  In this paper, we propose a new method based on the Sliding Algorithm from
Lan(2016, 2019) for the convex composite optimization problem that includes two
terms: smooth one and non-smooth one. Our method uses the stochastic noised
zeroth-order oracle for the non-smooth part and the first-order oracle for the
smooth part. To the best of our knowledge, this is the first method in the
literature that uses such a mixed oracle for the composite optimization. We
prove the convergence rate for the new method that matches the corresponding
rate for the first-order method up to a factor proportional to the dimension of
the space or, in some cases, its squared logarithm. We apply this method for
the decentralized distributed optimization and derive upper bounds for the
number of communication rounds for this method that matches known lower bounds.
Moreover, our bound for the number of zeroth-order oracle calls per node
matches the similar state-of-the-art bound for the first-order decentralized
distributed optimization up to to the factor proportional to the dimension of
the space or, in some cases, even its squared logarithm.
