421 research outputs found
Stochastic Block Mirror Descent Methods for Nonsmooth and Stochastic Optimization
In this paper, we present a new stochastic algorithm, namely the stochastic
block mirror descent (SBMD) method for solving large-scale nonsmooth and
stochastic optimization problems. The basic idea of this algorithm is to
incorporate the block-coordinate decomposition and an incremental block
averaging scheme into the classic (stochastic) mirror-descent method, in order
to significantly reduce the cost per iteration of the latter algorithm. We
establish the rate of convergence of the SBMD method along with its associated
large-deviation results for solving general nonsmooth and stochastic
optimization problems. We also introduce different variants of this method and
establish their rate of convergence for solving strongly convex, smooth, and
composite optimization problems, as well as certain nonconvex optimization
problems. To the best of our knowledge, all these developments related to the
SBMD methods are new in the stochastic optimization literature. Moreover, some
of our results also seem to be new for block coordinate descent methods for
deterministic optimization
Distributed Stochastic Optimization over Time-Varying Noisy Network
This paper is concerned with distributed stochastic multi-agent optimization
problem over a class of time-varying network with slowly decreasing
communication noise effects. This paper considers the problem in composite
optimization setting which is more general in noisy network optimization. It is
noteworthy that existing methods for noisy network optimization are Euclidean
projection based. We present two related different classes of non-Euclidean
methods and investigate their convergence behavior. One is distributed
stochastic composite mirror descent type method (DSCMD-N) which provides a more
general algorithm framework than former works in this literature. As a
counterpart, we also consider a composite dual averaging type method (DSCDA-N)
for noisy network optimization. Some main error bounds for DSCMD-N and DSCDA-N
are obtained. The trade-off among stepsizes, noise decreasing rates,
convergence rates of algorithm is analyzed in detail. To the best of our
knowledge, this is the first work to analyze and derive convergence rates of
optimization algorithm in noisy network optimization. We show that an optimal
rate of in nonsmooth convex optimization can be obtained for
proposed methods under appropriate communication noise condition. Moveover,
convergence rates in different orders are comprehensively derived in both
expectation convergence and high probability convergence sense.Comment: 27 page
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
Mirror Descent and Convex Optimization Problems With Non-Smooth Inequality Constraints
We consider the problem of minimization of a convex function on a simple set
with convex non-smooth inequality constraint and describe first-order methods
to solve such problems in different situations: smooth or non-smooth objective
function; convex or strongly convex objective and constraint; deterministic or
randomized information about the objective and constraint. We hope that it is
convenient for a reader to have all the methods for different settings in one
place. Described methods are based on Mirror Descent algorithm and switching
subgradient scheme. One of our focus is to propose, for the listed different
settings, a Mirror Descent with adaptive stepsizes and adaptive stopping rule.
This means that neither stepsize nor stopping rule require to know the
Lipschitz constant of the objective or constraint. We also construct Mirror
Descent for problems with objective function, which is not Lipschitz
continuous, e.g. is a quadratic function. Besides that, we address the problem
of recovering the solution of the dual problem
Optimal Complexity and Certification of Bregman First-Order Methods
We provide a lower bound showing that the convergence rate of the
NoLips method (a.k.a. Bregman Gradient) is optimal for the class of functions
satisfying the -smoothness assumption. This assumption, also known as
relative smoothness, appeared in the recent developments around the Bregman
Gradient method, where acceleration remained an open issue. On the way, we show
how to constructively obtain the corresponding worst-case functions by
extending the computer-assisted performance estimation framework of Drori and
Teboulle (Mathematical Programming, 2014) to Bregman first-order methods, and
to handle the classes of differentiable and strictly convex functions.Comment: To appear in Mathematical Programmin
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