Hypo-elliptic simulated annealing

Transcription

1 Hypo-elliptic simulated annealing Christian Bayer 1 Josef Teichmann 2 Richard Warnung 3 1 Department of Mathematics Royal Institute of Technology, Stockholm 2 Department of Mathematics ETH Zurich 3 Raiffeisen Capital Investment, Vienna 07/30/2009 Berlin

2 Outline 1 Introduction Smoluchowski dynamics Simulated annealing in continuous time Motivation for hypo-elliptic simulated annealing 2 The theory of hypo-elliptic simulated annealing Setting of hypo-elliptic simulated annealing The gradient Hypo-elliptic simulated annealing 3 Numerical examples Example in R 3 Example on SO(3) 4 Conclusions

3 Smoluchowski dynamics (1) dy y t = 1 2 U(Y y t )dt + KTdW t Y y 0 = y Rn, U : R n R and W is an n-dimensional standard Wiener process Unique invariant measure given by the Gibbs measure µ KT (dy) = 1 e U(y) KT dy C T Ergodicity, i.e., Y y converges in law to µ KT Extensively used in physics and chemistry for the simulation of canonical ensemble (NVT), i.e., of an ensemble with constant temperature

4 Smoluchowski dynamics (1) dy y t = 1 2 U(Y y t )dt + KTdW t Y y 0 = y Rn, U : R n R and W is an n-dimensional standard Wiener process Unique invariant measure given by the Gibbs measure µ KT (dy) = 1 e U(y) KT dy C T Ergodicity, i.e., Y y converges in law to µ KT Extensively used in physics and chemistry for the simulation of canonical ensemble (NVT), i.e., of an ensemble with constant temperature

5 Smoluchowski dynamics (1) dy y t = 1 2 U(Y y t )dt + KTdW t Y y 0 = y Rn, U : R n R and W is an n-dimensional standard Wiener process Unique invariant measure given by the Gibbs measure µ KT (dy) = 1 e U(y) KT dy C T Ergodicity, i.e., Y y converges in law to µ KT Extensively used in physics and chemistry for the simulation of canonical ensemble (NVT), i.e., of an ensemble with constant temperature

6 Motivation of simulated annealing For T 0, the Gibbs measure µ KT (dy) = 1 e U(y) KT dy arg min U(y). C T y R n Choose a cooling schedule T = T(t) such that the instantaneously invariant measures µ KT(t) concentrate around arg min U(y) for t, the (time-inhomogeneous) Markov process dy y t = 1 2 U(Y y t )dt + KT(t)dW t remains ergodic.

7 Spectral gap Consider the infinitesimal generator L σ of the Smoluchowski SDE with constant σ = KT: L σ f = 1 2 U, f + σ 2 f. L σ is symmetric on L 2 (R n, µ σ ) and has a discrete, non-positive spectrum. Spectral gap λ σ = σ { } 2 inf f L 2 (µ σ ) f L 2 (µ σ ) = 1 and fdµ σ = 0 R n λ σ controls the convergence of the distribution of the Smoluchowski process to its invariant distribution.

8 Elliptic simulated annealing Assume that U is smooth, U and U converge to infinity for x, U 2 U is bounded from below. Theorem Given a decreasing cooling schedule σ = σ(t) 0, such that σ(t) = k/ log(t) for some constant k > c, where c = lim t σ(t) log(λ σ(t) ). Then the law of the process Y σ( ) t converges to a distribution supported on arg min y U(y).

9 Heisenberg groups construction We consider the Heisenberg groups as a prototype of the state space of hypo-elliptic simulated annealing. denotes the space of non-commutative polynomials of degree 2 in p variables {e 1,..., e p }. A 2 p Forms a nil-potent associative algebra of degree two. g 2 p denotes the Lie algebra generated by {e 1,..., e p } (with respect to [x, y] = xy yx, x, y A 2 p ). Define exp : g 2 p A 2 p by its (nil-potent) power series and set G 2 p exp ( g 2 p).

10 Heisenberg groups properties exp : g 2 p Gp 2 is a global chart of the Lie group G2 p. Representation as matrix group, e.g., for p = 2: 1 a c G b a, b, c R dim(gp) 2 = (p + 1)p/2 Brownian motion X t defined by X 0 = 1 Gp 2 and p dx t = X t e i dbt i i=1 Natural to use X t as driving noise for simulated annealing on Gp 2 or g2 p

11 Heisenberg groups properties exp : g 2 p Gp 2 is a global chart of the Lie group G2 p. Representation as matrix group, e.g., for p = 2: 1 a c G b a, b, c R dim(gp) 2 = (p + 1)p/2 Brownian motion X t defined by X 0 = 1 Gp 2 and p dx t = X t e i dbt i i=1 Natural to use X t as driving noise for simulated annealing on Gp 2 or g2 p

12 Elliptic vs. hypo-elliptic simulated annealing Starting point of elliptic simulated annealing A small stochastic perturbation of a classical gradient flow allows the flow to overcome local minima (having the Gibbs measure as invariant distribution). Decreasing the perturbation slowly enough induces convergence to arg min U. Starting point of hypo-elliptic simulated annealing Given a driving, hypo-elliptic process. Want to construct a Markov process with invariant measure given by the Gibbs measure. How does the appropriate drift look like?

13 Elliptic vs. hypo-elliptic simulated annealing Starting point of elliptic simulated annealing A small stochastic perturbation of a classical gradient flow allows the flow to overcome local minima (having the Gibbs measure as invariant distribution). Decreasing the perturbation slowly enough induces convergence to arg min U. Starting point of hypo-elliptic simulated annealing Given a driving, hypo-elliptic process. Want to construct a Markov process with invariant measure given by the Gibbs measure. How does the appropriate drift look like?

14 Homogeneous spaces Definition Given a Lie group G. A finite-dimensional smooth manifold M such that there is a right-action r : M G M, i.e., r(r(x, g), h) = r(x, gh), x M, g, h G, r is transitive, i.e., x M, g r(x, g) is surjective, is called homogeneous space. Projection: for a fixed point o M, define π : G M by π(g) = r(o, g).

15 Setting of hypo-elliptic simulated annealing Assumption A 1 G is a finite-dimensional, connected Lie group with Lie algebra g and right-invariant Haar measure η. 2 M is a compact homogeneous space w.r.t. G with a positive finite measure η M invariant w.r.t. the right action of G on M. 3 U C (M; [0, [). Remark Assumption A is satisfied by compact Lie groups like SO(3), acting on themselves.

16 Heisenberg torus G 2 p is not compact. Define a discrete sub-group of G 2 p. l 2 p = { e i i = 1,..., p } { 1 2 [e i, e j ] i < j } Z g 2 P Lp 2 exp(l 2 p) Gp 2 M L 2 p G 2 p = { L 2 p g g G2 p } is called Heisenberg torus (not a group!). Construct a right-invariant measure η M from the Lebesgue measure on T p(p+1)/2 using M T p(p+1)/2. E.g.,, for p = 2, set e 3 = 1 2 [e 1, e 2 ] and define φ : g 2 2 T3 by φ(z 1 e 1 + z 2 e 2 + z 3 e 3 ) = ([z 1 ], [z 2 ], [z 3 [z 2 ]z 1 + [z 1 ]z 2 ]), where [z] z mod 1. Get a diffeom. φ exp 1 : L 2 2 G2 2 T3. M together with η M satisfies Assumption A.

17 Hypo-ellipticity Assumption B Let n = dim(g) and assume that there are d < n left-invariant vector-fields V 1,..., V d on G, which already generate g (Hörmander condition). Define a stochastic process X t on G with X 0 = 1 G and d dx t = V i (X t ) dbt i. i=1 X t is hypo-elliptic, i.e., has a smooth transition density. For G = G 2 p, d = p < p(p+1) 2 = n and V i (x) = xe i, i = 1,..., p.

18 Hypo-ellipticity Assumption B Let n = dim(g) and assume that there are d < n left-invariant vector-fields V 1,..., V d on G, which already generate g (Hörmander condition). Define a stochastic process X t on G with X 0 = 1 G and d dx t = V i (X t ) dbt i. i=1 X t is hypo-elliptic, i.e., has a smooth transition density. For G = G 2 p, d = p < p(p+1) 2 = n and V i (x) = xe i, i = 1,..., p.

19 Sub-Riemannian gradient Drift 1 2 U is the direction of steepest descent in a Riemannian environment. Noise X t can traverse the whole space, but locally only horizontal directions are possible. Carré-du-champs operator d Γ(f, g)(x) = V i f(x)v i g(x), i=1 x G Γ(U, ) : f Γ(U, f) defines a left-invariant, horizontal vector-field.

20 Sub-Riemannian gradient Drift 1 2 U is the direction of steepest descent in a Riemannian environment. Noise X t can traverse the whole space, but locally only horizontal directions are possible. Carré-du-champs operator d Γ(f, g)(x) = V i f(x)v i g(x), i=1 x G Γ(U, ) : f Γ(U, f) defines a left-invariant, horizontal vector-field.

21 Hypo-elliptic Smoluchowski dynamics σ = σ(t) cooling schedule. Push the vector fields V 1,..., V d and Γ(U, ) to vector fields V M 1,..., V M d and ΓM (U, ) on M using π : G M. Define the Smoluchowski dynamics on M dy t = 1 2 ΓM (U, )(Y t )dt + σ(t) d i=1 V M i (Y t ) db i t. Locally invariant Gibbs measure µ σ (dx) = 1 ( exp U(x) ) η M (dx). C σ σ

22 Hypo-elliptic simulated annealing Theorem (Baudoin, Hairer and Teichmann) Let U 0 = min x M U(x). There are constants R, c > 0 such that the simulated annealing process Y t with σ(t) = c log(r + t) satisfies P(Y t A δ ) D µ σ(t) (A δ ), δ > 0, where A δ = { x M U(x) U0 + δ }. Remark For a non-compact state space M, there are additional boundedness conditions, e.g., on U(x) d(x, x0 ) 2.

23 Set-up in R 3 1 Since g 2 2 R3, hypo-elliptic simulated annealing in G 2 2 can be interpreted as hypo-elliptic simulated annealing in R V 1 (x) = 0, V 2(x) = 1. x 2 x 1 This corresponds to the sub-riemannian gradient x1 U(x) x 2 x3 U(x) Γ(U, )(x) = x2 U(x) + x 1 x3 U(x) x 1 x2 U(x) x 2 x1 U(x) + (x x2 2 ) x 3 U(x).

24 Set-up in R 3 2 We test a variant of the Rastrigin potential U(x) = i=1 ( x 2 i 10 cos(2πx i ) ). Note that min U = 0 attained at y min = (0, 0, 0). Remark The Riemannian gradient U grows linearly in x, whereas the sub-riemannian gradient Γ(U, ) grows like x 3, which requires a much finer time-resolution for the approximation of the SDE.

25 Results table Elliptic simulated annealing t E( Y t ) E(U(Y t )) min ω U(Y t (ω)) Hypo-elliptic simulated annealing t E( Y t ) E(U(Y t )) min ω U(Y t (ω)) Y 0 = (5, 5, 5), Y 0 = 8.66, U(Y 0 ) = 50.25, y min = (0, 0, 0) c = 15 E denotes an average over 500 sampled paths

26 Results histogram for t = 1 Starting value: Y 0 = (5, 5, 5), c = 15, y min = (0, 0, 0).

27 Results histogram for t = 4000 Starting value: Y 0 = (5, 5, 5), c = 15, y min = (0, 0, 0).

28 Results histogram for t = 4000 for too fast cooling Starting value: Y 0 = (5, 5, 5), c = 3, y min = (0, 0, 0).

29 Set-up Represent SO(3) S 3 Choose to vector fields from so(3) Potential V 1 =, V = U(x) = (x 1 + 1) 4 2 cos(2π(x 1 + 1)) + x 2 2 cos(πx 2)+ + x cos(2πx 3) + x 2 4 cos(πx 4).

30 Set up 2 Potential U(x) = (x 1 + 1) 4 2 cos(2π(x 1 + 1)) + x 2 2 cos(πx 2)+ + x cos(2πx 3) + x 2 4 cos(πx 4). Use a geometrical approximation scheme for the solution of the SDE, i.e., X n SO(3) for every n min y U(y) = 0 attained at y min = ( 1, 0, 0, 0) We start at y 0 = (1, 0, 0, 0). The scheme is simpler than any elliptic simulated annealing scheme.

31 Set up 2 Potential U(x) = (x 1 + 1) 4 2 cos(2π(x 1 + 1)) + x 2 2 cos(πx 2)+ + x cos(2πx 3) + x 2 4 cos(πx 4). Use a geometrical approximation scheme for the solution of the SDE, i.e., X n SO(3) for every n min y U(y) = 0 attained at y min = ( 1, 0, 0, 0) We start at y 0 = (1, 0, 0, 0). The scheme is simpler than any elliptic simulated annealing scheme.

32 Numerical results Hypo-elliptic simulated annealing, c = 1.4 t E( Y t y min ) E(U(Y t )) min ω Y t (ω) Hypo-elliptic simulated annealing, c = 5 t E( Y t y min ) E(U(Y t )) min ω Y t (ω) Y 0 = (1, 0, 0, 0), y min = ( 1, 0, 0, 0), Y 0 y min = 2, U(Y 0 ) = 16

33 Histogram t = 2, c =

34 Histogram t = 62, c =

35 Histogram t = 65534, c =

36 Histogram t = , c =

37 Conclusions Construction of hypo-elliptic Smoluchowski (or Ornstein-Uhlenbeck) processes with Gibbs measure as invariant measure Simulated annealing for hypo-elliptic Smoluchowski processes both theoretical and experimental justification Additional numerical cost due to instability in genuinely elliptic situations (e.g., when vector fields need to be non-linear for hypo-elliptic simulated annealing) Competitive in certain situations of constraint optimization

38 References F. Baudoin, M. Hairer, J. Teichmann. Ornstein-Uhlenbeck processes on Lie groups, J. Func. Anal., 255(4), B. Driver, T. Melcher. Hypoelliptic heat kernel inequalities on the Heisenberg group, J. Func. Anal., 221(2), R. Holley, S. Kusuoka, D. Stroock. Asymptotics of the spectral gap with applications to the theory of simulated annealing, J. Func. Anal., 83(2), S. Kirkpatrick, C. D. Gelett, M. P. Vecchi. Optimization by simulated annealing, Science 220, May L. Miclo. Recuit simulé sur R n. Etude del l évolution de l énergie libre., Ann. Inst. H. Poincaré Prob. Stat., 28(2), 1992.