Bounding the Convergence of Mixing and Consensus Algorithms Simon - - PowerPoint PPT Presentation
Bounding the Convergence of Mixing and Consensus Algorithms Simon Apers 1 , Alain Sarlette 1,2 & Francesco Ticozzi 3,4 1 Ghent University, 2 INRIA Paris, 3 University of Padova, 4 Dartmouth College arXiv:1711.06024,1705.08253,1712.01609
1Ghent University, 2INRIA Paris, 3University of Padova, 4Dartmouth College
arXiv:1711.06024,1705.08253,1712.01609
dynamics on graphs:
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dynamics on graphs:
under appropriate conditions: dynamics will “mix” (converge, equilibrate)
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dynamics on graphs:
under appropriate conditions: dynamics will “mix” (converge, equilibrate) time scale = “mixing time”
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example: random walk on dumbbell graph
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example: random walk on dumbbell graph
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example: random walk on dumbbell graph
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mixing time: example: random walk on dumbbell graph
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example: random walk on dumbbell graph
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example: random walk on dumbbell graph conductance bound:
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example: random walk on dumbbell graph conductance bound:
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example: random walk on dumbbell graph conductance bound:
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proof idea: example: random walk on dumbbell graph
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conductance bound:
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example: random walk on dumbbell graph
however, diameter = 3 can we do any better ?
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example: random walk on dumbbell graph
however, diameter = 3 can we do any better ? yes: improve central hub
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example: random walk on dumbbell graph
however, diameter = 3 can we do any better ? yes: improve central hub
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example: random walk on dumbbell graph
however, diameter = 3 can we do any better ? yes: improve central hub
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example: random walk on dumbbell graph
however, diameter = 3 can we do any better ? yes: improve central hub
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example: random walk on dumbbell graph
example: random walk on dumbbell graph
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example: random walk on dumbbell graph however, diameter = 3 can we do any better ?
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example: random walk on dumbbell graph however, diameter = 3 can we do any better ? not using simple Markov chains:
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example: random walk on dumbbell graph however, diameter = 3 can we do any better ? not using simple Markov chains: what if we allow time dependence? memory? quantum dynamics?
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example: random walk on dumbbell graph however, diameter = 3 can we do any better ? not using simple Markov chains: what if we allow time dependence? memory? quantum dynamics?
e.g. non-backtracking random walks, lifted Markov chains, simulated annealing, polynomial filters, quantum walks,...
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stochastic process
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stochastic process
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stochastic process
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stochastic process
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stochastic process
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stochastic process examples of linear, local and invariant stochastic processes:
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stochastic process
examples of linear, local and invariant stochastic processes:
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stochastic process
examples of linear, local and invariant stochastic processes:
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stochastic process
examples of linear, local and invariant stochastic processes:
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stochastic process
examples of linear, local and invariant stochastic processes:
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stochastic process main theorem: any linear, local and invariant stochastic process has a mixing time
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stochastic process main theorem: any linear, local and invariant stochastic process has a mixing time
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stochastic process main theorem: any linear, local and invariant stochastic process has a mixing time
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main theorem: any linear, local and invariant stochastic process has a mixing time
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main theorem: any linear, local and invariant stochastic process has a mixing time proof:
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main theorem: any linear, local and invariant stochastic process has a mixing time proof: 1) we build a Markov chain simulator
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main theorem: any linear, local and invariant stochastic process has a mixing time proof: 1) we build a Markov chain simulator 2) we prove the theorem for Markov chain simulator
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1) Markov chain simulator of linear, local and invariant stochastic process:
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1) Markov chain simulator of linear, local and invariant stochastic process:
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1) Markov chain simulator of linear, local and invariant stochastic process: proof: max-flow min-cut argument
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1) Markov chain simulator of linear, local and invariant stochastic process: proof: max-flow min-cut argument
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1) Markov chain simulator of linear, local and invariant stochastic process: proof: max-flow min-cut argument
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1) Markov chain simulator of linear, local and invariant stochastic process: proof: max-flow min-cut argument
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1) Markov chain simulator of linear, local and invariant stochastic process: proof: max-flow min-cut argument
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1) Markov chain simulator of linear, local and invariant stochastic process:
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1) Markov chain simulator of linear, local and invariant stochastic process:
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if stochastic process is linear and local, then this transition rule simulates the process: 1) Markov chain simulator of linear, local and invariant stochastic process:
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1) Markov chain simulator of linear, local and invariant stochastic process:
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! rule is non-Markovian: depends on initial state and time 1) Markov chain simulator of linear, local and invariant stochastic process:
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classic trick: give walker a timer and a memory of initial state ! rule is non-Markovian: depends on initial state and time 1) Markov chain simulator of linear, local and invariant stochastic process:
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classic trick: give walker a timer and a memory of initial state = MC on enlarged state space (“lifted MC”) ! rule is non-Markovian: depends on initial state and time 1) Markov chain simulator of linear, local and invariant stochastic process:
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classic trick: give walker a timer and a memory of initial state = MC on enlarged state space (“lifted MC”) ! rule is non-Markovian: depends on initial state and time 1) Markov chain simulator of linear, local and invariant stochastic process:
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classic trick: give walker a timer and a memory of initial state = MC on enlarged state space (“lifted MC”) ! rule is non-Markovian: depends on initial state and time 1) Markov chain simulator of linear, local and invariant stochastic process:
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simulates up to time T 1) Markov chain simulator of linear, local and invariant stochastic process:
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second trick: if process is invariant, then we can “amplify” simulates up to time T 1) Markov chain simulator of linear, local and invariant stochastic process:
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second trick: if process is invariant, then we can “amplify” = restart the simulation every time timer reaches T simulates up to time T 1) Markov chain simulator of linear, local and invariant stochastic process:
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second trick: if process is invariant, then we can “amplify” = restart the simulation every time timer reaches T simulates up to time T 1) Markov chain simulator of linear, local and invariant stochastic process: proposition: the (asymptotic) mixing time of this amplified simulator closely relates to the (asymptotic) mixing time of the original process
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2) Markov chain simulator obeys a conductance bound:
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2) Markov chain simulator obeys a conductance bound: simulator is Markov chain on enlarged state space:
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2) Markov chain simulator obeys a conductance bound: simulator is Markov chain on enlarged state space: + conductance cannot be increased by lifting
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2) Markov chain simulator obeys a conductance bound: simulator is Markov chain on enlarged state space: + conductance cannot be increased by lifting = main theorem: any linear, local and invariant stochastic process has a mixing time
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main theorem: any linear, local and invariant stochastic process has a mixing time example 1: dumbbell graph
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main theorem: any linear, local and invariant stochastic process has a mixing time example 1: dumbbell graph
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any linear, local and invariant stochastic process on the dumbbell graph has a mixing time
main theorem: any linear, local and invariant stochastic process has a mixing time example 2: binary tree
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main theorem: any linear, local and invariant stochastic process has a mixing time example 2: binary tree
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main theorem: any linear, local and invariant stochastic process has a mixing time example 2: binary tree any linear, local and invariant stochastic process on the binary tree has the same mixing time as a random walk
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main theorem: any linear, local and invariant stochastic process has a mixing time example 3: finite time convergence
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main theorem: any linear, local and invariant stochastic process has a mixing time example 3: finite time convergence what is the least number of local, symmetric stochastic matrices whose product has rank one?
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main theorem: any linear, local and invariant stochastic process has a mixing time example 3: finite time convergence what is the least number of local, symmetric stochastic matrices whose product has rank one? = mixing time of time-inhomogeneous symmetric Markov chain
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main theorem: any linear, local and invariant stochastic process has a mixing time example 4: quantum walks
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main theorem: any linear, local and invariant stochastic process has a mixing time example 4: quantum walks first bound for the mixing time of general quantum Markov chains see details in [arXiv:1712.01609]
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main theorem: any linear, local and invariant stochastic process has a mixing time
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main theorem: any linear, local and invariant stochastic process has a mixing time
there exists a linear, local and invariant stochastic process that has a mixing time
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see Chen, Lovász and Pak (STOC’99)
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main theorem: any linear, local and invariant stochastic process has a mixing time
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main theorem: any linear, local and invariant stochastic process has a mixing time
there exists a linear and local process that has the trivial mixing time
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main theorem: any linear, local and invariant stochastic process has a mixing time
see Pavon and Ticozzi, Journal of Math.Ph. (‘10): there exists a linear and local process that has the trivial mixing time
main theorem: any linear, local and invariant stochastic process has a mixing time some open questions:
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main theorem: any linear, local and invariant stochastic process has a mixing time some open questions:
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main theorem: any linear, local and invariant stochastic process has a mixing time some open questions:
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main theorem: any linear, local and invariant stochastic process has a mixing time some open questions:
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main theorem: any linear, local and invariant stochastic process has a mixing time some open questions:
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