  In this paper, we present a generic framework that allows accelerating almost
arbitrary non-accelerated deterministic and randomized algorithms for smooth
convex optimization problems. The main approach of our envelope is the same as
in Catalyst (Lin et al., 2015): an accelerated proximal outer gradient method,
which is used as an envelope for a non-accelerated inner method for the
$\ell_2$ regularized auxiliary problem. Our algorithm has two key differences:
1) easily verifiable stopping criteria for inner algorithm; 2) the
regularization parameter can be tunned along the way. As a result, the main
contribution of our work is a new framework that applies to adaptive inner
algorithms: Steepest Descent, Adaptive Coordinate Descent, Alternating
Minimization. Moreover, in the non-adaptive case, our approach allows obtaining
Catalyst without a logarithmic factor, which appears in the standard Catalyst
(Lin et al., 2015, 2018).
