Technical Report on Work Packages 1-3 UoB Maths University of - - PowerPoint PPT Presentation

Overview Random Heuristic Search Convexity Other results Summary References

Technical Report on Work Packages 1-3 UoB Maths

University of Birmingham Chris Good, Nishanthan Kamaleson, David Parker, Mate Puljiz, Jonathan E. Rowe Brussels, 11th December 2014

• C. Good, N. Kamaleson, D. Parker, M. Puljiz, J. E. Rowe

UoB Maths: WP 1-3

Overview Random Heuristic Search Convexity Other results Summary References

Overview

1

Random Heuristic Search Finite horizon lumping Analytic Heuristic A game on a group

2

Convexity Decomposition Further Questions

3

Other results Universality Orbit structure

4

Summary

• C. Good, N. Kamaleson, D. Parker, M. Puljiz, J. E. Rowe

UoB Maths: WP 1-3

Overview Random Heuristic Search Convexity Other results Summary References

Random Heuristic Search

Recall

• Λn – the set of probabilities over a set of n elements (the unit

simplex in Rn)

• T : Λn → Λn – a heuristic function
• Given r ∈ N, RHS is an induced Markov chain M r

T with the

state set consisting of all rational vectors in Λn with denominator r and transitions given by P

1

r v → 1 r w

• = r!

w!

• T

1

r v

w

Theorem (Vose, 1999) Ξ: Λn → Λm is an aggregation if and only if it is a coarse graining

• f M r

T for all r ∈ N

• C. Good, N. Kamaleson, D. Parker, M. Puljiz, J. E. Rowe

UoB Maths: WP 1-3

Overview Random Heuristic Search Convexity Other results Summary References

Finite horizon

General Scheme (T not necessarily linear)

Ξ′

1 ∈ Mm0×n – initial partition of states

Rn Rn Rm0 T Ξ′

1

• C. Good, N. Kamaleson, D. Parker, M. Puljiz, J. E. Rowe

UoB Maths: WP 1-3

Overview Random Heuristic Search Convexity Other results Summary References

Finite horizon

General Scheme (T not necessarily linear)

Ξ′

1 ∈ Mm0×n – initial partition of states

T Ξ′

1

B R B R

• C. Good, N. Kamaleson, D. Parker, M. Puljiz, J. E. Rowe

UoB Maths: WP 1-3

Overview Random Heuristic Search Convexity Other results Summary References

Finite horizon

General Scheme (T not necessarily linear)

Factorise (refine) Ξ′

1 = Ξ1Ξ′ 2 and compute T1 : Rm1 → Rm1 s.t.

Ξ′

1T(p) = Ξ1T1(Ξ′ 2p), for all p ∈ Rn

T Ξ′

2

P G Y O Ξ′

2

T1 Ξ1 P G Y O Ξ1

• C. Good, N. Kamaleson, D. Parker, M. Puljiz, J. E. Rowe

UoB Maths: WP 1-3

Overview Random Heuristic Search Convexity Other results Summary References

Finite horizon

General Scheme (T not necessarily linear)

Rn RmN RmN−1 Rm3 Rm2 Rm1 Rn RmN RmN−1 Rm3 Rm2 Rm1 RmN RmN−1 RmN−2 Rm2 Rm1 Rm0 T TN TN−1 T3 T2 T1 Ξ′

N+1

ΞN ΞN−1 Ξ3 Ξ2 Ξ1 Ξ′

N+1

ΞN ΞN−1 Ξ4 Ξ3 Ξ2 ΞN ΞN−1 ΞN−2 Ξ3 Ξ2 Ξ1 Ξ′

1T k(p) = Ξ1T1Ξ2T2 . . . ΞkTk(Ξ′

kp) =

= ΞN−k+1TN−k+1 . . . ΞNTN(Ξ′

Np) = Ξ1Ξ2 . . . ΞNT k N(Ξ′ Np),

for 0 ≤ k ≤ N.

• C. Good, N. Kamaleson, D. Parker, M. Puljiz, J. E. Rowe

UoB Maths: WP 1-3

Overview Random Heuristic Search Convexity Other results Summary References

Finite horizon

General Scheme (T not necessarily linear)

Rn RmN RmN−1 Rm3 Rm2 Rm1 Rn RmN RmN−1 Rm3 Rm2 Rm1 RmN−2 Rm2 Rm1 Rm0 T TN TN−1 T3 T2 T1 Ξ′

N+1

ΞN−1 ΞN−1 Ξ3 Ξ2 Ξ1 Ξ′

N+1

id ΞN−1 Ξ4 Ξ3 Ξ2 ΞN−2 Ξ3 Ξ2 Ξ1

Stopping criteria: ΞN is the identity. Ξ′

N+1 is a proper coarse graining

• C. Good, N. Kamaleson, D. Parker, M. Puljiz, J. E. Rowe

UoB Maths: WP 1-3

Overview Random Heuristic Search Convexity Other results Summary References

Finite horizon

Generalisation of Vose’s theorem Theorem Let N be a fixed positive integer. Let T : Λn → Λn be a heuristic and Ξ′

1 : Λn → Λm0 an aggregation. Also assume that there exist

(Ξi)1≤i≤N following the scheme described above. Then for any r ∈ N the same relations hold for M r

T and the induced maps and

aggregations. Intuitively: To find a finite step aggregation of the simulated Markov Chain, it suffices to do so for a heuristic function. Setting N = 1 and Ξ′

1 to be a compatible aggregation gives Vose’s

theorem.

• C. Good, N. Kamaleson, D. Parker, M. Puljiz, J. E. Rowe

UoB Maths: WP 1-3

Overview Random Heuristic Search Convexity Other results Summary References

Finite horizon

The ideas presented here draw upon:

• Partition minimisation (Paige & Tarjan, 1987)
• Improved part. min. techniques developed by UoB Comp Sci
• Memoization techniques developed by Jena
• Finite approximations (Smyth, 1995)
• UoB Comp Sci developed algorithms that fit within this scheme
• It turns out that many other (non-linear) problems in practice

admit analogous ’partition minimisation’ algorithm that follow this scheme e.g. hashlife, the framework developed by Jena, in general, models based on short-distance interactions

• C. Good, N. Kamaleson, D. Parker, M. Puljiz, J. E. Rowe

UoB Maths: WP 1-3

Overview Random Heuristic Search Convexity Other results Summary References

Aggregation of heuristics

Last year

• A criterion for aggregations of a heuristic given by a polynomial

map. This year Theorem Let T(p) =

v∈Zn

+

1 v!αvpv be an absolutely convergent series

with an infinite radius of convergence defining analytic function on

• Rn. An aggregation of variables Ξ: Rn → Rm is a valid coarse

graining if and only if Ξ(v) = Ξ(w) implies Ξ(αv) = Ξ(αw) for all v, w ∈ Zn

+.

Gives an algorithm to check for coarse grainings.

• C. Good, N. Kamaleson, D. Parker, M. Puljiz, J. E. Rowe

UoB Maths: WP 1-3

Overview Random Heuristic Search Convexity Other results Summary References

A game on a group

Last year

• Set of labels G = {0, 1, . . . , n − 1} = Zn
• Two particles of types x, y ∈ G combine to produce a particle of

type x + y (mod n)

• A second degree reaction
• Aggregations ≈ Subgroups
• C. Good, N. Kamaleson, D. Parker, M. Puljiz, J. E. Rowe

UoB Maths: WP 1-3

Overview Random Heuristic Search Convexity Other results Summary References

A game on a group

This year

• (G, · ) – a topological group
• ΛG – the set of probability measures on G
• The dynamics is given on ΛG
• To produce the (n+1)st generation, two independent random

samples X and Y are drawn from the current, nth generation, and the distribution law of their group product X · Y is the next generation

• µ → µ ∗ µ for any µ ∈ ΛG

Theorem The compatible aggregations of this dynamics are in a 1 − 1 correspondence with normal closed subgroups of G. The aggregation refinement corresponds to set inclusion.

• C. Good, N. Kamaleson, D. Parker, M. Puljiz, J. E. Rowe

UoB Maths: WP 1-3

Overview Random Heuristic Search Convexity Other results Summary References

Decomposition of a Markov Chain

Proposition (Simon) Any Markov chain transition matrix M (a column stochastic matrix) decomposes into a convex combination of deterministic functions (0-1 column stochastic matrices). M = α1f1 + · · · + αkfk, where αi ∈ (0, 1] and k

i=1 αi = 1.

The decomposition need not be unique. A direct consequence of Krein-Milman’s Theorem. Question How do aggregations relate to this decomposition?

• C. Good, N. Kamaleson, D. Parker, M. Puljiz, J. E. Rowe

UoB Maths: WP 1-3

Overview Random Heuristic Search Convexity Other results Summary References

Is there always a compatible decomposition?

Proposition Given a transition matrix M and an aggregation Ξ there exists a decomposition of M in deterministic components M = α1f1 + · · · + αkfk, αi ∈ (0, 1],

k

• i=1

αi = 1 such that Ξ is compatible with each fi, for 1 ≤ i ≤ k.

• The proof is constructive - gives an algorithm to obtain the

desired decomposition.

• The decomposition allows a more efficient simulation.
• C. Good, N. Kamaleson, D. Parker, M. Puljiz, J. E. Rowe

UoB Maths: WP 1-3

Overview Random Heuristic Search Convexity Other results Summary References

Converse?

Question Given a Markov chain M, is there a decomposition M = α1f1 + · · · + αkfk, αi ∈ (0, 1],

k

• i=1

αi = 1 such that Ξ is compatible with M if and only if it is compatible with each fi, 1 ≤ i ≤ k? No!

• C. Good, N. Kamaleson, D. Parker, M. Puljiz, J. E. Rowe

UoB Maths: WP 1-3

Overview Random Heuristic Search Convexity Other results Summary References

Why not?

Lemma The lattice of aggregations of a deterministic function f : S → S

• ver a set of states S is a (complete) sub-lattice of the partition

lattice of the set S. (Both operations, meet and join, are inherited from partitions) As the intersection of finitely many complete sub-lattices is still a complete sub-lattice; if the converse were true, the aggregation lattice of any transition matrix would be a complete sub-lattice of the partition lattice of S. But last year we saw that there are some Markov chain for which this is not satisfied.

• C. Good, N. Kamaleson, D. Parker, M. Puljiz, J. E. Rowe

UoB Maths: WP 1-3

Overview Random Heuristic Search Convexity Other results Summary References

What are the obstacles?

P - the family (a lattice) of all partitions of [n] := {1, 2, . . . , n}. LM ⊆ P - the set of all aggregations compatible with M. LM is a complete lattice, but not necessarily a sub-lattice of P (meet is not preserved). Given a decomposition M = α1f1 + · · · + αkfk

k

• i=1

Lfi ⊆ LM The previous proposition can be stated as

• d

k(d)

• i=1

Lf (d)

i

= LM where d runs through all the convex decompositions of M.

• C. Good, N. Kamaleson, D. Parker, M. Puljiz, J. E. Rowe

UoB Maths: WP 1-3

Overview Random Heuristic Search Convexity Other results Summary References

What are the obstacles?

Question For which M there exists a d such that k(d)

i=1 L(d) fi

= LM? An easy result going in that direction: Proposition Let M be a transition matrix of a Markov chain. An aggregation coarse grains this system (ie. LM = P) if and only if there exist αi ∈ (0, 1], for 0 ≤ i ≤ n, such that M = α0I +

n

• i=1

αiCi, where n

i=0 αi = 1 and for each 1 ≤ i ≤ n, Ci stands for the

constant function mapping any state to the state i.

• C. Good, N. Kamaleson, D. Parker, M. Puljiz, J. E. Rowe

UoB Maths: WP 1-3

Overview Random Heuristic Search Convexity Other results Summary References

Universality

Smyth’s inverse limit construction in a way mimics the well-know fact of universality of Cantor set in category of compact metric

• spaces. This led us to ask

Question Is there a universal (compact, metric) dynamical system that coarse grains onto any other? NO! Theorem (Nunnally, 1967) No single dynamical system on a compact metric space can coarse grain simultaneously onto each irrational rotation.

• C. Good, N. Kamaleson, D. Parker, M. Puljiz, J. E. Rowe

UoB Maths: WP 1-3

Overview Random Heuristic Search Convexity Other results Summary References

Universality

However, Theorem (Anderson, 1963) Given a dynamical system (X, f ) on a compact metric space, there exists a continuous onto map Ξ: C → X and a map ˜ f : C → C on the Cantor set C that lifts f i.e. the following diagram commutes C X C X Ξ ˜ f Ξ f

• C. Good, N. Kamaleson, D. Parker, M. Puljiz, J. E. Rowe

UoB Maths: WP 1-3

Overview Random Heuristic Search Convexity Other results Summary References

Orbit structure

(X, T) – a minimal dynamical system A system is minimal if it contains no closed subsystems, in other words each orbit is dense. We can characterise coarse grainings of such systems in terms of their orbit structure.

• C. Good, N. Kamaleson, D. Parker, M. Puljiz, J. E. Rowe

UoB Maths: WP 1-3

Overview Random Heuristic Search Convexity Other results Summary References

Orbit structure

Theorem Let (X, T) and (Y , S) be two minimal dynamics on compact metric spaces X and Y . The following conditions are equivalent (i) (X, T) and (Y , S) are semi-conjugated (the first coarse grains onto the second), (ii) there exist two points x ∈ X and y ∈ Y such that whenever the subsequence (T nk(x))k converges, (T nk(y))k converges. (iii) for any x ∈ X there exists y ∈ Y such that whenever the subsequence (T nk(x))k converges, (T nk(y))k converges.

• C. Good, N. Kamaleson, D. Parker, M. Puljiz, J. E. Rowe

UoB Maths: WP 1-3

Overview Random Heuristic Search Convexity Other results Summary References

Summary of Key Achievements for Work Package 1

Collaboration between UoB (Maths), UoB (Comp Sci), Chalmers, Jena and Sheffield has led to: Generalising the example from the first year Structural obstructions preventing (non-linear) coarse-grainings (feeds into WP2) Finite horizon lumping (feeds into the objectives of WP3, WP4 and WP5) Analytic heuristic

• C. Good, N. Kamaleson, D. Parker, M. Puljiz, J. E. Rowe

UoB Maths: WP 1-3

Overview Random Heuristic Search Convexity Other results Summary References

Summary of Key Achievements for Work Package 2

This Work Package is joint with Sheffield. Structural characterisation of dynamics of linear maps (WP1). Opens a new direction exploring coarse-graining of minimal systems. Convex decomposition of a Markov process. This produced an algorithm which yields a decomposition compatible with a certain lumping. Monotonicity results in delayed Gambler’s Ruin game (with Sheffield and UoB(CS); feeds into the objectives of WP6)

• C. Good, N. Kamaleson, D. Parker, M. Puljiz, J. E. Rowe

UoB Maths: WP 1-3

Overview Random Heuristic Search Convexity Other results Summary References

Summary of Key Achievements for Work Package 3

This Work Package is joint with UoB (Comp Sci), Chalmers. Different notions of lumping. Finite horizon coarse graining(WP 1-2) and finite approximations are further explored with UoB Comp Sci which resulted in practical algorithms reported in WP5.

• C. Good, N. Kamaleson, D. Parker, M. Puljiz, J. E. Rowe

UoB Maths: WP 1-3

Overview Random Heuristic Search Convexity Other results Summary References

The End!

Thank you!

• C. Good, N. Kamaleson, D. Parker, M. Puljiz, J. E. Rowe

UoB Maths: WP 1-3

Overview Random Heuristic Search Convexity Other results Summary References

References I

[1] R. D. Anderson, On raising flows and mappings, Bull. Amer. Math. Soc. 69 (1963), 259–264. MR0144324 (26 #1870) [2] C. Good, S. Greenwood, R. W. Knight, D. W. McIntyre, and S. Watson, Characterizing continuous functions on compact spaces, Adv. Math. 206 (2006), no. 2, 695–728. MR2263719 (2007i:54027) [3] Chris Good, David Parker, Mate Puljiz, and Jonathan E. Rowe, WP1: Theory of hierarchical structure — D1.1: Algebraic structure of hierarchical decompositions, 2014. http://www.hieratic.eu/deliverables.php, Accessed: 01/08/2014. [4] Ellard Nunnally, There is no universal-projecting homeomorphism of the Cantor set, Colloq. Math. 17 (1967), 51–52. MR0214020 (35 #4872) [5] Robert Paige and Robert E. Tarjan, Three partition refinement algorithms, SIAM J. Comput. 16 (1987), no. 6, 973–989. MR917035 (89h:68069) [6] Barry Simon, Convexity, Cambridge Tracts in Mathematics, vol. 187, Cambridge University Press, Cambridge, 2011. An analytic viewpoint. MR2814377 (2012d:46002)

• C. Good, N. Kamaleson, D. Parker, M. Puljiz, J. E. Rowe

UoB Maths: WP 1-3

Overview Random Heuristic Search Convexity Other results Summary References

References II

[7] M. B. Smyth, Semi-metrics, closure spaces and digital topology, Theoret.

• Comput. Sci. 151 (1995), no. 1, 257–276. Topology and completion in

semantics (Chartres, 1993). MR1362155 (97a:68177) [8] Michael D. Vose, The simple genetic algorithm, Complex Adaptive Systems, MIT Press, Cambridge, MA, 1999. Foundations and theory, A Bradford Book. MR1713436 (2000h:65024)

• C. Good, N. Kamaleson, D. Parker, M. Puljiz, J. E. Rowe

UoB Maths: WP 1-3