10,689 research outputs found
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
From Infinite to Finite Programs: Explicit Error Bounds with Applications to Approximate Dynamic Programming
We consider linear programming (LP) problems in infinite dimensional spaces
that are in general computationally intractable. Under suitable assumptions, we
develop an approximation bridge from the infinite-dimensional LP to tractable
finite convex programs in which the performance of the approximation is
quantified explicitly. To this end, we adopt the recent developments in two
areas of randomized optimization and first order methods, leading to a priori
as well as a posterior performance guarantees. We illustrate the generality and
implications of our theoretical results in the special case of the long-run
average cost and discounted cost optimal control problems for Markov decision
processes on Borel spaces. The applicability of the theoretical results is
demonstrated through a constrained linear quadratic optimal control problem and
a fisheries management problem.Comment: 30 pages, 5 figure
Distributed Multi-Agent Optimization with State-Dependent Communication
We study distributed algorithms for solving global optimization problems in
which the objective function is the sum of local objective functions of agents
and the constraint set is given by the intersection of local constraint sets of
agents. We assume that each agent knows only his own local objective function
and constraint set, and exchanges information with the other agents over a
randomly varying network topology to update his information state. We assume a
state-dependent communication model over this topology: communication is
Markovian with respect to the states of the agents and the probability with
which the links are available depends on the states of the agents. In this
paper, we study a projected multi-agent subgradient algorithm under
state-dependent communication. The algorithm involves each agent performing a
local averaging to combine his estimate with the other agents' estimates,
taking a subgradient step along his local objective function, and projecting
the estimates on his local constraint set. The state-dependence of the
communication introduces significant challenges and couples the study of
information exchange with the analysis of subgradient steps and projection
errors. We first show that the multi-agent subgradient algorithm when used with
a constant stepsize may result in the agent estimates to diverge with
probability one. Under some assumptions on the stepsize sequence, we provide
convergence rate bounds on a "disagreement metric" between the agent estimates.
Our bounds are time-nonhomogeneous in the sense that they depend on the initial
starting time. Despite this, we show that agent estimates reach an almost sure
consensus and converge to the same optimal solution of the global optimization
problem with probability one under different assumptions on the local
constraint sets and the stepsize sequence
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
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