final report - probability.ca

1.4 Roberts and Rosenthal (1998): The MALA Algorithm
Results similar to the paper discussed in the previous section can be obtained for other Metropolis-Hastings algorithms.
Following the approach of [RGG97], Roberts and Rosenthal (1998) [RR98] demonstrate that the Metropolis-
Adjusted Langevin Algorithm (MALA, see 1.2.1), applied to target distributions of the form (1.8), is optimized
by fixing the variance of the proposal distribution to be O ( d −1/3) , which yields an overall asymptotically optimal
acceptance rate close to 0.574 and requires O ( d 1/3) steps to converge. This compares favorably to the RWM algorithm,
which require O (d) steps to explore the same class of target measures. The trade off being the need to
calculate the gradient of the Markov chain at every iteration, adding to the computational burden.
1.4.1 Target distribution and assumptions
Consider Metropolis-adjusted discrete approximations {X t } to the Langevin diffusion for π as above, with
( )
π x (d) , d =
d∏
f (x i ) =
i=1
d∏
exp {g (x i )} . (1.12)
Assume X 0 ∼ π, i.e., the chain starts out in stationarity. Furthermore, we assume that g is a C 8 function with
derivatives g (i) satisfying
∣
|g (x)| , ∣g (i) (x) ∣ ≤ M 0 (x) ,
for i ∈ [1, 8] for some polynomial M 0 (·), and that
ˆ
R
i=1
x k f (x) dx < ∞,
for k = 1, 2, 3, . . . . To be able apply results stemming from SDE theory, we assume that g ′ is a Lipschitz function.
To compare the discrete approximations to limiting continuous-time processes, it is helpful to define the discrete
approximations as jump processes { with } exponential holding times. In particular, let {J t } be a Poisson process
with rate d 1/3 , and let Γ (d) = Γ (d)
t be the d-dimensional jump process defined by Γ (d)
t = X Jt , where we set
t≥0
σd 2 = l2 nd −1/3 in the definitions above, with l an arbitrary positive constant. It is assumed throughout that {X t }
is non-explosive. Let
ˆ ˆ
( { })
π (Y ) q (Y , X)
α (l, d) = π (x) q (x, y) α (x, y, d) dxdy = E min
π (X) q (X, Y ) , 1 , (1.13)
denote the π-average acceptance rate of the algorithm generating Γ.
The first primary result of the paper gives an analytic formula for the asymptotic acceptance probability of the
algorithm.
1.4.2 Main Results
Theorem 1.13. The sequence (1.13) converges as d → ∞,
lim α (l, d) = α (l) ,
d→∞
with α (l) = 2Φ ( −Kl 3 /2 ) , where Φ (x) = 1 √
2π
´ x
−∞ e−t2 /2 dt, and
(
)
K = √ 5g ′′′ (X 2 ) − 3g ′′ (X) 3
E > 0,
48
where the expectation is taken over X having density f = exp (g).
The second result of the paper shows that for the target density π as given above and large values of d, a
corresponding MALA algorithm for the target π with proposal variances given by σ (d), and l = σ d d 1/6 , the speed
of the process is given by h (l) = 2l 2 Φ ( −Kl 3 /2 ) . Furthermore, the
)
optimal value ˆl of l which maximizes this speed
function is such that the asymptotic acceptance probability α
(ˆl ≈ 0.574.
14