  In this paper, we present a conditional gradient type (CGT) method for
solving a class of composite optimization problems where the objective function
consists of a (weakly) smooth term and a (strongly) convex regularization term.
While including a strongly convex term in the subproblems of the classical
conditional gradient (CG) method improves its rate of convergence, it does not
cost per iteration as much as general proximal type algorithms. More
specifically, we present a unified analysis for the CGT method in the sense
that it achieves the best-known rate of convergence when the weakly smooth term
is nonconvex and possesses (nearly) optimal complexity if it turns out to be
convex. While implementation of the CGT method requires explicitly estimating
problem parameters like the level of smoothness of the first term in the
objective function, we also present a few variants of this method which relax
such estimation. Unlike general proximal type parameter free methods, these
variants of the CGT method do not require any additional effort for computing
(sub)gradients of the objective function and/or solving extra subproblems at
each iteration. We then generalize these methods under stochastic setting and
present a few new complexity results. To the best of our knowledge, this is the
first time that such complexity results are presented for solving stochastic
weakly smooth nonconvex and (strongly) convex optimization problems.
