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, 11 th 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 Random Heuristic Search 1 Finite horizon lumping Analytic Heuristic A game on a group Convexity 2 Decomposition Further Questions Other results 3 Universality Orbit structure Summary 4 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 - Λ n – the set of probabilities over a set of n elements (the unit simplex in R n ) - T : Λ n → Λ n – 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 Λ n with denominator r and transitions given by � 1 r v → 1 = r ! � 1 � � �� w T P r w r v w ! Theorem (Vose, 1999) Ξ: Λ n → Λ m is an aggregation if and only if it is a coarse graining of 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 ∈ M m 0 × n – initial partition of states R n T R n Ξ ′ 1 R m 0 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 ∈ M m 0 × n – initial partition of states B R T B R 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) 2 and compute T 1 : R m 1 → R m 1 s.t. Factorise (refine) Ξ ′ 1 = Ξ 1 Ξ ′ Ξ ′ 1 T ( p ) = Ξ 1 T 1 (Ξ ′ 2 p ) , for all p ∈ R n Ξ ′ 2 P G Y O T T 1 Y P G O Ξ 1 Ξ ′ 2 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 ′ Ξ N − 1 Ξ N Ξ 4 Ξ 3 Ξ 2 N +1 R n R m N R m N − 1 R m 3 R m 2 R m 1 T N T N − 1 T 3 T 2 T 1 T R n R m N R m N − 1 R m 3 R m 2 R m 1 Ξ ′ Ξ N − 1 Ξ N Ξ 3 Ξ 2 Ξ 1 N +1 R m N R m N − 1 R m N − 2 R m 2 R m 1 R m 0 Ξ N Ξ N − 1 Ξ N − 2 Ξ 3 Ξ 2 Ξ 1 1 T k ( p ) = Ξ 1 T 1 Ξ 2 T 2 . . . Ξ k T k (Ξ ′ Ξ ′ k p ) = N p ) = Ξ 1 Ξ 2 . . . Ξ N T k = Ξ N − k +1 T N − k +1 . . . Ξ N T N (Ξ ′ N (Ξ ′ N p ) , 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) Ξ ′ Ξ N − 1 Ξ 4 Ξ 3 Ξ 2 id N +1 R n R m N R m N − 1 R m 3 R m 2 R m 1 T N T N − 1 T 3 T 2 T 1 T R n R m N R m N − 1 R m 3 R m 2 R m 1 Ξ ′ N +1 Ξ N − 1 Ξ 3 Ξ 2 Ξ 1 Ξ N − 1 R m N − 2 R m 2 R m 1 R m 0 Ξ N − 2 Ξ 3 Ξ 2 Ξ 1 Stopping criteria: Ξ N 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 : Λ n → Λ n be a heuristic and Ξ ′ 1 : Λ n → Λ m 0 an aggregation. Also assume that there exist (Ξ i ) 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 v ! α v p v be an absolutely convergent series 1 Let T ( p ) = � v ∈ Z n + with an infinite radius of convergence defining analytic function on R n . An aggregation of variables Ξ: R n → R m is a valid coarse graining if and only if Ξ( v ) = Ξ( w ) implies Ξ( α v ) = Ξ( α w ) for all v , w ∈ Z n + . 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 } = Z n - 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 - Λ G – the set of probability measures on G - The dynamics is given on Λ G - To produce the (n+1) st generation, two independent random samples X and Y are drawn from the current, n th generation, and the distribution law of their group product X · Y is the next generation - µ �→ µ ∗ µ for any µ ∈ Λ G 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 = α 1 f 1 + · · · + α k f k , where α i ∈ (0 , 1] and � k i =1 α i = 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 k � M = α 1 f 1 + · · · + α k f k , α i ∈ (0 , 1] , α i = 1 i =1 such that Ξ is compatible with each f i , 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 k � M = α 1 f 1 + · · · + α k f k , α i ∈ (0 , 1] , α i = 1 i =1 such that Ξ is compatible with M if and only if it is compatible with each f i , 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 over 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
Recommend
More recommend
