3,878 research outputs found
Stochastic Quasi-Fej\'er Block-Coordinate Fixed Point Iterations with Random Sweeping
This work proposes block-coordinate fixed point algorithms with applications
to nonlinear analysis and optimization in Hilbert spaces. The asymptotic
analysis relies on a notion of stochastic quasi-Fej\'er monotonicity, which is
thoroughly investigated. The iterative methods under consideration feature
random sweeping rules to select arbitrarily the blocks of variables that are
activated over the course of the iterations and they allow for stochastic
errors in the evaluation of the operators. Algorithms using quasinonexpansive
operators or compositions of averaged nonexpansive operators are constructed,
and weak and strong convergence results are established for the sequences they
generate. As a by-product, novel block-coordinate operator splitting methods
are obtained for solving structured monotone inclusion and convex minimization
problems. In particular, the proposed framework leads to random
block-coordinate versions of the Douglas-Rachford and forward-backward
algorithms and of some of their variants. In the standard case of block,
our results remain new as they incorporate stochastic perturbations
Stochastic Variance Reduction Methods for Saddle-Point Problems
We consider convex-concave saddle-point problems where the objective
functions may be split in many components, and extend recent stochastic
variance reduction methods (such as SVRG or SAGA) to provide the first
large-scale linearly convergent algorithms for this class of problems which is
common in machine learning. While the algorithmic extension is straightforward,
it comes with challenges and opportunities: (a) the convex minimization
analysis does not apply and we use the notion of monotone operators to prove
convergence, showing in particular that the same algorithm applies to a larger
class of problems, such as variational inequalities, (b) there are two notions
of splits, in terms of functions, or in terms of partial derivatives, (c) the
split does need to be done with convex-concave terms, (d) non-uniform sampling
is key to an efficient algorithm, both in theory and practice, and (e) these
incremental algorithms can be easily accelerated using a simple extension of
the "catalyst" framework, leading to an algorithm which is always superior to
accelerated batch algorithms.Comment: Neural Information Processing Systems (NIPS), 2016, Barcelona, Spai
Stochastic Approximations and Perturbations in Forward-Backward Splitting for Monotone Operators
We investigate the asymptotic behavior of a stochastic version of the
forward-backward splitting algorithm for finding a zero of the sum of a
maximally monotone set-valued operator and a cocoercive operator in Hilbert
spaces. Our general setting features stochastic approximations of the
cocoercive operator and stochastic perturbations in the evaluation of the
resolvents of the set-valued operator. In addition, relaxations and not
necessarily vanishing proximal parameters are allowed. Weak and strong almost
sure convergence properties of the iterates is established under mild
conditions on the underlying stochastic processes. Leveraging these results, we
also establish the almost sure convergence of the iterates of a stochastic
variant of a primal-dual proximal splitting method for composite minimization
problems
A stochastic inertial forward-backward splitting algorithm for multivariate monotone inclusions
We propose an inertial forward-backward splitting algorithm to compute the
zero of a sum of two monotone operators allowing for stochastic errors in the
computation of the operators. More precisely, we establish almost sure
convergence in real Hilbert spaces of the sequence of iterates to an optimal
solution. Then, based on this analysis, we introduce two new classes of
stochastic inertial primal-dual splitting methods for solving structured
systems of composite monotone inclusions and prove their convergence. Our
results extend to the stochastic and inertial setting various types of
structured monotone inclusion problems and corresponding algorithmic solutions.
Application to minimization problems is discussed
A Class of Randomized Primal-Dual Algorithms for Distributed Optimization
Based on a preconditioned version of the randomized block-coordinate
forward-backward algorithm recently proposed in [Combettes,Pesquet,2014],
several variants of block-coordinate primal-dual algorithms are designed in
order to solve a wide array of monotone inclusion problems. These methods rely
on a sweep of blocks of variables which are activated at each iteration
according to a random rule, and they allow stochastic errors in the evaluation
of the involved operators. Then, this framework is employed to derive
block-coordinate primal-dual proximal algorithms for solving composite convex
variational problems. The resulting algorithm implementations may be useful for
reducing computational complexity and memory requirements. Furthermore, we show
that the proposed approach can be used to develop novel asynchronous
distributed primal-dual algorithms in a multi-agent context
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