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Optimization in mean field random models

Optimization in mean field random models. Johan Wästlund Linköping University Sweden. Statistical Mechanics. Each particle has a spin Energy = Hamiltonian depends on spins of interacting particles Ising model: Spins ± 1, H = # interacting pairs of opposite spin .

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Optimization in mean field random models

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  1. Optimization in mean field random models Johan Wästlund Linköping University Sweden

  2. Statistical Mechanics • Each particle has a spin • Energy = Hamiltonian depends on spins of interacting particles • Ising model: Spins ±1, H = # interacting pairs of opposite spin

  3. Statistical Mechanics • Spin configuration  has energy H() • Gibbs measure depends on temperature T: • T→∞ random state • T→0 ground state, i.e. minimizing H()

  4. Statistical Mechanics • Thermodynamic limit N →∞ • Average energy? (suitably normalized)

  5. Disordered Systems • Spin glasses • AuFe random alloy • Fe atoms interact

  6. Disordered Systems • Random interactions between Fe atoms • Sherrington-Kirkpatrick model:

  7. Disordered Systems • Quenched random variables gi,j • S-K is a mean field model: No correlation betweeen quenched variables • NP hard to find ground state given gi,j

  8. Computer Science • Test / evaluate heuristics for NP-hard problems • Average case analysis • Random problem instances

  9. Combinatorial Optimization • Minimum Matching / Assignment • Minimum Spanning Tree • Traveling Salesman • Shortest Path • … • Points with given distances, minimize total length of configuration

  10. Statistical Physics / Computer Science • Feasible solution • Cost of solution • Cost of minimal solution • Artificial parameter T • Gibbs measure • N→∞ • Spin configuration • Hamiltonian • Ground state energy • Temperature • Gibbs measure • Thermodynamic limit

  11. Mean field models • Replica-cavity method has given good results for mean field models • Parisi solution of S-K model • The same methods can be applied to combinatorial optimization problems in mean field models

  12. Mean field models of distance • N points • Abstract geometry • Inter-point distances given by i. i. d. random variables • Exponential distribution easiest to analyze (pseudodimension 1)

  13. Matching • Set of edges giving a pairing of all points

  14. Spanning tree • Network connecting all points

  15. Traveling salesman • Tour visiting all points

  16. Mean field limits • No normalization needed! (pseudodimension 1) • Matching: 2/12≈0.822 (Mézard & Parisi 1985, rigorous proof by Aldous 2000) • Spanning tree: (3) = 1+1/8+1/27+… ≈1.202 (Frieze 1985) • Traveling salesman: 2.0415… (Krauth-Mézard-Parisi 1989), now established rigorously!

  17. Cavity results • Non-rigorous method • Aldous derived equivalent equations with the Poisson-Weighted Infinite Tree (PWIT)

  18. Cavity results • Non-rigorous quantity X = cost of minimal solution – cost of minimal solution with the root removed • Define X1, X2, X3,… similarly on sub-trees • Leads to the equation • Xidistributed like X, i are times of events in rate 1 Poisson process

  19. Cavity results • Analytically, this is equivalent to where

  20. Cavity results • Explicit solution • Ground state energy

  21. Cavity results • Note that the integral is equal to the area under the curve when f(u) is plotted against f(-u) • In this case, f satisfies the equation

  22. Cavity results

  23. K-L matching

  24. K-L matching • Similarly, the K-L matching problem leads to the equations: •  has rate K and  has rate L • min[K] stands for K:th smallest

  25. K-L matching • Shown by Parisi (2006) that this system has an essentially unique solution • The ground state energy is given by where x and y satisfy an explicit equation • For K=L=2, this equation is • Unfortunately the cavity method is not rigorous

  26. The exponential bipartite assignment problem n

  27. The exponential bipartite assignment problem • Exact formula conjectured by Parisi (1998) • Suggests proof by induction • Researchers in discrete math, combinatorics and graph theory became interested • Generalizations…

  28. Generalizations • by Coppersmith & Sorkin to incomplete matchings • Remarkable paper by M. Buck, C. Chan & D. Robbins (2000) • Introduces weighted vertices • Extremely close to proving Parisi’s conjecture!

  29. Incomplete matchings n m

  30. Weighted assignment problems • Weights 1,…,m, 1,…, n on vertices • Edge cost exponential of rate ij • Conjectured formula for the expected cost of minimum assignment • Formula for the probability that a vertex participates in solution (trivial for less general setting!)

  31. a3 a1 a2 The Buck-Chan-Robbins urn process • Balls are drawn with probabilities proportional to weight

  32. Proofs of the conjectures • Two independent proofs of the Parisi and Coppersmith-Sorkin conjectures were announced on March 17, 2003 (Nair, Prabhakar, Sharma and Linusson, Wästlund)

  33. Annealing • Powerful idea: Let T→0, forcing the system to converge to its ground state • Replica-cavity approach • Simulated annealing meta-algorithm (optimization by random local moves)

  34. In the mean field model: Underlying rate 1 variables Yi • ri plays the same role as T • Local temperature • Associate weight to vertices rather than edges

  35. Cavity/annealing method • Relax by introducing an extra vertex • Let the weight of the extra vertex go to zero • Example: Assignment problem with 1=…=m=1, 1=…=n=1, and m+1 =  • p = P(extra vertex participates) • p/n = P(edge (m+1,n) participates)

  36. Annealing • p/n = P(edge (m+1,n) participates) • When →0, this is • Hence • By Buck-Chan-Robbins urn theorem,

  37. Annealing • Hence • Inductively this establishes the Coppersmith-Sorkin formula

  38. Results with annealing • Much simpler proofs of Parisi, Coppersmith-Sorkin, Buck-Chan-Robbins formulas • Exact results for higher moments • Exact results and limits for optimization problems on the complete graph

  39. The 2-dimensional urn process • 2-dimensional time until k balls have been drawn

  40. Limit shape as n→∞ • Matching: • TSP/2-factor:

  41. Mean field TSP • If the edge costs are i.i.d and satisfy P(l<t)/t→1 as t→0 (pseudodimension 1), then as n →∞, • A. Frieze proved that whp a 2-factor can be patched to a tour at small cost

  42. Further exact formulas

  43. LP-relaxation of matching in the complete graph Kn

  44. Future work • Explain why the cavity method gives the same equation as the limit shape in the urn process • Establish more detailed cavity predictions • Use proof method of Nair-Prabhakar-Sharma in more general settings

  45. Thank you!

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