Convergence Rate of Frank-Wolfe for Non-Convex Objectives

We give a simple proof that the Frank-Wolfe algorithm obtains a stationary point at a rate of $O(1/\sqrt{t})$ on non-convex objectives with a Lipschitz continuous gradient. Our analysis is affine invariant and is the first, to the best of our knowledge, giving a similar rate to what was already proven for projected gradient methods (though on slightly different measures of stationarity).Comment: 6 page

Reducing Collision Risk in Multi-Agent Path Planning: Application to Air traffic Management

To minimize collision risks in the multi-agent path planning problem with stochastic transition dynamics, we formulate a Markov decision process congestion game with a multi-linear congestion cost. Players within the game complete individual tasks while minimizing their own collision risks. We show that the set of Nash equilibria coincides with the first-order KKT points of a non-convex optimization problem. Our game is applied to a historical flight plan over France to reduce collision risks between commercial aircraft.Comment: 6 pages, 2 figure

Stochastic Frank-Wolfe Methods for Nonconvex Optimization

We study Frank-Wolfe methods for nonconvex stochastic and finite-sum optimization problems. Frank-Wolfe methods (in the convex case) have gained tremendous recent interest in machine learning and optimization communities due to their projection-free property and their ability to exploit structured constraints. However, our understanding of these algorithms in the nonconvex setting is fairly limited. In this paper, we propose nonconvex stochastic Frank-Wolfe methods and analyze their convergence properties. For objective functions that decompose into a finite-sum, we leverage ideas from variance reduction techniques for convex optimization to obtain new variance reduced nonconvex Frank-Wolfe methods that have provably faster convergence than the classical Frank-Wolfe method. Finally, we show that the faster convergence rates of our variance reduced methods also translate into improved convergence rates for the stochastic setting

Riemannian Optimization via Frank-Wolfe Methods

We study projection-free methods for constrained Riemannian optimization. In particular, we propose the Riemannian Frank-Wolfe (RFW) method. We analyze non-asymptotic convergence rates of RFW to an optimum for (geodesically) convex problems, and to a critical point for nonconvex objectives. We also present a practical setting under which RFW can attain a linear convergence rate. As a concrete example, we specialize Rfw to the manifold of positive definite matrices and apply it to two tasks: (i) computing the matrix geometric mean (Riemannian centroid); and (ii) computing the Bures-Wasserstein barycenter. Both tasks involve geodesically convex interval constraints, for which we show that the Riemannian "linear oracle" required by RFW admits a closed-form solution; this result may be of independent interest. We further specialize RFW to the special orthogonal group and show that here too, the Riemannian "linear oracle" can be solved in closed form. Here, we describe an application to the synchronization of data matrices (Procrustes problem). We complement our theoretical results with an empirical comparison of Rfw against state-of-the-art Riemannian optimization methods and observe that RFW performs competitively on the task of computing Riemannian centroids.Comment: Under Review. Largely revised version, including an extended experimental section and an application to the special orthogonal group and the Procrustes proble

Fusion of Head and Full-Body Detectors for Multi-Object Tracking

In order to track all persons in a scene, the tracking-by-detection paradigm has proven to be a very effective approach. Yet, relying solely on a single detector is also a major limitation, as useful image information might be ignored. Consequently, this work demonstrates how to fuse two detectors into a tracking system. To obtain the trajectories, we propose to formulate tracking as a weighted graph labeling problem, resulting in a binary quadratic program. As such problems are NP-hard, the solution can only be approximated. Based on the Frank-Wolfe algorithm, we present a new solver that is crucial to handle such difficult problems. Evaluation on pedestrian tracking is provided for multiple scenarios, showing superior results over single detector tracking and standard QP-solvers. Finally, our tracker ranks 2nd on the MOT16 benchmark and 1st on the new MOT17 benchmark, outperforming over 90 trackers.Comment: 10 pages, 4 figures; Winner of the MOT17 challenge; CVPRW 201