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

dynamics on graphs: ● diffusion ● rumour spreading ● weight balancing ● quantum walks ● ... 2

dynamics on graphs: ● diffusion ● rumour spreading ● weight balancing ● quantum walks ● ... under appropriate conditions: dynamics will “mix” (converge, equilibrate) 2

dynamics on graphs: ● diffusion ● rumour spreading ● weight balancing ● quantum walks ● ... under appropriate conditions: dynamics will “mix” (converge, equilibrate) time scale = “mixing time” 2

example: random walk on dumbbell graph 3

example: random walk on dumbbell graph 3

example: random walk on dumbbell graph 3

example: random walk on dumbbell graph mixing time: 3

example: random walk on dumbbell graph 4

example: random walk on dumbbell graph conductance bound: 4

example: random walk on dumbbell graph conductance bound: 4

example: random walk on dumbbell graph conductance bound: 4

example: random walk on dumbbell graph conductance bound: proof idea: 5

example: random walk on dumbbell graph 6

example: random walk on dumbbell graph however, diameter = 3 can we do any better ? 6

example: random walk on dumbbell graph however, diameter = 3 can we do any better ? yes: improve central hub 6

example: random walk on dumbbell graph however, diameter = 3 can we do any better ? yes: improve central hub 6

example: random walk on dumbbell graph however, diameter = 3 can we do any better ? yes: improve central hub 6

example: random walk on dumbbell graph however, diameter = 3 can we do any better ? yes: improve central hub 6

example: random walk on dumbbell graph 7

example: random walk on dumbbell graph however, diameter = 3 can we do any better ? 7

example: random walk on dumbbell graph however, diameter = 3 can we do any better ? not using simple Markov chains: 7

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? 7

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, 7 polynomial filters, quantum walks,...

stochastic process 8

stochastic process 8

stochastic process ● linear 8

stochastic process ● linear ● local 8

stochastic process ● linear ● local ● invariant 8

stochastic process examples of linear, local and invariant stochastic processes: 9

stochastic process examples of linear, local and invariant stochastic processes: ● Markov chains, time-averaged MCs, time-inhomogeneous invariant MCs 9

stochastic process examples of linear, local and invariant stochastic processes: ● Markov chains, time-averaged MCs, time-inhomogeneous invariant MCs ● lifted MCs, non-backtracking RWs on regular graphs 9

stochastic process examples of linear, local and invariant stochastic processes: ● Markov chains, time-averaged MCs, time-inhomogeneous invariant MCs ● lifted MCs, non-backtracking RWs on regular graphs ● imprecise Markov chains, sets of doubly-stochastic matrices 9

stochastic process examples of linear, local and invariant stochastic processes: ● Markov chains, time-averaged MCs, time-inhomogeneous invariant MCs ● lifted MCs, non-backtracking RWs on regular graphs ● imprecise Markov chains, sets of doubly-stochastic matrices ● quantum walks and quantum Markov chains 9

stochastic process main theorem: any linear, local and invariant stochastic process has a mixing time 10

stochastic process main theorem: any linear, local and invariant stochastic process has a mixing time 10

stochastic process on dumbell graph: main theorem: any linear, local and invariant stochastic process has a mixing time 10

main theorem: any linear, local and invariant stochastic process has a mixing time 11

main theorem: any linear, local and invariant stochastic process has a mixing time proof: 11

main theorem: any linear, local and invariant stochastic process has a mixing time proof: 1) we build a Markov chain simulator 11

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 11

1) Markov chain simulator of linear, local and invariant stochastic process: 12

1) Markov chain simulator of linear, local and invariant stochastic process: 12

1) Markov chain simulator of linear, local and invariant stochastic process: proof: max-flow min-cut argument 12

1) Markov chain simulator of linear, local and invariant stochastic process: proof: max-flow min-cut argument 12

1) Markov chain simulator of linear, local and invariant stochastic process: proof: max-flow min-cut argument 12

1) Markov chain simulator of linear, local and invariant stochastic process: proof: max-flow min-cut argument 12

1) Markov chain simulator of linear, local and invariant stochastic process: proof: max-flow min-cut argument 12

1) Markov chain simulator of linear, local and invariant stochastic process: 13

1) Markov chain simulator of linear, local and invariant stochastic process: 13

1) Markov chain simulator of linear, local and invariant stochastic process: if stochastic process is linear and local, then this transition rule simulates the process: 13

1) Markov chain simulator of linear, local and invariant stochastic process: 14

1) Markov chain simulator of linear, local and invariant stochastic process: ! rule is non-Markovian: depends on initial state and time 14

1) Markov chain simulator of linear, local and invariant stochastic process: ! rule is non-Markovian: depends on initial state and time classic trick: give walker a timer and a memory of initial state 14

1) Markov chain simulator of linear, local and invariant stochastic process: ! rule is non-Markovian: depends on initial state and time classic trick: give walker a timer and a memory of initial state = MC on enlarged state space (“lifted MC”) 14

1) Markov chain simulator of linear, local and invariant stochastic process: ! rule is non-Markovian: depends on initial state and time classic trick: give walker a timer and a memory of initial state = MC on enlarged state space (“lifted MC”) 14

1) Markov chain simulator of linear, local and invariant stochastic process: ! rule is non-Markovian: depends on initial state and time classic trick: give walker a timer and a memory of initial state = MC on enlarged state space (“lifted MC”) 14

1) Markov chain simulator of linear, local and invariant stochastic process: simulates up to time T 15

1) Markov chain simulator of linear, local and invariant stochastic process: simulates up to time T second trick: if process is invariant, then we can “amplify” 15

1) Markov chain simulator of linear, local and invariant stochastic process: simulates up to time T second trick: if process is invariant, then we can “amplify” = restart the simulation every time timer reaches T 15

1) Markov chain simulator of linear, local and invariant stochastic process: simulates up to time T second trick: if process is invariant, then we can “amplify” = restart the simulation every time timer reaches T proposition: the (asymptotic) mixing time of this amplified simulator closely relates to 15 the (asymptotic) mixing time of the original process

2) Markov chain simulator obeys a conductance bound: 16

2) Markov chain simulator obeys a conductance bound: simulator is Markov chain on enlarged state space: 16

2) Markov chain simulator obeys a conductance bound: simulator is Markov chain on enlarged state space: + conductance cannot be increased by lifting 16

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 16

main theorem: any linear, local and invariant stochastic process has a mixing time example 1: dumbbell graph 17

main theorem: any linear, local and invariant stochastic process has a mixing time example 1: dumbbell graph any linear, local and invariant stochastic process on the dumbbell graph has a mixing time 17

main theorem: any linear, local and invariant stochastic process has a mixing time example 2: binary tree 18

main theorem: any linear, local and invariant stochastic process has a mixing time example 2: binary tree 18