introduction and discussion of a paper at BiPS reading group

1. Exponential ergodicity of the bouncy particle
sampler
Comments
G. Degliannidis, A. Bouchard-Cˆot´e, A. Doucet

2. plan
Motivations
A new bouncy particle sampler
Main results

3. Marketing arguments
Non-reversible MCMC algorithms based on piecewise deterministic
Markov processes perform well empirically
Quantitative convergence rates only available under stringent
assumptions:
Mesquita and Hespanha (2010) show geometric ergodicity for
targets with exponentially decaying tails,
Monmarch´e (2016) gives sharp results for compact state-space
Bierkens et al. (2016a,b) consider targets on the real line
Establishing exponential ergodicity and a CLT under weaker
conditions of theoretical and practical interest

4. Bouncy particle sampler
Bouncy particle sampler (BPS) piecewise deterministic MCMC
scheme proposed in Peters and Wells (2012)
Perform empirically very well when compared to other
state-of-the-art MCMC algorithms (Bouchard-Cˆot´e et al.,
2015)
BPS is the scaling limit of the (discrete-time) reflective slice
sampling algorithm (Vanetti et al., 2017)

5. Bouncy particle sampler
Target defined up to a constant
π(x) = γ(x) Z
with energy U(x) = − log γ(x)
Simulation of continuous-time piecewise linear trajectory (xt)t with
each segment in trajectory specified by
initial position x(i)
length τi+1
velocity v(i)

6. Bouncy particle sampler
Simulation of continuous-time piecewise linear trajectory (xt)t with
each segment in trajectory specified by
initial position x(i)
length τi+1
velocity v(i)
length specified by inhomogeneous Poisson point process with
intensity function
λ(x, v) = max{0, < U(x), v >}

7. Bouncy particle sampler
Simulation of continuous-time piecewise linear trajectory (xt)t with
each segment in trajectory specified by
initial position x(i)
length τi+1
velocity v(i)
length specified by inhomogeneous Poisson point process with
intensity function
λ(x, v) = max{0, < U(x), v >}
new velocity after bouncing given by Newtonian elastic collision
R(x)v = v − 2
< U(x), v >
|| U(x)||2
U(x)
[arXiv:1510.02451]

8. Bouncy particle sampler
[arXiv:1510.02451]

9. Basic bouncy particle sampler
velocity refreshment at the arrival times of a homogeneous Poisson
point process
[arXiv:1510.02451]

10. Simulation algorithms
Simulation using a time-scale transformation requiring
log-concave
Simulation using adaptive thinning requiring upper bounds on
inhomogeneous Poisson process
Simulation using superposition and thinning based on
decomposition of energy function
The piecewise deterministic Markov process has invariant
distribution π whenever λref > 0
[arXiv:1510.02451, Proposition 1]

11. Simulation algorithms
Simulation using a time-scale transformation requiring
log-concave
Simulation using adaptive thinning requiring upper bounds on
inhomogeneous Poisson process
Simulation using superposition and thinning based on
decomposition of energy function
The piecewise deterministic Markov process has invariant
distribution π whenever λref > 0
[arXiv:1510.02451, Proposition 1]

12. BPS versus HMC
[arXiv:1510.02451]

13. New BPS
Aiming at simulating from a target ˜π(dx) by adding auxiliary
direction v with uniform distribution on unit sphere, with
π(dx, dv) = ˜π(dx)dv
BPS targets π by a
π-invariant
non-reversible
piecewise deterministic
Markov process (Zt)t≥0
Refreshment times as inhomogeneous Poisson point process with
intensity function
λref
(x) + λ(x, v) = λref
(x) + max{0, < U(x), v >}

14. New BPS
Aiming at simulating from a target ˜π(dx) by adding auxiliary
direction v with uniform distribution on unit sphere, with
π(dx, dv) = ˜π(dx)dv
BPS targets π by a
π-invariant
non-reversible
piecewise deterministic
Markov process (Zt)t≥0
Refreshment times as inhomogeneous Poisson point process with
intensity function
λref
(x) + λ(x, v) = λref
(x) + max{0, < U(x), v >}

15. New BPS
Aiming at simulating from a target ˜π(dx) by adding auxiliary
direction v with uniform distribution on unit sphere, with
π(dx, dv) = ˜π(dx)dv
BPS targets π by a
π-invariant
non-reversible
piecewise deterministic
Markov process (Zt)t≥0
Refreshment times as inhomogeneous Poisson point process with
intensity function
λref
(x) + λ(x, v) = λref
(x) + max{0, < U(x), v >}

16. New BPS algorithm
[arXiv:1705.04579]

17. Infinitesimal generator and kernel
Lf (x, v) =< x f (x, v), v > +¯λ(x, v)[Kf (x, v) − f (x, v)]
and
K((x, v), (dy, dw)) =
λref(x)
¯λ(x, v)
δx (dy)ψ(dw) +
λ(x, v)
¯λ(x, v)
δx (dy)δv (dw)

18. Main results: V -uniform ergodicity
Assumptions
(A0) Hessian of U(x) locally Lipschitz
(A1) lim∞
exp{U(x)/2}
| U(x)|
> 0
(A3) Assumption on the drift function
V (x, v) =
exp{U(x)/2}
¯λ(x, −v)1/2
≥ c

19. Main results: V -uniform ergodicity
Tail behaviours
BPS with a properly chosen constant refreshment rate
exponentially ergodic for targets with tails that decay at least
as fast as an exponential, and at most as fast as a Gaussian.
In addition uniform bound on Hessian imposes some regularity
on the curvature of the target
When gradient grows faster than linearly in the tails
(thinned-tailed) any constant refreshment rate will eventually
be negligible. Hence need to scale the refreshment rate
accordingly in order for it to remain non-negligible in the tails.
For targets with tails thicker than exponential, lack of
exponential ergodicity of gradient-based methods such as
MALA and HMC, is natural as vanishing gradient induces
random-walk like behaviour in tails.

20. Main results: V -uniform ergodicity
Tail behaviours
For thin-tailed targets, modify λref(x) into
λref
(x) = λref
+
| U(x)|
max{1, |x| }
For thick-tailed targets, modify the target by a diffeomorphic
transform of X ∼ π

21. Central Limit Theorem
Estimator
ˆgT =
1
T
T
0
g(Zs)ds
Under same regularity conditions as for V -uniform ergodicity
√
T{g − π(g)}
L
−→ N(0, σ2
g )

22. Comparison table

23. Some comments
no indication that the obtained conditions on the refreshment
rate are sharp
third case requires isotopically transformed version of the BPS
(which would also apply to other simulation schemes)
difficult to spot practical side (like the exploitation of the
continuous path)
further details of Algorithm 1 implementation would be needed