Walking in the Shadow: A New Perspective on Descent Directions for Constrained Minimization

Descent directions such as movement towards Frank-Wolfe vertices, away steps, in-face away steps and pairwise directions have been an important design consideration in conditional gradient descent (CGD) variants. In this work, we attempt to demystify the impact of movement in these directions towards attaining constrained minimizers. The best local direction of descent is the directional derivative of the projection of the gradient, which we refer to as the $\textit{shadow}$ of the gradient. We show that the continuous-time dynamics of moving in the shadow are equivalent to those of PGD however non-trivial to discretize. By projecting gradients in PGD, one not only ensures feasibility but is also able to "wrap" around the convex region. We show that Frank-Wolfe (FW) vertices in fact recover the maximal wrap one can obtain by projecting gradients, thus providing a new perspective on these steps. We also claim that the shadow steps give the best direction of descent emanating from the convex hull of all possible away-steps. Viewing PGD movements in terms of shadow steps gives linear convergence, dependent on the number of faces. We combine these insights into a novel $S\small{HADOW}$-$CG$ method that uses FW steps (i.e., wrap around the polytope) and shadow steps (i.e., optimal local descent direction), while enjoying linear convergence. Our analysis develops properties of the curve formed by projecting a line on a polytope, which may be of independent interest, while providing a unifying view of various descent directions in the CGD literature

Recursive Frank-Wolfe algorithms

In the last decade there has been a resurgence of interest in Frank-Wolfe (FW) style methods for optimizing a smooth convex function over a polytope. Examples of recently developed techniques include {\em Decomposition-invariant Conditional Gradient} (DiCG), {\em Blended Condition Gradient} (BCG), and {\em Frank-Wolfe with in-face directions} (IF-FW) methods. We introduce two extensions of these techniques. First, we augment DiCG with the {\em working set} strategy, and show how to optimize over the working set using {\em shadow simplex steps}. Second, we generalize in-face Frank-Wolfe directions to polytopes in which faces cannot be efficiently computed, and also describe a generic recursive procedure that can be used in conjunction with several FW-style techniques. Experimental results indicate that these extensions are capable of speeding up original algorithms by orders of magnitude for certain applications

Avoiding bad steps in Frank Wolfe variants

The analysis of Frank Wolfe (FW) variants is often complicated by the presence of different kinds of "good" and "bad" steps. In this article we aim to simplify the convergence analysis of some of these variants by getting rid of such a distinction between steps, and to improve existing rates by ensuring a sizable decrease of the objective at each iteration. In order to do this, we define the Short Step Chain (SSC) procedure, which skips gradient computations in consecutive short steps until proper stopping conditions are satisfied. This technique allows us to give a unified analysis and converge rates in the general smooth non convex setting, as well as a linear convergence rate under a Kurdyka-Lojasiewicz (KL) property. While this setting has been widely studied for proximal gradient type methods, to our knowledge, it has not been analyzed before for the Frank Wolfe variants under study. An angle condition, ensuring that the directions selected by the methods have the steepest slope possible up to a constant, is used to carry out our analysis. We prove that this condition is satisfied on polytopes by the away step Frank-Wolfe (AFW), the pairwise Frank-Wolfe (PFW), and the Frank-Wolfe method with in face directions (FDFW).Comment: See arXiv:2008.09781 for an extended version of the pape

Conditional Gradient Methods

The purpose of this survey is to serve both as a gentle introduction and a coherent overview of state-of-the-art Frank--Wolfe algorithms, also called conditional gradient algorithms, for function minimization. These algorithms are especially useful in convex optimization when linear optimization is cheaper than projections. The selection of the material has been guided by the principle of highlighting crucial ideas as well as presenting new approaches that we believe might become important in the future, with ample citations even of old works imperative in the development of newer methods. Yet, our selection is sometimes biased, and need not reflect consensus of the research community, and we have certainly missed recent important contributions. After all the research area of Frank--Wolfe is very active, making it a moving target. We apologize sincerely in advance for any such distortions and we fully acknowledge: We stand on the shoulder of giants.Comment: 238 pages with many figures. The FrankWolfe.jl Julia package (https://github.com/ZIB-IOL/FrankWolfe.jl) providces state-of-the-art implementations of many Frank--Wolfe method