3,085 research outputs found
A Distributed Frank-Wolfe Algorithm for Communication-Efficient Sparse Learning
Learning sparse combinations is a frequent theme in machine learning. In this
paper, we study its associated optimization problem in the distributed setting
where the elements to be combined are not centrally located but spread over a
network. We address the key challenges of balancing communication costs and
optimization errors. To this end, we propose a distributed Frank-Wolfe (dFW)
algorithm. We obtain theoretical guarantees on the optimization error
and communication cost that do not depend on the total number of
combining elements. We further show that the communication cost of dFW is
optimal by deriving a lower-bound on the communication cost required to
construct an -approximate solution. We validate our theoretical
analysis with empirical studies on synthetic and real-world data, which
demonstrate that dFW outperforms both baselines and competing methods. We also
study the performance of dFW when the conditions of our analysis are relaxed,
and show that dFW is fairly robust.Comment: Extended version of the SIAM Data Mining 2015 pape
Optimization Methods for Inverse Problems
Optimization plays an important role in solving many inverse problems.
Indeed, the task of inversion often either involves or is fully cast as a
solution of an optimization problem. In this light, the mere non-linear,
non-convex, and large-scale nature of many of these inversions gives rise to
some very challenging optimization problems. The inverse problem community has
long been developing various techniques for solving such optimization tasks.
However, other, seemingly disjoint communities, such as that of machine
learning, have developed, almost in parallel, interesting alternative methods
which might have stayed under the radar of the inverse problem community. In
this survey, we aim to change that. In doing so, we first discuss current
state-of-the-art optimization methods widely used in inverse problems. We then
survey recent related advances in addressing similar challenges in problems
faced by the machine learning community, and discuss their potential advantages
for solving inverse problems. By highlighting the similarities among the
optimization challenges faced by the inverse problem and the machine learning
communities, we hope that this survey can serve as a bridge in bringing
together these two communities and encourage cross fertilization of ideas.Comment: 13 page
Reflection methods for user-friendly submodular optimization
Recently, it has become evident that submodularity naturally captures widely
occurring concepts in machine learning, signal processing and computer vision.
Consequently, there is need for efficient optimization procedures for
submodular functions, especially for minimization problems. While general
submodular minimization is challenging, we propose a new method that exploits
existing decomposability of submodular functions. In contrast to previous
approaches, our method is neither approximate, nor impractical, nor does it
need any cumbersome parameter tuning. Moreover, it is easy to implement and
parallelize. A key component of our method is a formulation of the discrete
submodular minimization problem as a continuous best approximation problem that
is solved through a sequence of reflections, and its solution can be easily
thresholded to obtain an optimal discrete solution. This method solves both the
continuous and discrete formulations of the problem, and therefore has
applications in learning, inference, and reconstruction. In our experiments, we
illustrate the benefits of our method on two image segmentation tasks.Comment: Neural Information Processing Systems (NIPS), \'Etats-Unis (2013
Analysis of Stochastic Frank-Wolfe methods for product domains
In this thesis we will study in detail the convergence of some stochastic variants of the Frank-Wolfe algorithm when we are in presence of a product domain.In this thesis we will study in detail the convergence of some stochastic variants of the Frank-Wolfe algorithm when we are in presence of a product domain
- …