242 research outputs found
Analysis of top to bottom- shuffles
A deck of cards is shuffled by repeatedly moving the top card to one of
the bottom positions uniformly at random. We give upper and lower bounds
on the total variation mixing time for this shuffle as ranges from a
constant to . We also consider a symmetric variant of this shuffle in which
at each step either the top card is randomly inserted into the bottom
positions or a random card from the bottom positions is moved to the top.
For this reversible shuffle we derive bounds on the mixing time. Finally,
we transfer mixing time estimates for the above shuffles to the lazy top to
bottom- walks that move with probability 1/2 at each step.Comment: Published at http://dx.doi.org/10.1214/10505160500000062 in the
Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute
of Mathematical Statistics (http://www.imstat.org
Location-Aided Fast Distributed Consensus in Wireless Networks
Existing works on distributed consensus explore linear iterations based on
reversible Markov chains, which contribute to the slow convergence of the
algorithms. It has been observed that by overcoming the diffusive behavior of
reversible chains, certain nonreversible chains lifted from reversible ones mix
substantially faster than the original chains. In this paper, we investigate
the idea of accelerating distributed consensus via lifting Markov chains, and
propose a class of Location-Aided Distributed Averaging (LADA) algorithms for
wireless networks, where nodes' coarse location information is used to
construct nonreversible chains that facilitate distributed computing and
cooperative processing. First, two general pseudo-algorithms are presented to
illustrate the notion of distributed averaging through chain-lifting. These
pseudo-algorithms are then respectively instantiated through one LADA algorithm
on grid networks, and one on general wireless networks. For a grid
network, the proposed LADA algorithm achieves an -averaging time of
. Based on this algorithm, in a wireless network with
transmission range , an -averaging time of
can be attained through a centralized algorithm.
Subsequently, we present a fully-distributed LADA algorithm for wireless
networks, which utilizes only the direction information of neighbors to
construct nonreversible chains. It is shown that this distributed LADA
algorithm achieves the same scaling law in averaging time as the centralized
scheme. Finally, we propose a cluster-based LADA (C-LADA) algorithm, which,
requiring no central coordination, provides the additional benefit of reduced
message complexity compared with the distributed LADA algorithm.Comment: 44 pages, 14 figures. Submitted to IEEE Transactions on Information
Theor
Spectral gap of nonreversible Markov chains
We define the spectral gap of a Markov chain on a finite state space as the
second-smallest singular value of the generator of the chain, generalizing the
usual definition of spectral gap for reversible chains. We then define the
relaxation time of the chain as the inverse of this spectral gap, and show that
this relaxation time can be characterized, for any Markov chain, as the time
required for convergence of empirical averages. This relaxation time is related
to the Cheeger constant and the mixing time of the chain through inequalities
that are similar to the reversible case, and the path argument can be used to
get upper bounds. Several examples are worked out. An interesting finding from
the examples is that the time for convergence of empirical averages in
nonreversible chains can often be substantially smaller than the mixing time.Comment: 40 pages. Minor corrections and simplifications in this revisio
Uniform Chernoff and Dvoretzky-Kiefer-Wolfowitz-type inequalities for Markov chains and related processes
We observe that the technique of Markov contraction can be used to establish
measure concentration for a broad class of non-contracting chains. In
particular, geometric ergodicity provides a simple and versatile framework.
This leads to a short, elementary proof of a general concentration inequality
for Markov and hidden Markov chains (HMM), which supercedes some of the known
results and easily extends to other processes such as Markov trees. As
applications, we give a Dvoretzky-Kiefer-Wolfowitz-type inequality and a
uniform Chernoff bound. All of our bounds are dimension-free and hold for
countably infinite state spaces
Non-reversible Metropolis-Hastings
The classical Metropolis-Hastings (MH) algorithm can be extended to generate
non-reversible Markov chains. This is achieved by means of a modification of
the acceptance probability, using the notion of vorticity matrix. The resulting
Markov chain is non-reversible. Results from the literature on asymptotic
variance, large deviations theory and mixing time are mentioned, and in the
case of a large deviations result, adapted, to explain how non-reversible
Markov chains have favorable properties in these respects.
We provide an application of NRMH in a continuous setting by developing the
necessary theory and applying, as first examples, the theory to Gaussian
distributions in three and nine dimensions. The empirical autocorrelation and
estimated asymptotic variance for NRMH applied to these examples show
significant improvement compared to MH with identical stepsize.Comment: in Statistics and Computing, 201
Near Optimal Bounds for Collision in Pollard Rho for Discrete Log
We analyze a fairly standard idealization of Pollard's Rho algorithm for
finding the discrete logarithm in a cyclic group G. It is found that, with high
probability, a collision occurs in steps,
not far from the widely conjectured value of . This
improves upon a recent result of Miller--Venkatesan which showed an upper bound
of . Our proof is based on analyzing an appropriate
nonreversible, non-lazy random walk on a discrete cycle of (odd) length |G|,
and showing that the mixing time of the corresponding walk is
- β¦