Sampling constrained probability distributions
using Spherical Augmentation
Shiwei Lan1
1 Department
of Statistics, University of Warwick, Coventry, UK
2 Department
of Statistics, University of California, Irvine, US
WCPM,
Lan, S. and Shahbaba, B. (UW and UCI)
Babak Shahbaba2
June 4, 2015
Spherical Augmentation
06/04/2015
1 / 51
Motivation
q = Inf
q=4
q=2
q=1
q = 0.5
q = 0.1
Lan, S. and Shahbaba, B. (UW and UCI)
Spherical Augmentation
06/04/2015
2 / 51
Background
Sampling from probability distributions with constraints is common:
Lasso, Bridge, probit, copula, and Latent Dirichlet Allocation, etc.
Direct truncation is easily doable but computationally wasteful.
Neal (2010) discusses a modified HMC algorithm for which the
sampler bounces off the boundary once hitting it (Wall HMC).
Brubaker et al (2012, constrained HMC on implicit manifolds),
Pakman and Paninski (2012, exact HMC for truncated Gaussian),
Byrne and Girolami (2013, Geodesic Monte Carlo on embedded
manifolds), etc.
Lan, S. and Shahbaba, B. (UW and UCI)
Spherical Augmentation
06/04/2015
3 / 51
1
Review: from HMC to RHMC
2
Spherical Augmentation
Simple examples: ball and box
General q-norm constraints
Some functional constraints
3
Spherical Monte Carlo
Spherical HMC in the Cartesian coordinate
Spherical HMC in the spherical coordinate
Spherical LMC on the probability simplex
4
Experiments
5
Conclusion and future work
Lan, S. and Shahbaba, B. (UW and UCI)
Spherical Augmentation
06/04/2015
4 / 51
Review: from HMC to RHMC
Hamiltonian Monte Carlo
∂H
∂p
∂H
ṗ = −
∂θ
U (θ )
θ̇ =
θ
Position θ ∈ RD ⇐= variable of interest
Momentum p ∈ RD ⇐= fictitious, usually ∼ N (0, M)
Potential energy U(θ) ⇐= minus log of target density f (·)
Kinetic energy K (p) ⇐= minus log of momentum density
Hamiltonian H(θ, p) = U(θ) + K (p) ⇐= constant.
Lan, S. and Shahbaba, B. (UW and UCI)
Spherical Augmentation
06/04/2015
5 / 51
Review: from HMC to RHMC
Hamiltonian Monte Carlo
Definition 1 (Hamiltonian dynamics)
θ̇ =
∂
∂p H(θ, p)
=
M−1 p
∂
ṗ = − ∂θ
H(θ, p) = −∇θ U(θ)
Leapfrog: numerical integrator
p(t + ε/2) = p(t) − (ε/2)∇θ U(θ(t))
θ(t + ε) = θ(t) + εM−1 p(t + ε/2)
p(t + ε) = p(t + ε/2) − (ε/2)∇θ U(θ(t + ε))
Run for L steps and accept the joint proposal of z := (θ, p) with
α = min{1, exp(−H(z∗ ) + H(z))}
Lan, S. and Shahbaba, B. (UW and UCI)
Spherical Augmentation
06/04/2015
6 / 51
Review: from HMC to RHMC
Riemannian Hamiltonian Monte Carlo
On the manifold {f (·; θ)} with metric G (θ) = −Ex|θ [∇2θ log f (x; θ)]:
H(θ, p) = U(θ) + K (p, θ)
1
1
= − log π(θ) + log det G(θ) + pT G(θ)−1 p
2
2
1 T
≡ φ(θ) + p G(θ)−1 p
2
where p|θ ∼ N (0, G(θ)). Girolami and Calderhead (2011) propose:
Definition 2 (Riemannian Hamiltonian dynamics)
∂
H(θ, p) =
G(θ)−1 p
∂p
∂
1
ṗ = − H(θ, p) = −∇θ φ(θ) + pT G(θ)−1 ∂G(θ)G(θ)−1 p
∂θ
2
θ̇ =
Lan, S. and Shahbaba, B. (UW and UCI)
Spherical Augmentation
06/04/2015
7 / 51
Review: from HMC to RHMC
Lagrangian Monte Carlo
To resolve the implicitness of RHMC, Lan et al. (2012) propose
Definition 3 (Lagrangian Dynamics)
θ̇ = G(θ)−1 p
1
ṗ = −∇θ φ(θ) + pT G(θ)−1 ∂G(θ)G(θ)−1 p
2
w
w
p→v w
 Lagrangian Dynamics
θ̇ = v
v̇ = −vT Γ(θ)v − G(θ)−1 ∇θ φ(θ)
Not Hamiltonian dynamics of (θ, v)!
An explicit integrator can be found more efficient.
Lan, S. and Shahbaba, B. (UW and UCI)
Spherical Augmentation
06/04/2015
8 / 51
Review: from HMC to RHMC
Geometry helps!
RWM
θ2
HMC
RHMC
LMC
2
2
2
2
1.5
1.5
1.5
1.5
1
1
1
1
0.5
0.5
0.5
0.5
θ2
0
-0.5
θ2
0
-0.5
θ2
0
-0.5
0
-0.5
-1
-1
-1
-1
-1.5
-1.5
-1.5
-1.5
-2
-2
-1
0
1
2
-2
-2
θ1
Lan, S. and Shahbaba, B. (UW and UCI)
-1
0
1
2
-2
-2
-1
θ1
Spherical Augmentation
0
θ1
1
2
-2
-2
-1
0
1
2
θ1
06/04/2015
9 / 51
Spherical Augmentation
1
Review: from HMC to RHMC
2
Spherical Augmentation
Simple examples: ball and box
General q-norm constraints
Some functional constraints
3
Spherical Monte Carlo
Spherical HMC in the Cartesian coordinate
Spherical HMC in the spherical coordinate
Spherical LMC on the probability simplex
4
Experiments
5
Conclusion and future work
Lan, S. and Shahbaba, B. (UW and UCI)
Spherical Augmentation
06/04/2015
10 / 51
Spherical Augmentation
Simple examples: ball and box
Change of the domain: from unit ball B0D (1) to sphere S D
θD+1
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θD+1=(1−||θ||2)0.5
Lan, S. and Shahbaba, B. (UW and UCI)
Spherical Augmentation
θ
B
06/04/2015
11 / 51
Spherical Augmentation
Simple examples: ball and box
D
Change of the domain: from rectangle RD
0 to sphere S
x1
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π
Lan, S. and Shahbaba, B. (UW and UCI)
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xD = sinθ1…sinθD−1cosθD ●
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xD+1 = sinθ1…sinθD−1sinθD ●
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x
θ
Spherical Augmentation
06/04/2015
12 / 51
Spherical Augmentation
General q-norm constraints
Mapping q-norm constrained domain to unit ball
0<q<∞
q=∞
||β||q≤ 1
||θ||2≤ 1
||θ||2≤ 1
||β||∞
θ=β
||β||2
(q 2)
θ = sgn(β)|β|
Lan, S. and Shahbaba, B. (UW and UCI)
Spherical Augmentation
||β||∞≤ 1
06/04/2015
13 / 51
Spherical Augmentation
Some functional constraints
Some functional constraints
linear M linear constraints l ≤ Aβ ≤ u, with A an M × D matrix, β a
D-vector and l, u both M-vectors.
Assume M = D and AD×D invertible. A−1 l ≤ β ≤ A−1 u not true.
Sample η := Xβ with l ≤ η ≤ u and transform back to β = A−1 η.
adratic Quadratic constraints l ≤ β T Aβ + bT β ≤ u with l, u > 0 scalars.
√
Assume A = QΣQT > 0. Use β 7→ β ∗ = ΣQT (β + 12 A−1 b):
1
1
} : l ∗ ≤ kβ ∗ k22 = (β ∗ )T β ∗ ≤ u ∗ , l ∗ = l + bT A−1 b, u ∗ = u + bT A−1 b
4
4
It can further be mapped to unit ball:
√
√
T}→B : B0D ( u ∗ )\B0D ( l ∗ ) −→ B0D (1), β ∗ 7→ θ =
Lan, S. and Shahbaba, B. (UW and UCI)
Spherical Augmentation
√
β ∗ kβ ∗ k2 − l ∗
√
√
kβ ∗ k2 u ∗ − l ∗
06/04/2015
14 / 51
Spherical Augmentation
Some functional constraints
An example of linear constraints
0.5
β2
1.0
1.5
2.0
β2
1.0
0.5
β2
1.0
0.5
8
0.0
14
Estimate by s−SphHMC
0
β1
1
2
0
β1
1
2
−2
−1
0
β1
1
2
−2
Estimate by c−SphHMC
−1
0
β1
1
2
Estimate by s−SphHMC
−2
−1
0
β1
1
2
−2
−1
0
β1
1
2
1.5
0.0
0.5
β2
1.0
1.5
0.0
0.0
0.5
β2
1.0
0.5
0.7
0.6
0.0
0.
3
0.2
β2
1.0
0.5
0.1
β2
1.0
1.5
0.7
0.2
0.4
0.3
1.5
0.
5
2.0
Estimate by exact HMC
2.0
Truth
−1
0.0
0.0
−2
2.0
−1
2.0
−2
0.0
0.0
0.
1
0.
Estimate by c−SphHMC
1.5
2.0
1.5
0
14 .1
0.
8
0.12
0.12 0.08
0.1
2.0
1.5
β2
1.0
0.16
0.5
and 0 ≤ β1 + β2 ≤ 2
Estimate by exact HMC
2.0
0 ≤ −0.5β1 + β2 ≤ 2
Truth
−2
−1
0
β1
1
2
−2
−1
0
β1
1
2
" #
"
#
0
1 0.5
upper row: N (µ, Σ) with µ =
,Σ=
1
0.5 1
lower row: f (β) ∝
Lan, S. and Shahbaba, B. (UW and UCI)
sin2 Q(β)
Q(β) ,
Q(β) = 12 (β − µ)T Σ −1 (β − µ)
Spherical Augmentation
06/04/2015
15 / 51
Spherical Monte Carlo
1
Review: from HMC to RHMC
2
Spherical Augmentation
Simple examples: ball and box
General q-norm constraints
Some functional constraints
3
Spherical Monte Carlo
Spherical HMC in the Cartesian coordinate
Spherical HMC in the spherical coordinate
Spherical LMC on the probability simplex
4
Experiments
5
Conclusion and future work
Lan, S. and Shahbaba, B. (UW and UCI)
Spherical Augmentation
06/04/2015
16 / 51
Spherical Monte Carlo
Change of variables
Denote the original parameter as β and the constrained domain as D.
We use θ to denote the coordinate of sphere S D . Change variables
Change of variables
dβ D dθ S
f (β)dβ D =
f (θ) dθ S D
S
Z
Z
(3.1)
The energy functions will be changed to
dβ D dβ D φ(θ) = − log f (θ) − log = U(β(θ)) − log dθ S dθ S 1
H(θ, v) = φ(θ) + hv, viGSc (θ)
2
D
The Jacobian determinant dβ
dθ S can be used as weight afterwards.
We then consider partial Hamiltonian H ∗ (θ, v) = U(θ) + 12 hv, viGSc (θ)
Lan, S. and Shahbaba, B. (UW and UCI)
Spherical Augmentation
06/04/2015
17 / 51
Spherical Monte Carlo
Spherical HMC in the Cartesian coordinate
θD+1
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θD+1=(1−||θ||2)0.5
B
Spherical HMC
in the Cartesian coordinate
Lan, S. and Shahbaba, B. (UW and UCI)
Spherical Augmentation
06/04/2015
18 / 51
θ
Spherical Monte Carlo
Spherical HMC in the Cartesian coordinate
θD+1
Spherical HMC for ball type constraints
B0D (1) :=
∈ RD :
q{θ
PD 2
kθk2 =
i=1 θ i ≤ 1}
Lan, S. and Shahbaba, B. (UW and UCI)
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θD+1=(1−||θ||2)0.5
B
θ 7→θ̃ =(θ ,θD+1 )
S D := {θ̃ ∈ RD+1 :
−−−−−−−−
√−−−−−−→
θD+1 =± 1−kθk22
kθ̃k2 = 1}
Spherical Augmentation
06/04/2015
19 / 51
θ
Spherical Monte Carlo
Spherical HMC in the Cartesian coordinate
θD+1
Spherical HMC for ball type constraints
B0D (1) :=
∈ RD :
q{θ
PD 2
kθk2 =
i=1 θ i ≤ 1}
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θD+1=(1−||θ||2)0.5
B
θ 7→θ̃ =(θ ,θD+1 )
S D := {θ̃ ∈ RD+1 :
−−−−−−−−
√−−−−−−→
θD+1 =± 1−kθk22
kθ̃k2 = 1}
Change of variables
Z
dθ B f (θ)dθ B =
f (θ̃)
dθ Sc =
f (θ̃)|θD+1 |dθ Sc
D
D
dθ Sc B0D (1)
S+
S+
Z
Z
where f (θ̃) ≡ f (θ).
Lan, S. and Shahbaba, B. (UW and UCI)
Spherical Augmentation
06/04/2015
19 / 51
θ
Spherical Monte Carlo
Spherical HMC in the Cartesian coordinate
θD+1
Spherical HMC for ball type constraints
B0D (1) :=
∈ RD :
q{θ
PD 2
kθk2 =
i=1 θ i ≤ 1}
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θD+1=(1−||θ||2)0.5
B
θ 7→θ̃ =(θ ,θD+1 )
S D := {θ̃ ∈ RD+1 :
−−−−−−−−
√−−−−−−→
θD+1 =± 1−kθk22
kθ̃k2 = 1}
Change of variables
Z
dθ B f (θ)dθ B =
f (θ̃)
dθ Sc =
f (θ̃)|θD+1 |dθ Sc
D
D
dθ Sc B0D (1)
S+
S+
Z
Z
where f (θ̃) ≡ f (θ).
What We Want:
θ ∼ f (θ)dθ B
Lan, S. and Shahbaba, B. (UW and UCI)
drop θD+1
←−−−−−−−−−−−−−− What We Sample:
weigh it by |θD+1 | θ̃ ∼ f (θ̃)dθSc
Spherical Augmentation
06/04/2015
19 / 51
θ
Spherical Monte Carlo
Spherical HMC in the Cartesian coordinate
θD+1
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θD+1=(1−||θ||2)0.5
B
Here, the proper metric on S D is called canonical spherical metric:
Definition 4 (canonical spherical metric)
GSc (θ) = ID +
θθ T
θθ T
=
I
+
D
2
θD+1
1 − kθk22
(3.2)
T
For any vector ṽ = (v, vD+1 ) ∈ Tθ̃ S D := {ṽ ∈ RD+1 : θ̃ ṽ = 0},
GSc (θ) can be viewed as a way to express the length of ṽ in v:
vT GSc (θ)v = kvk22 +
vT θθ T v
(−θD+1 vD+1 )2
2
+
=
kvk
2
2
2
θD+1
θD+1
2
= kvk22 + vD+1
= kṽk22
Lan, S. and Shahbaba, B. (UW and UCI)
Spherical Augmentation
06/04/2015
20 / 51
θ
Spherical Monte Carlo
Spherical HMC in the Cartesian coordinate
θD+1
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θ =(θ,θD+1)
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Hamiltonian (Lagrangian) dynamics on sphere
On B0D (1)
θD+1=(1−||θ||2)0.5
B
On S D
H(θ, v) = U(θ) + K (v)
1
= − log f (θ) + vT Iv
2
−−−−−→
v ∼ N (0, ID )
−−−−−→
Lan, S. and Shahbaba, B. (UW and UCI)
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θ7→θ̃
v7→ṽ
Spherical Augmentation
H ∗ (θ̃, ṽ) = U(θ̃) + K (ṽ)
1
= − log f (θ) + vT GSc (θ)v
2
T
ṽ ∼ (ID+1 − θ̃ θ̃ )N (0, ID+1 )
06/04/2015
21 / 51
θ
Spherical Monte Carlo
Spherical HMC in the Cartesian coordinate
θD+1
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θ =(θ,θD+1)
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Hamiltonian (Lagrangian) dynamics on sphere
On B0D (1)
θD+1=(1−||θ||2)0.5
B
On S D
H(θ, v) = U(θ) + K (v)
1
= − log f (θ) + vT Iv
2
−−−−−→
v ∼ N (0, ID )
−−−−−→
θ7→θ̃
v7→ṽ
H ∗ (θ̃, ṽ) = U(θ̃) + K (ṽ)
1
= − log f (θ) + vT GSc (θ)v
2
T
ṽ ∼ (ID+1 − θ̃ θ̃ )N (0, ID+1 )
θ̇ = v
θ̇ = v
v̇ = −∇θ U(θ)
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−−−−−→
kθk2 ≤ 1
Lan, S. and Shahbaba, B. (UW and UCI)
v̇ = −vT ΓSc (θ)v − GSc (θ)−1 ∇θ U(θ)
q
θD+1 = 1 − kθk22 , vD+1 = −θ T v/θD+1
Spherical Augmentation
06/04/2015
21 / 51
θ
Spherical Monte Carlo
Spherical HMC in the Cartesian coordinate
θD+1
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θ =(θ,θD+1)
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Split Lagrangian dynamics on sphere
θD+1=(1−||θ||2)0.5
θ̇ = v
1
v̇ = − GSc (θ)−1 ∇θ U(θ)
2
Lan, S. and Shahbaba, B. (UW and UCI)
B
(3.3)
v̇ = −vT ΓSc (θ)v − GSc (θ)−1 ∇θ U(θ)
θ̇ = 0
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θ̇ = v
←−−−→
Spherical Augmentation
v̇ = −vT ΓSc (θ)v
06/04/2015
22 / 51
θ
Spherical Monte Carlo
Spherical HMC in the Cartesian coordinate
θD+1
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θ =(θ,θD+1)
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Split Lagrangian dynamics on sphere
θD+1=(1−||θ||2)0.5
θ̇ = v
1
v̇ = − GSc (θ)−1 ∇θ U(θ)
2
↓
θ̃(t) = θ̃(0)
B
(3.3)
v̇ = −vT ΓSc (θ)v − GSc (θ)−1 ∇θ U(θ)
θ̇ = 0
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θ̇ = v
←−−−→
v̇ = −vT ΓSc (θ)v
↓
θ̃(t) =θ̃(0) cos(kṽ(0)k2 t)
+
ṽ(0)
sin(kṽ(0)k2 t)
kṽ(0)k2
ṽ(t) = ṽ(0)
"" #
#
ṽ(t) = − θ̃(0)kṽ(0)k2 sin(kṽ(0)k2 t)
ID
t
T
−
−
θ̃(0)θ(0)
∇
U(θ(0))
θ
2 0T
+ ṽ(0) cos(kṽ(0)k2 t)
Lan, S. and Shahbaba, B. (UW and UCI)
Spherical Augmentation
06/04/2015
22 / 51
θ
Spherical Monte Carlo
Spherical HMC in the Cartesian coordinate
θD+1
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θD+1=(1−||θ||2)0.5
B
Denote z := (θ, v), z(tn ) as the true solution to (3.3) at time tn and z(n)
the numerical solution at n-th step. We have the following bound of the
error en = kz(tn ) − z(n) k:
Proposition 1
Assume f(θ, v) := vT ΓS (θ)v + GS (θ)−1 ∇θ U(θ) is smooth. Then
en+1 ≤ (1 + M1 ε + M2 ε2 )en + O(ε3 )
where Mk = ck supt∈[0,T ] k∇k f(θ(t), v(t))k, k = 1, 2 for some constants
ck > 0. ε = tn+1 − tn is the discretization step size. Further accumulating
the local errors for L = T /ε steps yields the global error
eL+1 ≤ (e M1 T + T )ε2
Lan, S. and Shahbaba, B. (UW and UCI)
Spherical Augmentation
06/04/2015
23 / 51
θ
Spherical Monte Carlo
Spherical HMC in the Cartesian coordinate
θD+1
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Algorithm 1 Spherical HMC in the Cartesian coordinate (c − SphHMC )
θD+1=(1−||θ||2)0.5
B
(1)
Initialize θ̃ at current θ̃ after transformation TD→S
Sample a new velocity value ṽ(1) ∼ N (0, ID+1 )
Set ṽ(1) ← ṽ(1) − θ̃
(1)
(θ̃
(1) T (1)
) ṽ
(1)
Calculate H(θ̃ , ṽ(1) ) = U(θ (1) ) + K (ṽ(1) )
for ` = 1 to L do
" #
!
1
ID
(`) (`) T
ṽ(`+ 2 ) = ṽ(`) − ε2
−
θ̃
(θ
)
∇θ U(θ (`) )
0T
θ̃
(`+1)
1
= θ̃
(`)
ṽ(`+ 2 ) ← −θ̃
ṽ(`+1)
1
cos(kṽ(`+ 2 ) kε) +
(`)
1
ṽ
(`+ 1 )
2
(`+ 1 )
2 k
kṽ
(`+ 12 )
1
sin(kṽ(`+ 2 ) kε)
1
1
)
)
(`+ 2 )
2 cos(kṽ
kṽ(`+ 2 "
k sin(kṽ
kε) + ṽ(`+!
kε)
#
1
I
T
(`+1) (`+1)
D
(`+ 2 )
ε
(`+1)
= ṽ
− 2
− θ̃
(θ
)
∇θ U(θ
)
0T
end for
(L+1) (L+1)
Calculate H(θ̃
, ṽ
) = U(θ (L+1) ) + K (ṽ(L+1) )
(L+1)
Calculate the acceptance probability α = min{1, exp[−H(θ̃
, ṽ(L+1) ) + H(θ̃
0
Accept or reject the proposal according to α for the next state θ̃
0
Calculate TS→D (θ̃ ) and the corresponding weight |dTS→D |
Lan, S. and Shahbaba, B. (UW and UCI)
Spherical Augmentation
(1)
, ṽ(1) )]}
06/04/2015
24 / 51
θ
Spherical Monte Carlo
Spherical HMC in the spherical coordinate
x1
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xD+1 = sinθ1…sinθD−1sinθD ●
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π
θ
Spherical HMC
in the spherical coordinate
Lan, S. and Shahbaba, B. (UW and UCI)
Spherical Augmentation
06/04/2015
25 / 51
x
Spherical Monte Carlo
Spherical HMC in the spherical coordinate
x1
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xD+1 = sinθ1…sinθD−1sinθD ●
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Spherical HMC for box type constraints
π
RD
0 :=
[0, π]D−1 × [0, 2π)
Lan, S. and Shahbaba, B. (UW and UCI)
θ7→x
−−−−−−−−
→
Q−−−−−
xd =cos θd
d−1
i=1
sin θi
Spherical Augmentation
θ
S D := {x ∈ RD+1 :
kxk2 = 1}
06/04/2015
26 / 51
x
Spherical Monte Carlo
Spherical HMC in the spherical coordinate
x1
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xD+1 = sinθ1…sinθD−1sinθD ●
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Spherical HMC for box type constraints
π
θ7→x
RD
0 :=
[0, π]D−1 × [0, 2π)
−−−−−−−−
→
Q−−−−−
xd =cos θd
d−1
i=1
sin θi
θ
S D := {x ∈ RD+1 :
kxk2 = 1}
Change of measure
Z
RD
0
Z
D−1
Y
dθ R0 dθ Sr =
f
(θ)
sind−D θd dθ Sr
=
f (θ)
dθ
D
D
Sr
S
S
Z
f (θ)dθ R0
d=1
where f (θ) = f (θ(x)) on S D .
Lan, S. and Shahbaba, B. (UW and UCI)
Spherical Augmentation
06/04/2015
26 / 51
x
Spherical Monte Carlo
Spherical HMC in the spherical coordinate
x1
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xD+1 = sinθ1…sinθD−1sinθD ●
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Spherical HMC for box type constraints
π
θ7→x
RD
0 :=
[0, π]D−1 × [0, 2π)
−−−−−−−−
→
Q−−−−−
xd =cos θd
d−1
i=1
sin θi
θ
S D := {x ∈ RD+1 :
kxk2 = 1}
Change of measure
Z
RD
0
Z
D−1
Y
dθ R0 dθ Sr =
f
(θ)
sind−D θd dθ Sr
=
f (θ)
dθ
D
D
Sr
S
S
Z
f (θ)dθ R0
d=1
where f (θ) = f (θ(x)) on S D .
What We Want:
θ ∼ f (θ)dθ R0
Lan, S. and Shahbaba, B. (UW and UCI)
weigh sample θ
D−1
d−D
by
θd
d=1 sin
←−−−−Q
−−−−−−−−−−
Spherical Augmentation
What We Sample:
θ ∼ f (θ)dθ Sr
06/04/2015
26 / 51
x
Spherical Monte Carlo
Spherical HMC in the spherical coordinate
x1
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xD+1 = sinθ1…sinθD−1sinθD ●
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Round spherical metric
π
θ
Here, the natural metric on S D is called round spherical metric:
Definition 5 (round spherical metric)
"
GSr (θ) = diag 1, sin2 θ1 , · · · ,
D−1
Y
#
sin2 θd
(3.4)
d=1
For any vector v ∈ Tθ RD
0 , we have
vT GSr (θ)v ≤ kvk22 ≤ kṽk22 = vT GSc (θ)v
Lan, S. and Shahbaba, B. (UW and UCI)
Spherical Augmentation
06/04/2015
27 / 51
x
Spherical Monte Carlo
Spherical HMC in the spherical coordinate
x1
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xD+1 = sinθ1…sinθD−1sinθD ●
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Hamiltonian (Lagrangian) dynamics on sphere
π
On RD
0
H(θ, v) = U(θ) + K (v)
1
= − log f (θ) + vT Iv
2
v ∼ N (0, ID )
Lan, S. and Shahbaba, B. (UW and UCI)
θ
On S D
θ7→x
−−−−−→
v7→ẋ
−−−−−→
Spherical Augmentation
H ∗ (x, ẋ) = U(x) + K (ẋ)
1
= − log f (θ) + vT GSr (θ)v
2
1
v(x, ẋ) ∼ GSr (θ)− 2 N (0, ID )
06/04/2015
28 / 51
x
Spherical Monte Carlo
Spherical HMC in the spherical coordinate
x1
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xD+1 = sinθ1…sinθD−1sinθD ●
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Hamiltonian (Lagrangian) dynamics on sphere
π
On RD
0
On S D
H(θ, v) = U(θ) + K (v)
1
= − log f (θ) + vT Iv
2
θ7→x
−−−−−→
v7→ẋ
v ∼ N (0, ID )
−−−−−→
H ∗ (x, ẋ) = U(x) + K (ẋ)
1
= − log f (θ) + vT GSr (θ)v
2
1
v(x, ẋ) ∼ GSr (θ)− 2 N (0, ID )
θ̇ = v
θ̇ = v
v̇ = −∇θ U(θ)
θ
−−−−−→
l≤θ≤u
Lan, S. and Shahbaba, B. (UW and UCI)
v̇ = −vT ΓSr (θ)v − GSr (θ)−1 ∇θ U(θ)
θ = θ(x),
Spherical Augmentation
v = v(x, ẋ)
06/04/2015
28 / 51
x
Spherical Monte Carlo
Spherical HMC in the spherical coordinate
x1
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● xD = sinθ1
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xD+1 = sinθ1…sinθD−1sinθD ●
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Split Lagrangian dynamics on sphere
π
θ
θ̇ = v
v̇ = −vT ΓSr (θ)v − GSr (θ)−1 ∇θ U(θ)
θ̇ = 0
1
v̇ = − GSr (θ)−1 ∇θ U(θ)
2
Lan, S. and Shahbaba, B. (UW and UCI)
θ̇ = v
←−−−→
Spherical Augmentation
v̇ = −vT ΓSr (θ)v
06/04/2015
29 / 51
x
Spherical Monte Carlo
Spherical HMC in the spherical coordinate
x1
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● xD = sinθ1
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xD+1 = sinθ1…sinθD−1sinθD ●
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Split Lagrangian dynamics on sphere
π
θ
θ̇ = v
v̇ = −vT ΓSr (θ)v − GSr (θ)−1 ∇θ U(θ)
θ̇ = 0
θ̇ = v
1
v̇ = − GSr (θ)−1 ∇θ U(θ)
2
↓
←−−−→
v̇ = −vT ΓSr (θ)v
↓
(θ(0), v(0)) −→ (x(0), ẋ(0))
θ(t) = θ(0)
t
v(t) = v(0) − ·
2
"
#
D−1
Y
diag 1, · · · ,
sin−2 θd ∇θ U(θ(0))
d=1
Lan, S. and Shahbaba, B. (UW and UCI)
↓
(x(t), ẋ(t)) = gr (x(0), ẋ(0))
↓
(θ(0), v(0)) ←− (x(0), ẋ(0))
Spherical Augmentation
06/04/2015
29 / 51
x
Spherical Monte Carlo
Spherical HMC in the spherical coordinate
x1
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xD+1 = sinθ1…sinθD−1sinθD ●
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π
θ
Algorithm 2 Spherical HMC in the spherical coordinate (s-SphHMC)
Initialize θ (1) at current θ after transformation TD→S
Sample a new velocity value v(1) ∼ N (0, ID )
(1)
(1) Qd−1
−1 (1)
(θi ), d = 1, · · · , D
Set vd ← vd
i=1 sin
(1) (1)
Calculate H(θ , v ) = U(θ (1) ) + K (v(1) )
for ` = 11 to L do
Q
d
(`+ )
(`)
∂
−2 (`)
vd 2 = vd − ε2 ∂θ
(θi ), d = 1, · · · , D
U(θ (`) ) d−1
i=1 sin
1
d
1
(θ (`+1) , v(`+ 2 ) ) ← T̃S→R0 ◦ gε ◦ T̃R0 →S (θ (`) , v(`+ 2 ) )
Q
d
(`+ 1 )
(`+1)
∂
−2 (`+1)
U(θ (`+1) ) d−1
(θi
), d = 1, · · · , D
vd
= vd 2 − ε2 ∂θ
i=1 sin
d
end for
Calculate H(θ (L+1) , v(L+1) ) = U(θ (L+1) ) + K (v(L+1) )
Calculate the acceptance probability α = min{1, exp[−H(θ (L+1) , v(L+1) ) + H(θ (1) , v(1) )]}
Accept or reject the proposal according to α for the next state θ 0
Calculate TS→D (θ 0 ) and the corresponding weight |dTS→D |
Lan, S. and Shahbaba, B. (UW and UCI)
Spherical Augmentation
06/04/2015
30 / 51
x
Spherical Monte Carlo
Spherical LMC on the probability simplex
Spherical LMC
on the probability simplex
Lan, S. and Shahbaba, B. (UW and UCI)
Spherical Augmentation
06/04/2015
31 / 51
Spherical Monte Carlo
Spherical LMC on the probability simplex
Spherical LMC on the probability simplex
A class of models having probability distributions defined on simplex
K
D
∆ := {π ∈ R | πk ≥ 0,
K
X
πd = 1}
k=1
Latent Dirichlet Allocation (LDA) (Blei et al., 2003) is a hierarchical
Bayesian model frequently used to model document topics.
1-norm constraint: identify the first (all positive) orthant with others.
√
T∆→√∆ : π 7→ θ = π maps the simplex to the sphere
√
K
∆ := {θ ∈ S K −1 |θk ≥ 0, ∀ k = 1, · · · , K } ⊂ S K −1
Lan, S. and Shahbaba, B. (UW and UCI)
Spherical Augmentation
06/04/2015
32 / 51
Spherical Monte Carlo
Spherical LMC on the probability simplex
Spherical LMC on the probability simplex
Prototype example: Dirichlet-Multinomial distribution
k = 1, · · · , K
p(xi = k|π) = πk ,
p(π) ∝
p(π|x) ∝
K
Y
k=1
K
Y
πkαk −1
πknk +αk −1 ,
k=1
Fisher metric on
√
nk =
N
X
I (xi = k), n =
i=1
K
X
nk
k=1
∆ coincides GSc (θ) on S K −1 up to a constant.
G∆ (π −K ) = n[diag(1/π −K ) + 11T /πK ]
G√∆ (θ) =
Lan, S. and Shahbaba, B. (UW and UCI)
dπ T
dπ −K
−K
G∆ (π −K ) T = 4nGSc (θ)
dθ −K
dθ −K
Spherical Augmentation
06/04/2015
33 / 51
Spherical Monte Carlo
Spherical LMC on the probability simplex
Spherical LMC on the probability simplex
Use G√∆ (θ) instead of GSc (θ) in c-SphHMC.
D
Include the volume adjustment term, dβ
dθ S in the Hamiltonian
1
H(θ, v) = φ(θ) + hv, viG√∆ (θ) ,
2
dβ D φ(θ) = U(θ) − log dθ S No afterward re-weight: online learning
above modifications
c-SphHMC −−−−−−−−−−−→ Spherical Lagrangian Monte Carlo.
SphLMC: stems from the Fisher metric on the simplex.
Lan, S. and Shahbaba, B. (UW and UCI)
Spherical Augmentation
06/04/2015
34 / 51
Spherical Monte Carlo
Spherical LMC on the probability simplex
Spherical LMC on the probability simplex
0.6
0.4
0.2
0
10
20
0.4
0.2
0
-0.2
0
-0.3
0.8
0.6
0.4
0.2
0
-0.2
0
10
20
0.8
0.6
0.4
0.2
0
-0.2
0
-0.5
-0.6
0
1
2
3
4
5
6
Category
Lan, S. and Shahbaba, B. (UW and UCI)
7
8
9
10
0.8
0.6
0.4
0.2
0
-0.2
0
10
Lag
Spherical Augmentation
10
20
-0.2
0
20
0.8
0.6
0.4
0.2
0
-0.2
0
10
Lag
10
20
0.6
0.4
0.2
0
-0.2
0
0.8
0.6
0.4
0.2
0
-0.2
0
10
20
20
0.6
0.4
0.2
0
0
10
Lag
20
0.8
0.6
0.4
0.2
0
-0.2
0
10
20
Lag
0.8
-0.2
10
Lag
Lag
Autocorrelation of p 9
RWMt
WallHMC
RLD
SphLMC
Truth
0
Lag
Autocorrelation of p 9
Lag
-0.4
0.2
0.8
Lag
Autocorrelation of p 2
-0.2
20
0.4
Lag
Autocorrelation of p 2
Autocorrelation of p 2
Lag
-0.1
10
0.6
Autocorrelation of p 2
0
0.6
SphLMC
0.8
Autocorrelation of p 9
-0.2
RLD
0.8
Autocorrelation of p 1
0.8
Autocorrelation of p 1
Autocorrelation of p 1
0
Autocorrelation of p 9
Estimated Probability adjusted by Truth
0.1
WallHMC
Autocorrelation of p 1
RWMt
0.2
20
0.8
0.6
0.4
0.2
0
-0.2
0
10
20
Lag
06/04/2015
35 / 51
Experiments
1
Review: from HMC to RHMC
2
Spherical Augmentation
Simple examples: ball and box
General q-norm constraints
Some functional constraints
3
Spherical Monte Carlo
Spherical HMC in the Cartesian coordinate
Spherical HMC in the spherical coordinate
Spherical LMC on the probability simplex
4
Experiments
5
Conclusion and future work
Lan, S. and Shahbaba, B. (UW and UCI)
Spherical Augmentation
06/04/2015
36 / 51
Experiments
Experiments
Definition 6 (Effective Sample Size)
For N samples, effective sample size is calculated as follows:
K
ESS = N[1 + 2Σk=1
ρ(k)]−1
where ρ(k) is the autocorrelation function with lag k, and K 1.
Performance measured by time-normalized ESS.
Interpreted as number of nearly independent samples.
Use the minimum ESS normalized by CPU time: min(ESS)/s.
Compare RWMt, Wall HMC, exact HMC, c-SphHMC, s-SphHMC,
RLD and SphLMC.
Lan, S. and Shahbaba, B. (UW and UCI)
Spherical Augmentation
06/04/2015
37 / 51
Experiments
Truncated Multivariate Gaussian
"
β1
∼N
β2
1 .5
0,
.5 1
#!
−2
−1
0
β1
1
2
2
1
−2
−1
β2
0
1
β2
0
−2
−1
0.02
1
−2
−1
β2
0
6
0.06
0.1
0.12
0.1
−2
−1
0
β1
1
2
1
Lan, S. and Shahbaba, B. (UW and UCI)
2
0
β1
1
2
2
−2
−1
β2
0
1
2
1
−1
−2
0
β1
−1
Estimate by s−SphHMC
β2
0
1
β2
0
−1
−2
−1
−2
Estimate by c−SphHMC
2
Estimate by exact HMC
−2
0 ≤ β2 ≤ 1
Estimate by Wall HMC
2
Estimate by RWMt
2
Truth
0 ≤ β1 ≤ 5,
,
−2
−1
0
β1
1
2
Spherical Augmentation
−2
−1
0
β1
1
2
06/04/2015
38 / 51
Experiments
Truncated Multivariate Gaussian
To evaluate efficiency, we increase the dimensionality for D = 10, 100
β ∼ N (0, Σ), Σij = 1/(1+|i−j|);
0 ≤ β1 ≤ 5, 0 ≤ βi ≤ 0.5, i 6= 1.
RWM: > 95% of times proposals rejected due to constraint violation.
Wall HMC: average wall hits 3.81 (L=2, D=10), 6.19 (L=5, D=100).
Dim
D=10
D=100
Method
RWMt
Wall HMC
exact HMC
c-SphHMC
s-SphHMC
RWMt
Wall HMC
exact HMC
c-SphHMC
s-SphHMC
AP
0.62
0.83
1.00
0.82
0.79
0.81
0.74
1.00
0.73
0.87
Lan, S. and Shahbaba, B. (UW and UCI)
s/iter
5.72E-05
1.19E-04
7.60E-05
2.53E-04
2.02E-04
5.45E-04
2.23E-03
4.65E-02
3.45E-03
2.30E-03
ESS(min,med,max)
(48,691,736)
(31904,86275,87311)
(1e+05,1e+05,1e+05)
(62658,85570,86295)
(76088,1e+05,1e+05)
(1,4,54)
(17777,52909,55713)
(97963,1e+05,1e+05)
(55667,68585,72850)
(74476,99670,1e+05)
Spherical Augmentation
Min(ESS)/s
7.58
2441.72
11960.29
2253.32
3429.56
0.01
72.45
19.16
146.75
294.31
06/04/2015
spdup
1.00
322.33
1578.87
297.46
452.73
1.00
5130.21
1356.64
10390.94
20839.43
39 / 51
Experiments
Bayesian Lasso: regularized regression
20
20
0
0.2
0.4
0.6
0.8
Shrinkage Factor
1
4
4
7 1 10
6
10
0
2
1 10
7
Coefficients
5
−20
−30
−20
−30
−30
−20
5
2
−10
8
10
0
Coefficients
1 10
2
−10
0
−10
Coefficients
10
4
20
6
3
3
9
30
Bayesian Lasso
Spherical HMC
30
Bayesian Lasso
Wall HMC
30
Bayesian Lasso
Gibbs Sampler
0
0.2
0.4
0.6
0.8
1
0
Shrinkage Factor
0.2
0.4
0.6
0.8
1
Shrinkage Factor
Obtain the coefficients β by minimizing the residual sum of squares
(RSS) subject to a constraint on the magnitude of β
X
min RSS(β),
RSS(β) :=
(yi − β0 − xiT β)2
kβk1 ≤t
i
Park and Casella (2008) use a Laplace prior: P(β) ∝ exp(−λ|β|)
Lan, S. and Shahbaba, B. (UW and UCI)
Spherical Augmentation
06/04/2015
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Experiments
1500
Bayesian Lasso
0
500
Min(ESS)/s
1000
Gibbs Sampler
Wall HMC
Spherical HMC
0.1
0.2
Lan, S. and Shahbaba, B. (UW and UCI)
0.3
0.4
0.5
0.6
Shrinking Factor
Spherical Augmentation
0.7
0.8
0.9
1
06/04/2015
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Experiments
Bayesian Bridge: regularized regression
Beysian Bridge
q=1.2
Beysian Bridge
q=0.8
20
20
0
0.2
0.4
0.6
0.8
Shrinkage Factor
1
4
6
10
5 1
0
7
1 10 8
0
0.2
0.4
0.6
0.8
−30
5
−20
2
−10
Coefficients
10
6
7 1 10
Coefficients
0
−10
2
−30
5
−20
−30
−20
0
−10
Coefficients
10
4
20
9
3
3
30
30
9
30
9
Beysian Bridge
Lasso (q=1)
1
0
Shrinkage Factor
0.2
0.4
0.6
0.8
1
Shrinkage Factor
Obtain the coefficients β by minimizing the residual sum of squares
(RSS) subject to a constraint on the magnitude of β
X
min RSS(β),
RSS(β) :=
(yi − β0 − xiT β)2
kβkq ≤t
i
Polson et al (2013) have Bayesian Bridge with complicated priors
Lan, S. and Shahbaba, B. (UW and UCI)
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06/04/2015
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Experiments
−1.5 −0.5 0.5
1.5
Reconstruction of quantized stationary Gaussian process
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40
Truth
RWMt
Wall HMC
exact HMC
c−SphHMC
s−SphHMC
−2 −1 0
1
2
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0
10
20
Given N values of a function
{f (xi )}N
i=1 ,
30
40
taken values in a set {qk }K
k=1
Assume this is a quantized projection of y (xi ) from a stationary GP
f (xi ) = qk , if zk ≤ y (xi ) < zk+1
The objective is to sample from the posterior distribution
|xi − xj |2
p(y|f) ∼ T N (0, Σ), Σij = σ 2 exp −
, σ 2 = 0.6, η 2 = 0.2
2η 2
Lan, S. and Shahbaba, B. (UW and UCI)
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06/04/2015
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Experiments
Reconstruction of quantized stationary Gaussian process
Method
RWMt
Wall HMC
exact HMC
c-SphHMC
s-SphHMC
AP
0.70
0.69
1.00
0.72
0.80
s/iter
7.11E-05
9.94E-04
1.00E-02
1.73E-03
1.09E-03
ESS(min,med,max)
(2,9,35)
(12564,24317,43876)
(72074,1e+05,1e+05)
(13029,26021,56445)
(14422,31182,81948)
Min(ESS)/s
0.22
114.92
65.31
68.44
120.59
spdup
1.00
534.48
303.76
318.32
560.86
Table: Comparing efficiency of RWMt, Wall HMC, exact HMC, c-SphHMC and
s-SphHMC in reconstructing a quantized stationary Gaussian process. AP is
acceptance probability, s/iter is seconds per iteration, ESS(min,med,max) is the
(minimal,median,maximal) effective sample size, and Min(ESS)/s is the minimal
ESS per second.
Lan, S. and Shahbaba, B. (UW and UCI)
Spherical Augmentation
06/04/2015
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Experiments
LDA on Wikipedia corpus
LDA(Blei et al. 2003) is a popular Bayesian model for topic modeling.
The model consists of K topicks πk , which are distributions over the
words in the collection, drawn from a Dirichlet prior Dir(β).
A document d is modeled by a mixture of topics, with mixing
proportion ηd ∼ Dir(α).
Documents are produced by drawing a topic assignment zdi i.i.d from
ηd for each word wdi in document d, and then drawing the word wdi
from the assigned topic πzdi .
Lan, S. and Shahbaba, B. (UW and UCI)
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06/04/2015
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Experiments
LDA on Wikipedia corpus
Conditioned on π, the documents are i.i.d, and the joint distribution
can be factorized (Patterson and Teh, 2013)
D
Y
p(w , z, π|α, β) = p(π|β)
p(wd , zd |α, π)
d=1
p(wd , zd |α, π) =
K
Y
k=1
W
Γ(α + ndk· ) Y ndkw
πkw
Γ(α)
w =1
To compare with sg-RLD (Patterson and Teh, 2013), apply SphLMC
√
to update θ = π with stochastic gradient for L = 1 decreasing ε
∗
∗
∗
gkw = [(nkw
+ β − 1/2)/θkw + θkw (nk·
+ W (β − 1/2))]/(2 ∗ nk·
)
∗ =
where nkw
|D|
|Dt |
P
Lan, S. and Shahbaba, B. (UW and UCI)
d∈Dt
Ezd |wd ,θ,α [ndkw ], and |Dt | = 50.
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06/04/2015
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Experiments
LDA on Wikipedia corpus
Online learn 50000 documents randomly downloaded from Wikipedia.
Vocabulary consists of approx. 8000 words from Project Gutenburg.
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4000
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2000
7.0
log−perplexity (circle)
7.5
8.0
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10000
sg−RLD
sg−wallLMC
sg−SphHMC
sg−SphLMC
●
Time (square)
8.5
Evaluate the performance in perplexity on 1000 held-out documents.
0
10000
20000
30000
number of documents
Lan, S. and Shahbaba, B. (UW and UCI)
Spherical Augmentation
40000
50000
06/04/2015
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Conclusion and future work
1
Review: from HMC to RHMC
2
Spherical Augmentation
Simple examples: ball and box
General q-norm constraints
Some functional constraints
3
Spherical Monte Carlo
Spherical HMC in the Cartesian coordinate
Spherical HMC in the spherical coordinate
Spherical LMC on the probability simplex
4
Experiments
5
Conclusion and future work
Lan, S. and Shahbaba, B. (UW and UCI)
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06/04/2015
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Conclusion and future work
Conclusion
Spherical Augmentation (SA) is a natural and efficient framework to
handle norm related constraints in statistical inference.
Spherical HMC and Spherical LMC demonstrate substantial
advantage over existing methods. SA can have more extensions.
Based on change of variables, SA defines the dynamics on sphere in 1
higher dimension by slack variable or embedding map. The resulting
sampler moves on sphere freely while implicitly handling constraints.
To account for the change of geometry, volume adjustment is needed
to re-weight samples (SphHMC) or added to Hamiltonian (SphLMC).
Lan, S. and Shahbaba, B. (UW and UCI)
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06/04/2015
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Conclusion and future work
Future work
Instead of Euclidean metric I on B0D (1), we can start from Fisher
2
metric GF (θ), and consider metric like GF (θ) + θθ T /θD+1
for
augmented space to facilitate exploring complicated structures.
Derive an acceptance rule that does not drop quickly as dimension
increases (Beskos et al., 2011).
Develop tune-free algorithms for spherical HMC (Hoffman and
Gelman, 2011).
Lan, S. and Shahbaba, B. (UW and UCI)
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06/04/2015
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Thank you !
Web:
http://www.ics.uci.edu/~slan/SphHMC/Intro.html
Lan, S. and Shahbaba, B. (UW and UCI)
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06/04/2015
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