A Network Gossip Algorithm using Lifted Markov Chain

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A Network Gossip Algorithm using Lifted Markov
Chain

• Jinwoo Shin
• Joint work with Kyomin Jung and Devavrat Shah
• Laboratory of Information and Decision Systems
• Massachusetts Institute of Technology
• ICASSP, Apr. 22th 2009

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Motivation

• Network in the 20th century- Telephone
  networks- Internet
• Next generation network?- Peer-to-peer
  networks- Social networks- Wireless sensor
  networks- Mobile (ad-hoc) networks

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Motivation

• Characteristics of these networks?
• - Lack of (reliable) infrastructure
• ? Algorithm should be distributed
  (i.e. local message-passing)
• - Dynamic topology
• ? Algorithm should be robust against
  network changes
• - Resource constraints
• ? Algorithm should be simple and
  efficient-computable
• Naturally, lead to Gossip algorithms.

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Distributed Averaging

• Setup- A network graph G(V,E) with Vn-
  Initial values xi at node i- Goal Design a
  Gossip algorithm to find the average
  at all nodes
• Applications- distributed linear estimation,
  distributed load balancing, distributed
  consensus, synchronization, etc.

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Gossip Algorithm for Averaging

• Linear distributed iterations- a popular
  approach since Tsitsiklis (1984)- setting it
  needs a Markov Chain P s.t. ? P is graph
  conformant i.e. P(i.j)gt0 only if (i,j)?E. ? P
  has the uniform stationary distribution i.e.
  11P.- updating mechanism ? xi(t) is a value
  of a node i at time t and initially xi(0) xi ?
  a node j transfers P(i,j)xj(t) to its neighbor i
  at time t ? xi(t1) ?jP(i,j)xj(t) - for every
  node i, xi(t) converges to xave

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Mixing time of MC
? E number of communications happens per each
iteration.

• The mixing time of MC is the time to approach
  stationarity from a worst initial state.
• The running time for the convergence is the
  mixing time H of P
• The total number of operations is HE.
• The key issue is finding a graph conformant MC P
  that has the smallest mixing time H and the
  uniform stationary distribution.
• Boyd et al. (2004) showed that the fastest mixing
  symmetric one can be found using SDP.
• Question How about the non-symmetric one?

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Example
1-1/n
E2
E1
1-1/n
1/2
1/n
1/2
1/n
E2 is a lifting of E1
? HO(n2) ? symmetric
? HO(nlog n) ? non-symmetric
? Observed by Diaconis et al. (1997)
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Lifting a Markov chain

• Definition is a lifted Markov chain of P
  if there exists a many-to-one function
  s.t.- are sets of states of
  respectively. -
  , where
  and is a stationary
  distribution of P
• Informally, is a lifted Markov chain of P
  if- a node of is a copy of a node of P-
  two copies have a positive transition prob. only
  if their original nodes have-
  the lifted MC can simulate the original
  stationary distribution

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Previous results

• Diaconis, Holmes and Neal (1997)- found a fast
  mixing lifted MC for the ring-like graph (E2).
• Chen, Lovasz and Pak (1999)1) for a general P,
  constructed a non-symmetric lifted MC that has
  the mixing time 1/F(P), where F(P) is the
  conductance of P. ? via lifting, the mixing
  time can be reduced up to its square
  root since 1/F(P)H1/F2(P)
• 2) showed that this is the fastest mixing
  lifted MC and symmetric lifted MC cannot reduce
  mixing time.

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Back to averaging

• Recall the performance guarantees- the running
  time H- the total number of operations HE
• Averaging using the lifted MC- Implementable via
  running multiple threads per each node.- The
  lifted MC reduces the running time.- However,
  the total number of operations may not be reduced
  due to the increased size E of the lifted MC.
• Revised goal- Find a fast mixing slim lifted MC

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Pseudo lifting

• We introduce the new notion of lifting, which
  will produce better performance than the old one
  for averaging.
• Basic motivation similar to the original
  lifting, but allow a small fractional loss in
  preserving the original stationary distribution
• Definition same as the original lifting except
  for the last condition- previous condition
  - new condition there exists a
  copy j of i s.t.

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Our result L1

• CLP(1999) lifting
• Mixing time 1/F(P)- Size n2/F(P)- D
  1/F(P)

• For a general P, we construct a pseudo lifted
  MC- its mixing time D,where D is a diameter
  of the underlying graph of P - its size nD
• The mixing time D is the optimal one achievable
  for a given underlying graph topology.
• Example Barbell-graph

? DO(1), but 1/F(P)O(n), which is a lower
bound of the mixing time of the original lifted
MC -gt The pseudo lifting is better than the
original one.
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Main idea for L1

• The essence of CLP- consider the complete graph
  which mixes in one step, and embed its topology
  to the original graph.- need to solve the
  uniform multi-commodity flow problem to maintain
  the properties of lifting.
• Our main ideas
• 1) we use a sparse expander graph instead of
  the complete graph, and it leads to the reduction
  of size.
• 2) by allowing a loss in preserving the
  original stationary distribution, we relax the
  uniform multi-commodity flow problem. It reduces
  the mixing time.

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Our result L2

• Doubling dimension- Definition The doubling
  dimension ? of a metric space is the least k s.t.
  any ball of radius R can be covered by 2k balls
  of radius R/2.- Example dim-k grid has ?k.
• For a general P, we construct a pseudo lifted
  MC- its mixing time D - its size
  Dn1-1/(?1), where ? is the doubling dimension
  of the underlying graph of P.
• Graphs which need lifting usually have small
  doubling dimension.

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Back to averaging

• Implementable using L2 construction - For every
  node i, exists a copy j of i such that xj(t)
  converges to (1-e)xave.- Possible to estimate
  xave by xj(t)/(1-e)
• The performance- the running time D - the
  total number of operations D2n1-1/(?1)
  

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Example 2-d Grid

• Consider a random walk on the 2-dimensional grid
  graph with n nodes.
• Before lifting - Mixing time O(n)- Size n
• After lifting using L2 (L1,CLP)- Mixing time
  n0.5 (n0.5,n0.5)- Size n1.17 (n1.5,n2.5 )
• Benefit of lifting for averaging - Running time
  n ? n0.5- Total number of operations n2 ?
  n1.67

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Summary - Comparison of results
? D 1/F(P)
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Conclusion

• Markov chain is one of fundamental tools for
  designing network algorithms. - Mixing time
  decides the running time.
• Lifting reduces the mixing time, but increase the
  size.- We construct lifting with the best
  possible mixing time and small size.
• Thank you!