches is analyzed with respect to the
centralized implementation with perfect CSI. Finally, numerical results show
that the proposed distributed methods greatly outperform conventional cell-free
massive MIMO precoding designs that rely solely on local CSI.
