Mini-bucket+MPLP/SAC

Rina Dechter3/25-2011

Solving Min-sum problems

• Exact by Bucket elimination

• Bounding scheme: MBE, MPLP, Soft arc-consistency.

• MBE: capitalize on large cluster processed exactly. Similar to i-consistency for large i (parameter z), directional, non-iterative.

• SAC, MPLP: equivalent to arc-consistency. Can be extended to cluster graphs, but may not be efficient. Based on cost shifting subject t equivalence-preserving transformations (EPT)

• Goal: MBE+MPLP/SAC can improve bounds of both in an efficient way.

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Mini-Bucket Approximation

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Semantics of Mini-Bucket: Splitting a Node

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Variables in different buckets are renamed and duplicated (Kask et. al., 2001), (Geffner et. al., 2007), (Choi, Chavira, Darwiche , 2007), (covering graphs, Ihler et. Al 2010.)

Optimal Soft Arc-consistency

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Algorithm 1: Project and UnaryProject for soft arc andnode consistency enforcing

Procedure Project(cS, i, a, α)c_i(a) ← c_i(a) ⊕ α;foreach (t ∈ (S) such that t[{i}] = a) doc_S(t) ← c_S(t) - α;

Procedure UnaryProject(i, α)c_∅ ← c_∅⊕ α;foreach (a ∈ di) doC_i(a) ← c_i(a) - α;

A variable i is said to be node consistent [Larrosa, 2002]if 1) for all values a ∈ di, ci(a) ⊕ c_∅ = k, 2) there exists avalue b ∈ di such that ci(b) = 0. A WCSP is node consistent(NC) if every variable is node consistent

A value b ∈ dj is a support for a value a ∈ di along cij if cij(a, b) = 0.

Variable i is arc consistent if every value a ∈ di has a support in every constraint cij ∈ C. A WCSP is arc consistent (AC) if every variable is arc and node consistent.

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Subtract from source in order to preserve the problem Equivalence Preserving Transformation

Shifting Costs (cost compensation)

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• Shifting costs from fAB to AShift(fAB,(A,w),1)

• Can be reversed (e.g. Shift(fAB,(A,w), -1))

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Arc EPT: shift cost in the scope of 1 cost function

Problem structure preserved

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OSAC: The Optimal SAC:A continuous linear formulation

• uA : cost shifted from A to f0

• pAaAB: cost shifted from fAB to (A,a)‏

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Subject to non negativity of costs

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Optimal Soft AC

• Solved by Linear Programming

• Polynomial time, rational costs (bounded arity r)

• Computes an optimal set of EPT (uA ,pAaAB) to apply

simultaneously

• Stronger than AC, DAC, FDAC, EDAC...(or any local consistency that preserves scopes)‏

(Cooper et al., IJCAI 2007)

(Boros & Hammer, Discrete Appl. Math. 2002)

(Schlesinger, Kibernetika 1976)

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The OSAC algorithmP’= OSAC(P)

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Combining MBE with OSAC

• MBE-OSAC(1)

1. Process buckets from i=n to 1.

• Partition into mini-bucket

• {mb_i1,…mb_ij} sum functions in each mini-bucket

• {mb’_i1,…mb’_ij} OSAC{mb_i1,…mb_ij}

• Lambda_ijmax_{X_i} mb’_ij

2. Return the sum of functions in the first bucket.

Properties:

1. The algorithm is exp(z) and solves LP problem in each bucket.

2. The output is a lower bound. But its quality may or may not improve over MBE (question: any guarantee? In practice?)

mbe-osac(2): shift costs based on all bucket variables and then mini-bucket

Input:

Output: messages placed in lower buckets based on their scope.

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2. For j=1 to r do (process each mini-bucket)

End.

Properties: May not improve on BME. Will it improve over OSAC? In practice?

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Linear relaxation-based schemes(MPLP class, Globerson and Jakkola)

Find:

Best upper bound by Equivalence preserving transformations:

There are several variations of scheme computing the optimizing shifts based on partial gradient descent, which differ by what is being kept constant. The 1.2 task is theDual of a linear relaxation of the original problem.

Is the cost shifted from f to value x_i of X_i.

MPLP block coordinated descent

Return updated functions:

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EMPLP and NMPLP

Convergence Properties

Proposition 3 If the fixed point of MPLP has bi(xi) such that for all i the function bi(xi) has ai , then x is the solution to the MAP problem and the LP relaxation is exact.

Proposition 4 When xi are binary, the MPLP fixed point can be used to obtain the primal optimum.

Generalized MPLP

MBE+MPLP

• MPE-MPLP can be combined in a similar ways as with OSAC. Which method can provide guarantee for improvement….

• Use generalized MPLP as a basis…

• Develop directly similar to Alex and Qiang for belief.

Generalized MPLP

Perhaps re-derive generalized MPLP where S (seperators) are just single-variables.We can do that by using a join-graph generated using the mini-bucket schematic method.In this case the seperators within each mini-bucket are the bucket-variable itself.

Considering a covering graph that corresponds to the mini-bucket structure, we can have a collection of shifting costs from mini-bucket to single variables and we canDerive the updating equations associated with the dual that corresponds to this problem.The execution of this algorithm in an ordered derived from the mini-bucket may be a mini-bucket-like algorithm that iterates. (This may be actually identical to Alex and Qiang)

THE ENDWho wants to grab this?