We propose a conditional gradient framework for a composite convex minimization template with broad applications. Our approach combines smoothing and homotopy techniques under the CGM framework, and provably achieves the optimal convergence rate. We demonstrate that the same rate holds if the linear subproblems are solved approximately with additive or multiplicative error. In contrast with the relevant work, we are able to characterize the convergence when the non-smooth term is an indicator function. Specific applications of our framework include the non-smooth minimization, semidefinite programming, and minimization with linear inclusion constraints over a compact domain. Numerical evidence demonstrates the benefits of our framework.

We analyze two novel randomized variants of the Frank-Wolfe (FW) or conditional gradient algorithm.While classical FW algorithms require solving a linear minimization problem over the domain at each iteration, the proposed method only requires to solve a linear minimization problem over a small \emph{subset} of the original domain.The first algorithm that we propose is a randomized variant of the original FW algorithm and achieves a $\mathcal{O}(1/t)$ sublinear convergence rate as in the deterministic counterpart. The second algorithm is a randomized variant of the Away-step FW algorithm, and again as its deterministic counterpart, reaches linear (i.e., exponential) convergence rate making it the first provably convergent randomized variant of Away-step FW. In both cases, while subsampling reduces the convergence rate by a constant factor, the linear minimization step can be a fraction of the cost of that of the deterministic versions, especially when the data is streamed. We illustrate computational gains of both algorithms on regression problems, involving both $\ell_1$ and latent group lasso penalties.

Two popular examples of first-order optimization methods over linear spaces are coordinate descent and matching pursuit algorithms, with their randomized variants. While the former targets the optimization by moving along coordinates, the latter considers a generalized notion of directions. Exploiting the connection between the two algorithms, we present a unified analysis of both, providing affine invariant sublinear $O(1/t)$ rates on smooth objectives and linear convergence on strongly convex objectives. As a byproduct of our affine invariant analysis of matching pursuit, our rates for steepest coordinate descent are the tightest known. Furthermore, we show the first accelerated convergence rate $O(1/t^2)$ for matching pursuit and steepest coordinate descent on convex objectives.

We propose and analyze a novel adaptive step size variant of the Davis-Yin three operator splitting, a method that can solve optimization problems composed of a sum of a smooth term for which we have access to its gradient and an arbitrary number of potentially non-smooth terms for which we have access to their proximal operator. The proposed method leverages local information of the objective function, allowing for larger step sizes while preserving the convergence properties of the original method. It only requires two extra function evaluations per iteration and does not depend on any step size hyperparameter besides an initial estimate. We provide a convergence rate analysis of this method, showing sublinear convergence rate for general convex functions and linear convergence under stronger assumptions, matching the best known rates of its non adaptive variant. Finally, an empirical comparison with related methods on 6 different problems illustrates the computational advantage of the adaptive step size strategy.