1,448 research outputs found
Incremental Stochastic Subgradient Algorithms for Convex Optimization
In this paper we study the effect of stochastic errors on two constrained
incremental sub-gradient algorithms. We view the incremental sub-gradient
algorithms as decentralized network optimization algorithms as applied to
minimize a sum of functions, when each component function is known only to a
particular agent of a distributed network. We first study the standard cyclic
incremental sub-gradient algorithm in which the agents form a ring structure
and pass the iterate in a cycle. We consider the method with stochastic errors
in the sub-gradient evaluations and provide sufficient conditions on the
moments of the stochastic errors that guarantee almost sure convergence when a
diminishing step-size is used. We also obtain almost sure bounds on the
algorithm's performance when a constant step-size is used. We then consider
\ram{the} Markov randomized incremental subgradient method, which is a
non-cyclic version of the incremental algorithm where the sequence of computing
agents is modeled as a time non-homogeneous Markov chain. Such a model is
appropriate for mobile networks, as the network topology changes across time in
these networks. We establish the convergence results and error bounds for the
Markov randomized method in the presence of stochastic errors for diminishing
and constant step-sizes, respectively
An Infeasible-Point Subgradient Method Using Adaptive Approximate Projections
We propose a new subgradient method for the minimization of nonsmooth convex
functions over a convex set. To speed up computations we use adaptive
approximate projections only requiring to move within a certain distance of the
exact projections (which decreases in the course of the algorithm). In
particular, the iterates in our method can be infeasible throughout the whole
procedure. Nevertheless, we provide conditions which ensure convergence to an
optimal feasible point under suitable assumptions. One convergence result deals
with step size sequences that are fixed a priori. Two other results handle
dynamic Polyak-type step sizes depending on a lower or upper estimate of the
optimal objective function value, respectively. Additionally, we briefly sketch
two applications: Optimization with convex chance constraints, and finding the
minimum l1-norm solution to an underdetermined linear system, an important
problem in Compressed Sensing.Comment: 36 pages, 3 figure
Convex optimization over intersection of simple sets: improved convergence rate guarantees via an exact penalty approach
We consider the problem of minimizing a convex function over the intersection
of finitely many simple sets which are easy to project onto. This is an
important problem arising in various domains such as machine learning. The main
difficulty lies in finding the projection of a point in the intersection of
many sets. Existing approaches yield an infeasible point with an
iteration-complexity of for nonsmooth problems with no
guarantees on the in-feasibility. By reformulating the problem through exact
penalty functions, we derive first-order algorithms which not only guarantees
that the distance to the intersection is small but also improve the complexity
to and for smooth functions. For
composite and smooth problems, this is achieved through a saddle-point
reformulation where the proximal operators required by the primal-dual
algorithms can be computed in closed form. We illustrate the benefits of our
approach on a graph transduction problem and on graph matching
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