  We consider strongly-convex-strongly-concave saddle-point problems with
general non-bilinear objective and different condition numbers with respect to
the primal and the dual variables. First, we consider such problems with smooth
composite terms, one of which having finite-sum structure. For this setting we
propose a variance reduction algorithm with complexity estimates superior to
the existing bounds in the literature. Second, we consider finite-sum
saddle-point problems with composite terms and propose several algorithms
depending on the properties of the composite terms. When the composite terms
are smooth we obtain better complexity bounds than the ones in the literature,
including the bounds of a recently proposed nearly-optimal algorithms which do
not consider the composite structure of the problem. If the composite terms are
prox-friendly, we propose a variance reduction algorithm that, on the one hand,
is accelerated compared to existing variance reduction algorithms and, on the
other hand, provides in the composite setting similar complexity bounds to the
nearly-optimal algorithm which is designed for non-composite setting. Besides
that, our algorithms allow to separate the complexity bounds, i.e. estimate,
for each part of the objective separately, the number of oracle calls that is
sufficient to achieve a given accuracy. This is important since different parts
can have different arithmetic complexity of the oracle, and it is desired to
call expensive oracles less often than cheap oracles. The key thing to all
these results is our general framework for saddle-point problems, which may be
of independent interest. This framework, in turn is based on our proposed
Accelerated Meta-Algorithm for composite optimization with probabilistic
inexact oracles and probabilistic inexactness in the proximal mapping, which
may be of independent interest as well.
