Efficient Stochastic Local Search for MPE Solving

1
Efficient Stochastic Local Search for MPE Solving

• Frank Hutter The University of British Columbia
  (UBC), Vancouver, Canada
• Joint work with Holger Hoos (UBC) and
• Thomas Stützle (Darmstadt University of
  Technology, Germany)

2
SLS general algorithmic framework for solving
combinatorial problems
3
MPE in graphical models many applications
4
Outline

• Most probable explanation (MPE) problem
• Problem definition
• Previous work
• SLS algorithms for MPE
• Illustration
• Previous SLS algorithms
• Guided Local Search (GLS) in detail
• From Guided Local Search to GLS
• Modifications
• Performance gains
• Comparison to state-of-the-art

5
MPE - problem definition(in most general
representation factor graphs)

• Given a factor graph
• Discrete Variables X X1, ..., Xn
• Factors ? ?1,...,?m over subsets of X
• A factor ?i over variables Vi µ X assigns a
  non-negative number to every complete
  instantiation vi of Vi
• Find
• Complete instantiation x1,...,xn maximizing
  ?i1m ?ix1,...,xn
• NP-hard (simple reduction from SAT)
• Also known as Max-product or Maximum a posteriori
  (MAP)

6
Previous approaches for solving MPE

• Variable elimination / Junction tree
• Exponential in the graphical models induced
  width
• Approximation with loopy belief propagation and
  its generalizations Yedidia, Freeman, Weiss 02
• Approximation with Mini Buckets (MB) Dechter
  Rish 97 ! also gives lower upper bound
• Search algorithms
• Local Search
• Branch and Bound with various MB heuristics
  Dechters group, 99 - 05UAI 03 BB with MB
  heuristic shown to be state-of-the-art

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Motivation for our work

• BB clearly outperforms best SLS algorithm so
  far, even on random problem instances
  Marinescu, Kask, Dechter, UAI 03
• MPE is closely related to weighted Max-SAT Park
  02
• For Max-SAT, SLS is state-of-the-art(at the very
  least for random problems)
• Why is SLS not state-of-the-art for MPE ?
• Additional problem structure inside the factors
• But for completely random problems ?
• SLS algos should be much better than they
  currently are
• We took the best SLS algorithm so far (GLS) and
  improved it

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Outline

• Most probable explanation (MPE) problem
• Problem definition
• Previous work
• SLS algorithms for MPE
• Illustration
• Previous SLS algorithms
• Guided Local Search (GLS) in detail
• From Guided Local Search to GLS
• Modifications
• Performance gains
• Comparison to state-of-the-art

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SLS for MPE illustration
2
1
0
0
Instantiation
X2 X3 X4 ?5
0 0 0 10
0 0 1 0.9
0 1 0 0
0 1 1 100
1 0 0 33.2
1 0 1 0
1 1 0 23.2
1 1 1 13.7
X1
X2
X4
X3
X1 ?1
0 0
1 21.2
2 0.1
?1
?2
?3
?4
?5
X1 X2 ?2
0 0 21
0 1 0.7
1 0 0
1 1 1
2 0 0.9
2 1 0.2
X3 ?4
0 0.9
1 0.1
X1 X3 ?3
0 0 1.1
0 1 23
1 0 0
1 1 0.7
2 0 2.7
2 1 42
?i1M ?i2,1,0,0 0.1 0.2 2.7 0.9 33.2
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SLS for MPE illustration
2
1!0
0
0
Instantiation
X2 X3 X4 ?5
0 0 0 10
0 0 1 0.9
0 1 0 0
0 1 1 100
1 0 0 33.2
1 0 1 0
1 1 0 23.2
1 1 1 13.7
X1
X2
X4
X3
X1 ?1
0 0
1 21.2
2 0.1
?1
?2
?3
?4
?5
X1 X2 ?2
0 0 21
0 1 0.7
1 0 0
1 1 1
2 0 0.9
2 1 0.2
X3 ?4
0 0.9
1 0.1
X1 X3 ?3
0 0 1.1
0 1 23
1 0 0
1 1 0.7
2 0 2.7
2 1 42
?i1M ?i2,0,0,0 ?i1M ?i2,1,0,0 0.9/0.2
10/33.2
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Previous SLS algorithms for MPE

• Iterative Conditional Modes Besag, 86
• Just greedy hill climbing
• Stochastic Simulation
• Sampling algorithm, very poor for optimization
• Greedy Stochastic Simulation Kask Dechter,
  99
• Outperforms the above simulated annealing by
  orders of magnitude
• Guided Local Search (GLS) Park 02
• (Iterated Local Search (ILS) Hutter 04)
• Outperforms Greedy Stochastic Simulation by
  orders of magnitude

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Guided Local Search (GLS) Voudouris 1997

• Subclass of Dynamic Local Search Hoos
  Stützle, 2004Iteratively1) Local search !
  local optimum2) Modify evaluation function

• In local optima penalize some solution features
• Solution features for MPE are partial assigments
• Evaluation fct. Objective fct. - sum of
  respective penalties
• Penalty update rule experimentally designed
• Performs very well across many problem classes

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GLS for MPE Park 2002

• Initialize penalties to 0
• Evaluation function
• Obj. function - sum of penalties of current
  instantiation
• ?i1m ?ix1,...,xn - ?i1p ?ix1,...,xn
• In local optimum
• Choose partial instantiations (according to GLS
  update rule)
• Increment their penalty by 1
• Every N? local optima
• Smooth all penalties by multiplying them with ? lt
  1
• Important to eventually optimize the original
  objective function

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Outline

• Most probable explanation (MPE) problem
• Problem definition
• Previous work
• SLS algorithms for MPE
• Illustration
• Previous SLS algorithms
• Guided Local Search (GLS) in detail
• From Guided Local Search to GLS
• Modifications
• Performance gains
• Comparison to state-of-the-art

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GLS ! GLSOverview of modified components

• Modified evaluation function
• Pay more attention to the actual objective
  function
• Improved caching of evaluation function
• Straightforward adaption from SAT caching schemes
• Tuning of smoothing parameter ?
• Over two orders of magnitude improvement !
• Initialization with Mini-Buckets instead of
  random
• Was shown to perform better by Kask Dechter,
  1999

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GLS ! GLS (1)Modified evaluation function

• GLS
• ?i1m ?ix1,...,xn - ?i1p ?ix1,...,xn
• Product of entries minus sum of penalties¼ zero
  minus sum of penaltiesAlmost neglecting
  objective function
• GLS
• ?i1m log(?ix1,...,xn) - ?i1p ?ix1,...,xn
• Use logarithmic objective function
• Very simple, but much better results
• Penalties are now just new temporary factors
  that decay over time!
• Could be improved by dynamic weighting of the
  penalties

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GLS ! GLS (1) Modified evaluation function

• Much faster in early stages of the search
• Speedups of about 1 order of magnitude

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GLS ! GLS (2)Speedups by caching

• Time complexity for a single best-improvement
  step
• Previously best caching ?(V DV ?V)
• Improved caching ?(Vimproving DV)

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GLS ! GLS (3)Tuning the smoothing factor ?

• Park 02 stated GLS to have no parameters
• Changing ? from Parks setting 0.8 to 0.99
• Sometimes from unsolvable to milliseconds
• Effect increases for large instances

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GLS ! GLS (4)Initialization with Mini-Buckets

• Sometimes a bit worse, sometimes much better
• Particularly helps for some structured instances

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Outline

• Most probable explanation (MPE) problem
• Problem definition
• Previous work
• SLS algorithms for MPE
• Illustration
• Previous SLS algorithms
• Guided Local Search (GLS) in detail
• From Guided Local Search to GLS
• Modifications
• Performance gains
• Comparison to state-of-the-art

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Comparison based on Marinescu, Kask, Dechter,
UAI 03

• Branch Bound with MB heuristic was
  state-of-the-art for MPE, even for random
  instances!
• Scales better than original GLS with
• Number of variables
• Domain size
• Both as anytime algorithm and in terms of time
  needed to find optimum
• On the same problem instances, we show that our
  new GLS scales better than their implementation
  with
• Number of variables
• Domain size
• Density
• Induced width

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Benchmark instances

• Randomly generated Bayes nets
• Graph structure completely random/grid networks
• Controlled number of variables domain size
• Random networks with controlled induced width
• Bayesian networks from Bayes net repository

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Original GLS vs. BB with MB heuristic relative
solution quality after 100 secondsfor random
grid networks of size NxN
A
Large
Small
Medium
A
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GLS vs. GLS and BB with MB heuristic relative
solution quality after 100 secondsfor random
grid networks of size NxN
Large
Small
Medium
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GLS vs. BB with MB heuristic Solution time
with increasing domain size on random networks
Large
Medium
Small
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Solution times with increasing induced width on
random networks
d-BBMB
s-BBMB
Orig GLS
GLS
A
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Results for Bayes net repository

• GLS shows overall best performance
• Only algorithm to solve Link network (in 1
  second!)
• Problems for Barley and especially Diabetes
• Preprocessing with partial variable elimination
  helps a lot
• Can reduce (variables) dramatically

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Conclusions

• SLS algorithms are competitive for MPE solving
• Scale very well, especially with induced width
• But they need careful design, analysis
  parameter tuning
• SLS and Machine Learning (ML) people should talk
• SLS can perform very well for some traditional ML
  problems
• Our C source code is online
• Please use it ?
• Theres also a Matlab interface

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Extensions in progress

• Real problem domains
• MRFs for stereo vision
• CRFs for sketch recognition
• Domain-dependent extensions
• Hierarchical SLS for problems in computer vision
• Automated parameter tuning
• Use Machine Learning to predict runtime for
  different settings of algorithm parameters
• Use parameter setting with lowest predicted
  runtime

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The End

• Thanks to
• Holger Hoos Thomas Stützle
• Radu Marinescu for their BB code
• You for your attention ?