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In this paper we consider a composite optimization problem that minimizes thesum of a weakly smooth function and a convex function with either a boundeddomain or a uniformly convex structure. In particular, we first present aparameter-dependent conditional gradient method for this problem, whose stepsizes require prior knowledge of the parameters associated with the H\"oldercontinuity of the gradient of the weakly smooth function, and establish itsrate of convergence. Given that these parameters could be unknown or known butpossibly conservative, such a method may suffer from implementation issue orslow convergence. We therefore propose a parameter-free conditional gradientmethod whose step size is determined by using a constructive local quadraticupper approximation and an adaptive line search scheme, without using anyproblem parameter. We show that this method achieves the same rate ofconvergence as the parameter-dependent conditional gradient method. Preliminaryexperiments are also conducted and illustrate the superior performance of theparameter-free conditional gradient method over the methods with some otherstep size rules.

A Parameter-Free Conditional Gradient Method for Composite Minimization under H\"older Condition