4,298 research outputs found
Generalized conditional gradient: analysis of convergence and applications
The objectives of this technical report is to provide additional results on
the generalized conditional gradient methods introduced by Bredies et al.
[BLM05]. Indeed , when the objective function is smooth, we provide a novel
certificate of optimality and we show that the algorithm has a linear
convergence rate. Applications of this algorithm are also discussed
Semi-proximal Mirror-Prox for Nonsmooth Composite Minimization
We propose a new first-order optimisation algorithm to solve high-dimensional
non-smooth composite minimisation problems. Typical examples of such problems
have an objective that decomposes into a non-smooth empirical risk part and a
non-smooth regularisation penalty. The proposed algorithm, called Semi-Proximal
Mirror-Prox, leverages the Fenchel-type representation of one part of the
objective while handling the other part of the objective via linear
minimization over the domain. The algorithm stands in contrast with more
classical proximal gradient algorithms with smoothing, which require the
computation of proximal operators at each iteration and can therefore be
impractical for high-dimensional problems. We establish the theoretical
convergence rate of Semi-Proximal Mirror-Prox, which exhibits the optimal
complexity bounds, i.e. , for the number of calls to linear
minimization oracle. We present promising experimental results showing the
interest of the approach in comparison to competing methods
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
Variance-Reduced and Projection-Free Stochastic Optimization
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 stochastic Frank-Wolfe variants which substantially improve
previous results in terms of the number of stochastic gradient evaluations
needed to achieve accuracy. For example, we improve from
to if the objective function
is smooth and strongly convex, and from to
if the objective function is smooth and
Lipschitz. The theoretical improvement is also observed in experiments on
real-world datasets for a multiclass classification application
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