We study an optimization program over nonnegative Borel measures that encourages sparsity in its solution. Efficient solvers for this program are in increasing demand, as it arises when learning from data generated by a continuum-of-subspaces' model, a recent trend with applications in signal processing, machine learning, and high-dimensional statistics. We prove that the conditional gradient method (CGM) applied to this infinite-dimensional program, as proposed recently in the literature, is equivalent to the exchange method (EM) applied to its Lagrangian dual, which is a semi-infinite program. In doing so, we formally connect such infinite-dimensional programs to the well-established field of semi-infinite programming. On the one hand, the equivalence established in this paper allows us to provide a rate of convergence for EM which is more general than those existing in the literature. On the other hand, this connection and the resulting geometric insights might in the future lead to the design of improved variants of CGM for infinite-dimensional programs, which has been an active research topic. CGM is also known as the Frank-Wolfe algorithm.
The conditional gradient method (CGM), also known as the Frank-Wolfe algorithm, is one of the earliest algorithms for constrained optimization @cite_12 . The version of the algorithm considered in this paper is the fully-corrective Frank-Wolfe algorithm, also known as the algorithm, in which the objective is optimized over the convex hull of all previous atoms @cite_49 @cite_8 . The algorithm was proposed for optimization over measures, the context considered in this paper, in @cite_3 .
@cite_3 @cite_49 @cite_12 @cite_8
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We present new results for the Frank---Wolfe method (also known as the conditional gradient method). We derive computational guarantees for arbitrary step-size sequences, which are then applied to various step-size rules, including simple averaging and constant step-sizes. We also develop step-size rules and computational guarantees that depend naturally on the warm-start quality of the initial (and subsequent) iterates. Our results include computational guarantees for both duality bound gaps and the so-called FW gaps. Lastly, we present complexity bounds in the presence of approximate computation of gradients and or linear optimization subproblem solutions. We present a blended conditional gradient approach for minimizing a smooth convex function over a polytope @math , that combines the Frank--Wolfe algorithm (also called conditional gradient) with gradient-based steps different from away steps and pairwise steps, however, still achieving linear convergence for strongly convex functions and good practical performance. Our approach retains all favorable properties of conditional gradient algorithms, most notably avoidance of projections onto @math and maintenance of iterates as sparse convex combinations of a limited number of extreme points of @math . The algorithm decreases measures of optimality (primal and dual gaps) rapidly, both in the number of iterations and in wall-clock time, outperforming even the efficient "lazified" conditional gradient algorithms of [arXiv:1410.8816]. Nota bene the algorithm is lazified itself. We also present a streamlined algorithm when @math is the probability simplex. The Frank-Wolfe (FW) optimization algorithm has lately re-gained popularity thanks in particular to its ability to nicely handle the structured constraints appearing in machine learning applications. However, its convergence rate is known to be slow (sublinear) when the solution lies at the boundary. A simple less-known fix is to add the possibility to take 'away steps' during optimization, an operation that importantly does not require a feasibility oracle. In this paper, we highlight and clarify several variants of the Frank-Wolfe optimization algorithm that have been successfully applied in practice: away-steps FW, pairwise FW, fully-corrective FW and Wolfe's minimum norm point algorithm, and prove for the first time that they all enjoy global linear convergence, under a weaker condition than strong convexity of the objective. The constant in the convergence rate has an elegant interpretation as the product of the (classical) condition number of the function with a novel geometric quantity that plays the role of a 'condition number' of the constraint set. We provide pointers to where these algorithms have made a difference in practice, in particular with the flow polytope, the marginal polytope and the base polytope for submodular optimization. This paper considers a fundamental class of convex matrix optimization problems with low-rank solutions. We show it is possible to solve these problem using far less memory than the natural size of the decision variable when the problem data has a concise representation. Our proposed method, SketchyCGM, is the first algorithm to offer provable convergence to an optimal point with an optimal memory footprint. SketchyCGM modifies a standard convex optimization method - the conditional gradient method - to work on a sketched version of the decision variable, and can recover the solution from this sketch. In contrast to recent work on non-convex methods for this problem class, SketchyCGM is a convex method; and our convergence guarantees do not rely on statistical assumptions.