5,237 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
Confronting brane inflation with Planck and pre-Planck data
In this paper, we compare brane inflation models with the Planck data and the
pre-Planck data (which combines WMAP, ACT, SPT, BAO and H0 data). The Planck
data prefer a spectral index less than unity at more than 5\sigma confidence
level, and a running of the spectral index at around 2\sigma confidence level.
We find that the KKLMMT model can survive at the level of 2\sigma only if the
parameter (the conformal coupling between the Hubble parameter and the
inflaton) is less than , which indicates a certain level
of fine-tuning. The IR DBI model can provide a slightly larger negative running
of spectral index and red tilt, but in order to be consistent with the
non-Gaussianity constraints from Planck, its parameter also needs fine-tuning
at some level.Comment: 10 pages, 8 figure
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