Ensemble Quasi-Newton HMC Xiao-Yong Jin and James Osborn Argonne - - PowerPoint PPT Presentation

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Ensemble Quasi-Newton HMC Xiao-Yong Jin and James Osborn Argonne - - PowerPoint PPT Presentation

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Ensemble Quasi-Newton HMC Xiao-Yong Jin and James Osborn Argonne National Laboratory July 23, 2018 The 36th International Symposium on Lattice Field Theory East Lansing, MI 1 Reduce critical slowing down Part of US DOE-funded

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Ensemble Quasi-Newton HMC

Xiao-Yong Jin and James Osborn Argonne National Laboratory

July 23, 2018 The 36th International Symposium on Lattice Field Theory East Lansing, MI

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Reduce critical slowing down

  • Part of US DOE-funded


Exascale Computing Project (ECP)

  • Support research in lattice QCD


to prepare for exascale

  • Reducing critical slowing down,


lead by Norman Christ,
 is part of the USQCD's effort in ECP

  • See Norman's slides


for a list of people actively involved

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Outline

  • Generate ensemble assisted Markov chains
  • Apply Quasi-Newton HMC
  • Test on 2D U(1) pure gauge theory


(work in progress)

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Generate multiple Markov chains

  • Can we exchange information between chains?
  • Use info from other chains
  • Extra info from itself (not explored in this talk)
  • Any advantage?

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⋯ 1 2 3 ⋯ 1 2 3 0′ 1′ 2′ 3′

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ℱ({2,3,0′}) ⋯ 1 2 3 0′ 1′ ⋯ 1 2 3 ⋯ 1 2 3 0′ ⋯ 1 2 3 3′ 2′ 1′ 0′ ⋯ 1 2 3 0′ 1′ 2′ ⋯ 1 2 3 0′ 1′ 2′ 3′ ℱ({1,2,3}) Reverse ℱ({3,0′,1′}) ℱ({0′,1′,2′})

Generate the next state

  • f each Markov chain

with information from

  • ther chains:

ℱ(a set of configs)

Detailed balance: evolve backward from (3′,2′,1′,0′)

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Ensemble assisted Markov chains: in parallel

  • Embedding Markov chains in Markov chains

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⋯ 0, 1 2, 3 0′, 1′ ⋯ 0, 1 2, 3 ⋯ 0, 1 2, 3 2′, 3′ 0′, 1′ ⋯ 0, 1 2, 3 0′, 1′ 2′, 3′ ℱ({2,3}) Reverse ℱ({0′,1′})

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Ensemble assisted Markov chains: multi-state

  • Embedding Markov chains in Markov chains

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⋯ 0, 0′, 1, 1′ 2, 2′, 3, 3′ 2′′, 2′′′, 3′′, 3′′′ 0′′, 0′′′, 1′′, 1′′′ ℱ({2, 2′, 3, 3′}) Reverse ℱ({0′′, 0′′′, 1′′, 1′′′}) ⋯ 0, 0′, 1, 1′ 2, 2′, 3, 3′ 0′′, 0′′′, 1′′, 1′′′ ⋯ 0, 0′, 1, 1′ 2, 2′, 3, 3′ 0′′, 0′′′, 1′′, 1′′′ 2′′, 2′′′, 3′′, 3′′′ ⋯ 0, 0′, 1, 1′ 2, 2′, 3, 3′

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What kind of information from other chains?

  • How do we generate the next state?
  • Modify MD evolution


“Quasi-Newton MCMC” — Zhang & Sutton (2011)
 “Ensemble precondition” — Matthews et al (2016)
 “Quasi-Newton Langevin” — Simsekli et al (2016)
 “Magnetic HMC” — Tripuraneni et al (2016)
 “Wormhole” — Lan et al (2013)

  • Modify Metropolis-Hastings


“Multi-try” — Liu, Liang, and Wong (2000)

  • Other techniques? Machine learning!!!

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Quasi-Newton method for HMC Hamiltonian

  • BFGS approximation of the Hessian:


Update an old approximation to a new one

  • Approximate Hessian from configs of other MC


Repeatedly apply the update according to

  • Use the approximate Hessian for the mass matrix
  • Note: Fourier acceleration≃Local free field Hessian

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G′ = G + yy† y†s − Gss†G s†Gs

s = ln U′U† step y = ∇S(U′) − ∇S(U) yield

H = S(U) + 1 2 p†G−1p G′s = y

Nstream

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Quasi-Newton method

  • Avoids the slow down of the steepest decent in

narrow valleys

  • Caveat in the current study:
  • The approximated Hessian is global
  • We do not use the current location

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Benefits of rank-2 update (BFGS style)

  • Factorizable matrix (Brodlie et al 1973)
  • Initializing random momenta
  • Exactly invertible
  • MD evolution
  • Computing the kinetic energy

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G′ = G + ww† − zz† → G′ = (1 − uv†)G(1 − vu†) G′−1 = (1 − vu† v†u − 1)G−1(1 − uv† v†u − 1 )

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Gauge fixing of 2D U(1) lattice

  • Removes exact zero modes from the real Hessian
  • Frozen degrees of freedom take the same values
  • We choose maximal tree gauge fixing
  • Fix two more non-gauge degree of freedom

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Regulate the approximated Hessian matrix

  • Remove low modes in the approximate global Hessian
  • Add one more term to keep the rank-2 update
  • Works in practice, but not a strict bound
  • Caveat:
  • Mildly violates
  • Still no upper bound

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G′ = G + yy† y†s − (1 − λ s†s s†Gs ) Gss†G s†Gs G′s = y

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Test on 2D U(1) theory (work in progress)

  • Fixed , lattice size 32 × 32
  • Serial version of the ensemble Markov chain
  • Second order Omelyan integrator (did not tune λ)
  • Look at the autocorrelation of the topological

susceptibility,

  • Topological charge,

  • Topological charge is exact integer with periodic

boundary conditions

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Q = 1 2π ∑

x

Arg □x β = 5.8 ⟨Q2/V⟩ Arg : ℂ ↦ (−π, π)

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Acceptance tuning

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  • HMC

8 streams QNHMC 𝜇=0.1 8 streams QNHMC 𝜇=0.01 16 streams QNHMC 𝜇=0.01

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Autocorrelation of topological susceptibility

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  • Trajectory length has no effect on HMC

<2× for Gfix HMC (update half lattice)

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Autocorrelation of topological susceptibility

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  • Cost grows if allow lower eigenmodes

We need more tuning

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Summary & Outlook

  • We devise an algorithm creating multiple Markov chains in parallel


Allow exchange of information while generating the Markov chains

  • We modify HMC to use information from neighboring Markov chains


BFGS approximated Hessian as the mass matrix of the MD Hamiltonian
 Use a custom regulator for the approximated Hessian for stability

  • We still need more tuning and testing (parameters / observables)
  • Ways to improve the algorithm
  • Exploit the ensemble of Markov chains (multi-scale?)
  • Other method for constructing the mass matrix
  • Use other information / observables to augment MD / Metropolis
  • Machine learning!

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