Frank-Wolfe Algorithms for Saddle Point Problems

We extend the Frank-Wolfe (FW) optimization algorithm to solve constrained smooth convex-concave saddle point (SP) problems. Remarkably, the method only requires access to linear minimization oracles. Leveraging recent advances in FW optimization, we provide the first proof of convergence of a FW-type saddle point solver over polytopes, thereby partially answering a 30 year-old conjecture. We also survey other convergence results and highlight gaps in the theoretical underpinnings of FW-style algorithms. Motivating applications without known efficient alternatives are explored through structured prediction with combinatorial penalties as well as games over matching polytopes involving an exponential number of constraints.Comment: Appears in: Proceedings of the 20th International Conference on Artificial Intelligence and Statistics (AISTATS 2017). 39 page

Convex Combination of Constraint Vectors for Set-membership Affine Projection Algorithms

Set-membership affine projection (SM-AP) adaptive filters have been increasingly employed in the context of online data-selective learning. A key aspect for their good performance in terms of both convergence speed and steady-state mean-squared error is the choice of the so-called constraint vector. Optimal constraint vectors were recently proposed relying on convex optimization tools, which might some- times lead to prohibitive computational burden. This paper proposes a convex combination of simpler constraint vectors whose performance approaches the optimal solution closely, utilizing much fewer computations. Some illustrative examples confirm that the sub-optimal solution follows the accomplishments of the optimal one

Greedy Algorithms for Cone Constrained Optimization with Convergence Guarantees

Greedy optimization methods such as Matching Pursuit (MP) and Frank-Wolfe (FW) algorithms regained popularity in recent years due to their simplicity, effectiveness and theoretical guarantees. MP and FW address optimization over the linear span and the convex hull of a set of atoms, respectively. In this paper, we consider the intermediate case of optimization over the convex cone, parametrized as the conic hull of a generic atom set, leading to the first principled definitions of non-negative MP algorithms for which we give explicit convergence rates and demonstrate excellent empirical performance. In particular, we derive sublinear ($\mathcal{O}(1/t)$) convergence on general smooth and convex objectives, and linear convergence ($\mathcal{O}(e^{-t})$) on strongly convex objectives, in both cases for general sets of atoms. Furthermore, we establish a clear correspondence of our algorithms to known algorithms from the MP and FW literature. Our novel algorithms and analyses target general atom sets and general objective functions, and hence are directly applicable to a large variety of learning settings.Comment: NIPS 201