We develop subgradientand gradient-based methods for minimizing strongly convex functions under a notion which generalizes the standard Euclidean strong convexity. We propose a unifying framework for subgradient methods which yields two kinds of methods, namely, the Proximal Gradient Method (PGM) and the Conditional Gradient Method (CGM), unifying several existing methods. The unifying framewor...

There is renewed interest in formulating integration as a statistical inference problem, motivated by obtaining a full distribution over numerical error that can be propagated through subsequent computation. Current methods, such as Bayesian Quadrature, demonstrate impressive empirical performance but lack theoretical analysis. An important challenge is therefore to reconcile these probabilisti...

Point source localisation is generally modelled as a Lasso-type problem on measures. However, optimisation methods in non-Hilbert spaces, such the space of Radon measures, are much less developed than Hilbert spaces. Most numerical algorithms for point based Frank-Wolfe conditional gradient method, which ad hoc convergence theory developed. We develop extensions proximal-type to spaces This inc...

Recently, the Frank-Wolfe optimization algorithm was suggested as a procedure to obtain adaptive quadrature rules for integrals of functions in a reproducing kernel Hilbert space (RKHS) with a potentially faster rate of convergence than Monte Carlo integration (and “kernel herding” was shown to be a special case of this procedure). In this paper, we propose to replace the random sampling step i...

The famous Frank–Wolfe theorem ensures attainability of the optimal value for quadratic objective functions over a (possibly unbounded) polyhedron if the feasible values are bounded. This theorem does not hold in general for conic programs where linear constraints are replaced by more general convex constraints like positive-semidefiniteness or copositivity conditions, despite the fact that the...

The Frank-Wolfe optimization algorithm has recently regained popularity for machine learning applications due to its projection-free property and its ability to handle structured constraints. However, in the stochastic learning setting, it is still relatively understudied compared to the gradient descent counterpart. In this work, leveraging a recent variance reduction technique, we propose two...

The Frank-Wolfe method (a.k.a. conditional gradient algorithm) for smooth optimization has regained much interest in recent years in the context of large scale optimization and machine learning. A key advantage of the method is that it avoids projections the computational bottleneck in many applications replacing it by a linear optimization step. Despite this advantage, the known convergence ra...