Positive Curvature and Hamiltonian Monte CarloChristof Seiler, Simon Rubinstein-Salzedo, and Susan Holmes

Department of Statistics, Stanford University

GoalRunning time estimates for sampling from high di-mensional probability distributions using Hamilto-nian Monte Carlo (HMC).

Hamiltonian Monte Carlo

Bayesian posterior distribution:π(q | data) = 1

Z L(data | q) π(q)parameter q ∈ Rd, d� observations, andunknown normalizing constant Z.

Expectation of f : Rd→ R

I := Eπ(f ) =∫f (q) π(q | data) dq

Empirical mean:̂I := 1

T

T∑

t=1Qt, Q1, . . . , QT ∼ π(q | data)

simulated using HMC for running time THMC algorithm: Metropolis-Hastings with proposal

step following Hamiltonian trajectoriesError and running time: Bound the error | ̂I − I| as

a function of the running time T

Sectional Curvature

Sectional curvature Sec describes the nature ofgeodesics

x

�

✓1 � "2

2Secx(u, v) + O("3 + "2�)

◆

Secx(u, v) > 0

Positive curvature: geodesics meetFlat and negative curvature: geodesics deviate

Probability Meets Geometry

Coarse Ricci curvature κ describes the deviationfrom independence of subsequent draws fromMarkov chain Monte Carlo•The larger κ the more independent•The larger κ the smaller T for the same error•We calculate the κ of HMC

Concentration inequality (JO10)P(| ̂I − I| ≥ r‖f‖Lip

)≤ 2e−r2/(16V 2(κ,T ))

Jacobi Metric

Jacobi metric gh links Hamiltonians to Riemanniangeometry

Hamiltonian function:h = V (q) + K(p)

Potential energy:V (q) = − log π(q | data) + C

Kinetic energy:K(p) = 1

2 〈p, p〉Hamiltonian equations:

dq

dt= ∂H

∂pand dp

dt= −∂H

∂q

Jacobi-Maupertuis Principle:

q(t)

TqX

p(0) q(0)

Hamiltonian trajectories q(t) with fixed h aregeodesics on Riemannian manifolds X with theJacobi metric:

gh(p, p) = 2 (h− V ) 〈p, p〉.

HMC Meets Geometry

Reference metric 〈·, ·〉Conformal scaling of a metric 2 (h− V )Equivalence table:

HMC Riemannian geometryposterior distribution scaling of reference metriccovariance of proposal reference metricpick from proposal geodesics

Gaussian Example

•HMC sampling from a Gaussian in Rd with zeromean and covariance matrix Λ−1

•Calculate sectional curvature of Jacobi metric• Sec is random because p is random•Calculate coarse Ricci curvature from SecOur Lemma on positive curvature of HMC:

P(d2 Sec ≥ K1) ≥ 1−K2e−K3√d

Our Corollary in high dimensions:

Sec ≥ Tr(Λ)d3 as d→∞

Our Proposal on coarse Ricci curvature of HMC:

κ ≥ Tr(Λ)6d2 + O

1d2

as d→∞

Numerical simulations: Sample Sec’s uniformly ateach step, q(0), . . . , q(T − 1)

Histogram of sectional curvatures (d = 100)

Freq

uenc

y

0.00005 0.00015 0.00025 0.00035

020

000

4000

060

000

expectation E(Sec)sample mean

Histogram of sectional curvatures (d = 10)

Freq

uenc

y

−3 −2 −1 0 1 2 3 4

020

000

4000

060

000

8000

0

expectation E(Sec)sample mean

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(d = 10)

(d = 1000)

0

0-3 3

10-6

Computational Anatomy

ConclusionLinking HMC to geodesics of the Jacobi metric en-ables us to analyze HMC through the rich mathe-matical framework of Riemannian geometry.

References

(JO10) A. Joulin and Y. Ollivier. Curvature, Concentrationand Error Estimates for Markov Chain Monte Carlo. Ann.Probab., 2010.

(Pin75) O. C. Pin. Curvature and Mechanics. Advances inMath., 1975.

Acknowledgments

CS is supported by a postdoctoral fellowship from the Swiss Na-tional Science Foundation and a travel grant from the France-Stanford Center. SRS and SH are supported by NIH grant R01-GM086884.