Hit-and-run for numerical integration Daniel Rudolf University Jena - - PowerPoint PPT Presentation

Hit-and-run for numerical integration

Daniel Rudolf

University Jena

February 2012, Sydney

supported by

Daniel Rudolf (University Jena) Hit-and-run for numerical integration February 2012, Sydney 1 / 12

The problem

Let D ⊂ Rd. Compute Eπ̺f =

• D

f(x) π̺(dx) =

• D

f(x) · c̺(x) dx, where c̺ > 0 is a density. Function evaluations of f and ̺ are possible.

The normalizing constant 1 c =

• D

̺(x) dx is not known and hard to compute.

Equivalent formulation: Compute S(f, ̺) =

• D f(x)̺(x) dx
• D ̺(x) dx

for (f, ̺) ∈ F(D).

Daniel Rudolf (University Jena) Hit-and-run for numerical integration February 2012, Sydney 2 / 12

Markov chain Monte Carlo

Idea: Run a suitable Markov chain (Xn)n∈N with limit distribution π̺(A) =

• A ̺(x) dx
• D ̺(x) dx

and compute Sn,n0(f, ̺) = 1 n

n

• j=1

f(Xj+n0). Error: e(Sn,n0, (f, ̺)) = (E |S(f, ̺) − Sn,n0(f, ̺)|2)1/2. Goal: Error bounds? Polynomially in d? (D ⊂ Rd)

Daniel Rudolf (University Jena) Hit-and-run for numerical integration February 2012, Sydney 3 / 12

Markov chains

Suppose ν is an initial distribution and K is a Markov kernel.

• For f ∈ L1(π̺) the Markov operator is defined by

Pf(x) =

• D

f(y) K(x, dy) and S(f, ̺) =

• D

f(x) π̺(dx).

• There exists an L2-spectral gap if

β := P − SL2(π̺)→L2(π̺) < 1.

• For suitable burn-in n0 one has

e(Sn,n0, (f, ̺))2 ≤ 4 f2

4

n(1 − β). (see Rudolf 2011).

Daniel Rudolf (University Jena) Hit-and-run for numerical integration February 2012, Sydney 4 / 12

Hit-and-run algorithm

Compute S(f, ̺) =

• D f(x)̺(x) dx
• D ̺(x) dx

. Let B(0, R) be the ball with radius R around 0. Hit-and-run algorithm:

• assume x1, . . . , xk are already computed;
• choose direction uk uniformly distributed on ∂B(0, 1);
• return xk+1 ∈ Ak = {α ∈ R | xk + αuk ∈ D} distributed with density

ℓk(α) = ̺(xk + ukα)

• Ak ̺(xk + ukt) dt .

Daniel Rudolf (University Jena) Hit-and-run for numerical integration February 2012, Sydney 5 / 12

Simulation for d = 2

Hit-and-run in the unit ball and the cube, time steps 104 and α = 20.

−1 −0.8 −0.6 −0.4 −0.2 0.2 0.4 0.6 0.8 1 −1 −0.8 −0.6 −0.4 −0.2 0.2 0.4 0.6 0.8 1 −1 −0.8 −0.6 −0.4 −0.2 0.2 0.4 0.6 0.8 1 −1 −0.8 −0.6 −0.4 −0.2 0.2 0.4 0.6 0.8 1

Figure: On the left ̺1(x) = eαx1 and on the right ̺2(x) = eα

d j=1 xj √ d

.

Daniel Rudolf (University Jena) Hit-and-run for numerical integration February 2012, Sydney 6 / 12

Densities with certain structure

The class Rd,R:

• B(0, 1) ⊂ D ⊂ B(0, R);
• ̺ is log-concave, for x, y ∈ D and 0 < λ < 1

̺(λx + (1 − λ)y) ≥ ̺(x)λ · ̺(y)1−λ.

• For c > 0 let L(c) =
• x ∈ Rd | ̺(x) ≥ c
• . Assume that

π̺(L(c)) ≥ 1 8 = ⇒ ∃˜ x ∈ D B(˜ x, 1) ⊂ L(c).

• For u ∈ ∂B(0, 1) and x ∈ D one can sample w.r.t. the density

ℓ(α) = ̺(x + uα)

• A ̺(x + ut)dt

where A = {α ∈ R | x + uα ∈ D} .

Daniel Rudolf (University Jena) Hit-and-run for numerical integration February 2012, Sydney 7 / 12

Hit-and-run algorithm

Theorem (Lovász and Vempala 06)

Assumptions:

• let ̺ ∈ Rd,R;
• initial distribution ν, with bounded dν

dπ̺;

• let νPn0 be the distribution after n0 steps of hit-and-run.

If n0 ≻ d2R2 log3

• dν

dπ̺

• ∞

dR ε

• ,

then νPn0 − π̺tv ≤ 2ε.

• Proof is based on an estimate of the s-conductance.

Daniel Rudolf (University Jena) Hit-and-run for numerical integration February 2012, Sydney 8 / 12

Question

Assume that there exist numbers α < 1, k ∈ N and M < ∞, such that for all ν with

• dν

dπ̺

• 2 < ∞ we have

νPn0 − π̺tv ≤ Mα

k

√n0.

Question:

• Does it imply the existence of an L2-spectral gap?
• Additional assumptions?
• Estimate of the L2-spectral gap?

Daniel Rudolf (University Jena) Hit-and-run for numerical integration February 2012, Sydney 9 / 12

Different approximation scheme

Theorem (similar to Belloni and Chernozhukov 07)

Assumptions:

• n independent runs of a suitable Markov chain,

after n0 steps one has X 1

n0, . . . , X n n0.

• let f be a bounded function.
• compute
• Sn,n0(f) = 1

n

n

• i=1

f(X i

n0).

Then e( Sn,n0, (f, ̺))2 ≤ 1 n f∞ + 2 f∞ νPn0 − π̺tv .

• Number of steps n · n0 for

Sn,n0, while n + n0 for Sn,n0.

Daniel Rudolf (University Jena) Hit-and-run for numerical integration February 2012, Sydney 10 / 12

Error bound

Theorem

Let

• Fd,R,C =
• (f, ̺) | sup ̺

inf ̺ ≤ C, ̺ ∈ Rd,R, f∞ ≤ 1

• .

Then e( Shar

n,n0,

Fd,R,C)2 ≤ 1 n + 4dRd+1C exp

• −3 · 10−10

n1/3 (dR)2/3

• .

Consequently comp(ε, d, Fd,R,C) ≺ d5 ε−2 [log ε−2]3 [log C]3 R2[log R]3.

• Initial state chosen with respect to uniform distribution in B(0, 1).
• Constants are very large, of the magnitude of 1030.

Daniel Rudolf (University Jena) Hit-and-run for numerical integration February 2012, Sydney 11 / 12

References

• Belloni and Chernozhukov 2009: On the computational complexity of

MCMC-based estimators in large samples, Ann. Statist. 4, 2011-2055.

• Lovász and Vempala 2006: Fast algorithms for logconcave functions:

Sampling, rounding, integration and optimization, Proceedings of the 47th Annual IEEE Symposium on Foundations of Computer Science (FOCS’06), 57–68, 2006.

• Mathé and Novak 2007: Simple Monte Carlo and the Metropolis

algorithm, J. Complexity 23, 673-696.

• Rudolf 2011: Explicit error bounds of Markov chain Monte Carlo, to

appear in Dissertationes Math.

Daniel Rudolf (University Jena) Hit-and-run for numerical integration February 2012, Sydney 12 / 12