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This paper studies first-order algorithms for solving fully compositeoptimization problems over convex and compact sets. We leverage the structureof the objective by handling its differentiable and non-differentiablecomponents separately, linearizing only the smooth parts. This provides us withnew generalizations of the classical Frank-Wolfe method and the ConditionalGradient Sliding algorithm, that cater to a subclass of non-differentiableproblems. Our algorithms rely on a stronger version of the linear minimizationoracle, which can be efficiently implemented in several practical applications.We provide the basic version of our method with an affine-invariant analysisand prove global convergence rates for both convex and non-convex objectives.Furthermore, in the convex case, we propose an accelerated method withcorrespondingly improved complexity. Finally, we provide illustrativeexperiments to support our theoretical results.

Linearization Algorithms for Fully Composite Optimization