Speeding up HMC with better integrators

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Speeding up HMC with

better integrators

Speeding up HMC with

better integrators

A D Kennedy and M A ClarkA D Kennedy and M A ClarkSchool of Physics & SUPA, The University of EdinburghSchool of Physics & SUPA, The University of Edinburgh

Boston UniversityBoston University

A D Kennedy and M A ClarkA D Kennedy and M A ClarkSchool of Physics & SUPA, The University of EdinburghSchool of Physics & SUPA, The University of Edinburgh

Boston UniversityBoston University


A D KennedyA D Kennedy 33

OutlineOutlineSymmetric symplectic integrators in HMCShadow Hamiltonians and Poisson bracketsTuning integrators using Poisson bracketsHessian or Force-Gradient integratorsSymplectic integrators and Poisson brackets on Lie groupsResults for single-link updates

Symmetric symplectic integrators in HMCShadow Hamiltonians and Poisson bracketsTuning integrators using Poisson bracketsHessian or Force-Gradient integratorsSymplectic integrators and Poisson brackets on Lie groupsResults for single-link updates



Symplectic IntegratorsSymplectic Integrators

212,H q p T p S q p S q

We are interested in finding the classical trajectory in phase space of a system described by the Hamiltonian

exp expd dp dqdt dt p dt q

The idea of a symplectic integrator is to write the time evolution operator (Lie derivative) as

ˆexp HH He

q p p q

exp S q T pp q




Define and so

that H T S

S S qp

T T pq

Since the kinetic energy T is a function only of p and the potential energy S is a function only of q, it follows that the action of and may be evaluated trivially

Se

Te

: , ,

: , ,

T

S

e f q p f q T p p

e f q p f q p S q



If A and B belong to any (non-commutative)

algebra then , where constructed from commutators of A and B (i.e.,

is in the Free Lie Algebra generated by A and

B )

A B A Be e e


1 2

1 2

1 2

221

1 2, , 10

1, , , ,

1 2 ! m

m

m

nm

n n k kk km

k k n

Bc c A B c c A B

n m

More precisely, where and

1

ln A Bn

n

e e c

1c A B

Baker-Campbell-Hausdorff (BCH) formula




Explicitly, the first few terms are

1 1 12 12 24

1720

ln , , , , , , , ,

, , , , 4 , , , ,

6 , , , , 4 , , , ,

2 , , , ,

A Be e A B A B A A B B A B B A A B

A A A A B B A A A B

A B A A B B B A A B

A B B A

, , , ,B B B B A B

In order to construct reversible integrators we use symmetric symplectic integrators

2 2 124

15760

ln , , 2 , ,

7 , , , , 28 , , , ,

12 , , , , 32 , , , ,

16 , , , , 8 , , , ,

A B Ae e e A B A A B B A B

A A A A B B A A A B

A B A A B B B A A B

A B B A B B B B A

B

The following identity follows directly from the BCH formula




From the BCH formula we find that the PQP symmetric symplectic integrator is given by

1 12 2

//

0( ) S STU e e e

3 5124

exp , , 2 , ,T S S S T T S T O

ˆ 2S THe e O

In addition to conserving energy to O (² ) such symmetric symplectic integrators are manifestly area preserving and reversible

2 4124exp , , 2 , ,T S S S T T S T O



Shadow HamiltoniansShadow Hamiltonians

This may be obtained by replacing the

commutators in the BCH expansion

of

with the Poisson bracket

,S T

ln S Te e

,S T

For each symplectic integrator there

exists a Hamiltonian H’ which is exactly

conserved



Conserved HamiltonianConserved Hamiltonian

For the PQP integrator we have

124

15760

' , , 2 , ,

7 , , , , 28 , , , ,

12 , , , 32 , , , ,

16 , , , 8 , , , ,

H T S S S T T S T

S S S S T T S S S T

S T S S T T T S S T

S T T S T T T T S T



Tuning HMCTuning HMCFor any (symmetric) symplectic integrator the conserved Hamiltonian is constructed from the same Poisson bracketsA procedure for tuning such integrators is

Measure the Poisson brackets during an HMC runOptimize the integrator (number of pseudofermions, step-sizes, order of integration scheme, etc.) offline using these measured valuesThis can be done because the acceptance rate (and instabilities) are completely determined by δH = H’ - H

For any (symmetric) symplectic integrator the conserved Hamiltonian is constructed from the same Poisson bracketsA procedure for tuning such integrators is

Measure the Poisson brackets during an HMC runOptimize the integrator (number of pseudofermions, step-sizes, order of integration scheme, etc.) offline using these measured valuesThis can be done because the acceptance rate (and instabilities) are completely determined by δH = H’ - H



Simple Example (Omelyan)Simple Example (Omelyan)

1 12 21 2Q QPP Pe e e e e Consider the PQPQP integrator

The conserved Hamiltonian is thus

2

3 56 6 1 1 6' , , , ,

12 24H H S S T T S T O

Measure the “operators” and minimize the cost by adjusting the parameter α

2

3 5, , , ,6 6 1 1 6

12 24SH OS T T S T

, ,12 ,

14 ,

T S T

S S T



Hessian IntegratorsHessian Integrators

We may therefore evaluate the integrator explicitly

We may therefore evaluate the integrator explicitly

3, ,S S Te

An interesting observation is that the Poisson bracket depends only of q

An interesting observation is that the Poisson bracket depends only of q

, ,S S T

The force for this integrator involves second derivatives of the actionUsing this type of step we can construct very efficient Force-Gradient integrators

The force for this integrator involves second derivatives of the actionUsing this type of step we can construct very efficient Force-Gradient integrators



Higher-Order IntegratorsHigher-Order Integrators

We can eliminate all the leading order Poisson brackets in the shadow Hamiltonian leaving errors of O (δτ2)The coefficients of the higher-order Poisson brackets are much smaller than those from the Campostrini integrator

We can eliminate all the leading order Poisson brackets in the shadow Hamiltonian leaving errors of O (δτ2)The coefficients of the higher-order Poisson brackets are much smaller than those from the Campostrini integrator



Beyond Scalar Field TheoryBeyond Scalar Field Theory

We need to extend the formalism beyond a scalar field theoryFermions are easy

† 1 1 †TrFS U U U M M1

1 1

U U

M M

M M

How do we extend all this fancy differential geometry formalism to gauge fields?



Hamiltonian MechanicsHamiltonian Mechanics

,A B A B

A Bp q q p

: 0d dq dp

Flat Manifold General

Symplectic 2-form

Hamiltonian vector field

Equations of motion

Poisson bracket

ˆ H HH

p q q p

H

dH i

,H H

q pp q

ˆdH

dt

ˆ ˆ, ( , )A B A B

Darboux theorem:

All manifolds are locally flat



Maurer-Cartan EquationsMaurer-Cartan Equations

12

i i j kjk

jk

d c

The left invariant forms dual to the

generators of a Lie algebra satisfy the

Maurer-Cartan equations

i



We can invent any Classical Mechanics we want…So we may therefore define a closed symplectic 2-form which globally defines an invariant Poisson bracket by

i i

i

d p

Fundamental 2-formFundamental 2-form

12

i i i i j kjk

i

dp p c

i i i i

i

dp p d



We may now follow the usual procedure to find the equations of motion:

Introduce a Hamiltonian function (0-form) H on the cotangent bundle (phase space) over the group manifold

Hamiltonian Vector FieldHamiltonian Vector Field

Define a vector field such thatH

dH i H

k ki ji ii j i

i jk

H HH e c p e H

p p p



Integral CurvesIntegral CurvesThe classical trajectories are then the integral curves of h:

,t t tQ P t th



Poisson BracketsPoisson Brackets

ˆ k ki ji ii j i

i jk

H HH e c p e H

p p p

Recall our Hamiltonian vector field

For H(q,p) = T(p) + S(q) we have vector fields

ˆ k ki jii j i

i jk

T TT e c p

p p p

ˆi i

i

S e Sp

2

if 2

i k k ji ji i

i jk

pp e c p p T p

p



More Poisson BracketsMore Poisson Brackets

12

ˆ ˆˆ ˆ, , ,i i i i j kjkS T S T dp p c S T

We thus compute the lowest-order Poisson bracket

Retrii

Sp e S PU

U

and the Hamiltonian vector corresponding to it

, ,, ,k k

i ji ii j ii jk

S T S TS T e c p e S T

p p p

k k ji i ji j i j ie S e c p e S p e e S

p



Even More Poisson BracketsEven More Poisson Brackets

, , ( ) ( )

, , ( )

, , , , 2 ( ) ( ) ( ) ( )

, , , , 3 ( ) ( ) ( )

( ) ( ) ( ) ( ) ( )

, , , , 0

, , , , 2

i i

i ji j

i ji j k k i k j k

i ijk j k k

i jk k i j k i i k k j

S S T e S e S

T S T p p e e S

T T S S T p p e e e S e S e e S e e S

S T T S T c p p e S e e S e e S

p p e S e e e S e e S e e S e e S

T T S S T

S T S S T e

( ) ( ) ( )

, , , , ( )

, , , , 0

i j i j

i j ki j k

S e S e e S

T T T S T p p p p e e e e S

S S S S T



IntegratorsIntegrators

2/ 2 / 2

22 2

2 26 3 62 2

3 3 3

36 2 2

Integrator Update Steps Shadow Hamiltonian

, , 2 , ,24

2 , , , ,24

Omelyan , ,72

Omelyan

S T S

T TS

S S ST T

S ST T

PQP e e e T S S S T T S T

QPQ e e e T S S S T T S T

SST e e e e e T S S S T

TST e e e e e

3

26 3 2T, S,T

24

S

T S



Campostrini IntegratorCampostrini Integrator

3 33 3

3 33 3 3 3

3 3 3 3

3 3 3

3

4 2 2 44 2 2 4612

4 2 2 14 2 2 2 4 2 2 2312 12

4 2 2 4 4 2 2 46 12

40 4+40 2+48 , , , , + 20 2+32 , , , ,

60 4+

Integrator Campostrini

Update Steps

Shadow Hamiltonian

ST

ST T

S T

T S

S S S S T T T S S T

e e

e e e

e e

3 3

3 3 3

80 2+104 , , , , 20 4+8 , , , ,

+ 180 4+240 2+312 , , , , 5 2+8 , , , ,4

34560

S T T S T T S S S T

S T S S T T T T S T



3

36 8 3

48 , ,

192

33 8 6

2259 , , , , 4224 , , , ,

768 , , , , 5616 , , , ,

3024 , , , , 896 , , , ,

Integrator Update Steps Shadow Hamiltonian

Force-Gradient 1

T S T

S S S T

T S T

T S

S S S S T T T S S T

S T T S T T S S S T

S T S S T T T T S T

e e e

e

e e e

3

6 2

48 , ,

72

62

46635520

41 , , , , +126 , , , ,

+72 , , , , +84 , , , ,

+36 , , , , +54 , , , ,4

155520

Force-Gradient 2

S T

S S S T

ST

T S

S S S S T T T S S T

S T T S T T S S S T

S T S S T T T T S T

e e

e

e e

Hessian IntegratorsHessian Integrators



One-Link ResultsOne-Link Results

-6

-5

-4

-3

-2

-1

0

1

2

3

4

5

0 1 2 3 4 5 6 7 8 9 10

t

x 1

07

dH

dH Shadow



Scaling BehaviourScaling Behaviour

-11

-10

-9

-8

-7

-6

-5

-4

-3

-2

-1

-1.3 -1.2 -1.1 -1.0 -0.9 -0.8 -0.7 -0.6 -0.5 -0.4 -0.3

-log10 Step Size

log

10 E

rror

PQPOmelyan {S,{S,T}}CampostriniForce-Gradient



ConclusionsConclusionsWe hope that very significant performance improvements can be obtained using Force-Gradient integratorsFor fermions one extra inversion of the Dirac operator is requiredPure gauge force terms and Poisson brackets get quite complicated to programReal-life speed-up factors will be measured really soon…

We hope that very significant performance improvements can be obtained using Force-Gradient integratorsFor fermions one extra inversion of the Dirac operator is requiredPure gauge force terms and Poisson brackets get quite complicated to programReal-life speed-up factors will be measured really soon…

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Speeding up HMC with better integrators

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