2,331 research outputs found
On the Convergence of Decentralized Gradient Descent
Consider the consensus problem of minimizing where
each is only known to one individual agent out of a connected network
of agents. All the agents shall collaboratively solve this problem and
obtain the solution subject to data exchanges restricted to between neighboring
agents. Such algorithms avoid the need of a fusion center, offer better network
load balance, and improve data privacy. We study the decentralized gradient
descent method in which each agent updates its variable , which is
a local approximate to the unknown variable , by combining the average of
its neighbors' with the negative gradient step .
The iteration is where the averaging coefficients form a symmetric doubly stochastic matrix
. We analyze the convergence of this
iteration and derive its converge rate, assuming that each is proper
closed convex and lower bounded, is Lipschitz continuous with
constant , and stepsize is fixed. Provided that where , the objective error at the averaged
solution, , reduces at a speed of
until it reaches . If are further (restricted) strongly
convex, then both and each converge
to the global minimizer at a linear rate until reaching an
-neighborhood of . We also develop an iteration for
decentralized basis pursuit and establish its linear convergence to an
-neighborhood of the true unknown sparse signal
"Multiproduct Duopoly with Vertical Differentiation"
The paper investigates a two-stage competition in a vertical differentiated industry, where each firm produces an arbitrary number of similar qualities and sells them to heterogeneous consumers. We show that, when unit costs of quality are increasing and quadratic, each firm has an incentive to provide an interval of qualities. The finding is in sharp contrast to the single-quality outcome when the market coverage is exogenously determined. We also show that allowing for an interval of qualities intensifies competition, lowers the profits of each firm and raises the consumer surplus and the social welfare in comparison to the single-quality duopoly.
On the Linear Convergence of the ADMM in Decentralized Consensus Optimization
In decentralized consensus optimization, a connected network of agents
collaboratively minimize the sum of their local objective functions over a
common decision variable, where their information exchange is restricted
between the neighbors. To this end, one can first obtain a problem
reformulation and then apply the alternating direction method of multipliers
(ADMM). The method applies iterative computation at the individual agents and
information exchange between the neighbors. This approach has been observed to
converge quickly and deemed powerful. This paper establishes its linear
convergence rate for decentralized consensus optimization problem with strongly
convex local objective functions. The theoretical convergence rate is
explicitly given in terms of the network topology, the properties of local
objective functions, and the algorithm parameter. This result is not only a
performance guarantee but also a guideline toward accelerating the ADMM
convergence.Comment: 11 figures, IEEE Transactions on Signal Processing, 201
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