  Over the past years, operator splitting methods have become ubiquitous for
non-smooth optimization owing to their simplicity and efficiency. In this
paper, we consider the Forward--Douglas--Rachford splitting method (FDR)
[10,40], and study both global and local convergence rates of this method. For
the global rate, we establish an $o(1/k)$ convergence rate in terms of a
Bregman divergence suitably designed for the objective function. Moreover, when
specializing to the case of Forward--Backward splitting method, we show that
convergence rate of the objective function of the method is actually $o(1/k)$
for a large choice of the descent step-size. Then locally, based on the
assumption that the non-smooth part of the optimization problem is partly
smooth, we establish local linear convergence of the method. More precisely, we
show that the sequence generated by FDR method first (i) identifies a smooth
manifold in a finite number of iteration, and then (ii) enters a local linear
convergence regime, which is for instance characterized in terms of the
structure of the underlying active smooth manifold. To exemplify the usefulness
of the obtained result, we consider several concrete numerical experiments
arising from applicative fields including, for instance, signal/image
processing, inverse problems and machine learning.
