Empirical Evaluation of Anytime Weighted AND/OR Best-First Search - - PowerPoint PPT Presentation

Empirical Evaluation of Anytime Weighted AND/OR Best-First Search for MAP

joint work with

Radu Marinescu and Rina Dechter

Summary

 Best first search (A*) is *best* combinatorial optimization

algorithm, but is not anytime. (First solution at termination

• nly). It also requires excessive memory, and therefore rarely

used for Graphical models.

 Depth-first branch-and-bound is less effective, but is

anytime and requires far less memory. It is the main search scheme for Graphical models.

 Recent work in path-finding heuristic search showed that

weighted A* facilitates anytime scheme with accuracy guarantees.

 Weighted A* finds approximate solution faster [Pohl 1970]  Anytime A* finds an approximate solution and improves it over time [Hansen, Zhou 2007]. Other weighted anytime best first schemes [Likhachev et al. 2003, Richter et al. 2010, van den Berg et al. 2011]

Summary: our work

 we adapt the weighted anytime best first search to

AND/OR search spaces. Specifically,

 We investigate the extensions of AOBF to anytime

schemes and compare with the most effective anytime scheme to date: Breadth-Rotating AOBB (BRAOBB).

 We also investigate the theoretical properties of the

search space explored by a weighted Best First search.

 Ongoing work: investigation of the potential of

Weighted Branch and Bound

Background: Weighted A*

A* search

• admissible heuristic
• evaluation function:

f(n)=g(n)+h(n)

• guaranteed optimal

solution, cost C*

Weighted A* search

• non-admissible heuristic
• Evaluation function:

f(n)=g(n)+w∙h(n)

• Guaranteed w-optimal

solution, cost C ≤ w∙C*

s t s t

explored search space smaller (hopefully) explored search space

Background: AND/OR Best First (AOBF)

[Marinescu, Dechter 2009]

 Time, space worst-case time complexity:  AND/OR tree:  AND/OR graph:  Mini-bucket heuristics are known to be efficient for

AND/OR search [Kask&Dechter 1999, Marinescu&Dechter, 2009]

 The i-bound parameter flexibly controls accuracy  Extreme case: Bucket Elimination produces exact

heuristic [Dechter 1999]

) (

h

k N O 

) (

* w

k m N O  

Background: BRAOBB [Otten, Dechter 2011]

 OR Branch-and-Bound is anytime.  But AND/OR breaks anytime behavior

• f depth-first scheme:

 First anytime solution delayed until

last subproblem starts processing

 Breadth-Rotating AOBB:

 Take turns processing subproblems.  Solve each subproblem depth-first.

rotate

Our contribution

 We adapted for the AND/OR search space 3 existing

anytime best first search schemes:

 ARA* [Likhachev, M.; Gordon, G.; and Thrun, S. NIPS’03]  ANA* [van den Berg, J.; Shah, R.; Huang, A.; and Goldberg, K. AAAI’11]  Anytime AO* [Bonet, B, Geffner H. AAAI’12]

 We proposed 3 original schemes:

 a simple iterative anytime weighted Best First scheme

wAOBF

 2 hybrid schemes that interleave Depth First search and

Best First search, using some ideas of ANA*

Anytime weighted AOBF (wAOBF)

 Weighted AOBF:

Run ordinary AOBF with evaluation function f(n)=g(n)+w∙h(n)

 Anytime weighted AOBF (wAOBF):

 ( is similar to Restarting weighted A* [Richter et al. 2010] )

 Consider some starting weight w  until w=1 or out of time

 with current weight w run Weighted AOBF from scratch to completion  output the solution found by Weighted A*  Decrease w by fixed positive value δ

Theorem: The cost of each solution is bounded by current weight:

C ≤ w ∙ C*

[Pohl 1970] a lot of computation is repeated -> wasteful!

Anytime Repairing AOBF (wR-AOBF)

 wAOBF repeats a lot of computations at each

iteration – wasteful!

 Solution: reuse results of previous iterations

Keep track of the partially explored AND/OR graph After w decreases update evaluation for all nodes

whose f(n) changed with new weight, bottom-up from leaves to the root

Identify a new best partial solution tree and

continue search

Anytime Nonparametric AOBF (wN-AOBF)

 how to choose weight? nobody knows, always done ad hoc  wN-AOBF doesn’t use input parameters  Main Idea:

 assign to each node n a function e(n)=(C- g(n))/h(n). where C is the

current best solution

 e(n) is equal to the maximum value of w such that f(n) ≤ C, so the

algorithm improves the solution as greedily as possible, automatically adapting the implicit value of w as the path quality increases.

 Original ANA* at each step expanded the node in OPEN with

max e(n). However, AOBF-style algorithms don’t keep explicit OPEN list and the implementation is more involved

Anytime Stochastic AOBF (p-AOBF)

 Main idea:

allow search to also expand nodes that may not

be on the optimal path

 Specifically:

at each step it expands a tip node that does not

belong to the current best partial solution graph with a fixed probability (1 − p)

 Parameter 0 ≤ p ≤1 allows a trade off between

exploration and exploitation of search space

 does not provide optimality guarantees

Experiments 1: anytime stochastic AOBF

 Algorithms:

 wAOBF  p-AOBF (p=0.2)  p-AOBF (p=0.8)  BRAOBB

 3 datasets:

 16 pedigree networks  17 binary grids  20 protein instances

 time limit - 1 hour  memory limit - 2 Gb

MPE task – higher values are better! Simple weight schedule substract(0.1): wi+1 = wi-0.1 w1=32

Experiments 1: anytime stochastic AOBF

Experiments 2: anytime nonparametric AOBF

 Algorithms:

 wAOBF  p-AOBF (p=0.5)  wN-AOBF  wR-AOBF  BRAOBB

 3 datasets:

 16 pedigree networks  17 binary grids  20 protein instances

 time limit - 1 hour  memory limit - 2 Gb

Simple weight schedule substract(0.1): wi+1 = wi-0.1 w1=32

Experiments 2: anytime nonparametric AOBF

Experiments 3: impact of the weight

 Algorithms:

 wAOBF  wR-AOBF

 3 datasets:

 16 pedigree networks  17 binary grids  20 protein instances

 time limit - 1 hour  memory limit - 2 Gb

Simple weight schedule substract(0.1): wi+1 = wi-0.1 w1=4

Experiments 3: impact of the weight

Experiments 3: impact of the weight

Experiments 3: impact of the weight

Experiments 4: alternative weight schedules

Experiments 4: alternative weight schedules

 Algorithms:

 wAOBF  wR-AOBF

 3 datasets of hardest instances:

 Pedigrees  Grids  WCSP

 time limit - 1 hour  memory limit - 2 Gb

5 weight schedules

Experiments 4: alternative weight schedules

Experiments 4: alternative weight schedules

Experiments 4: alternative weight schedules

Experiments 5: large memory limit

 Algorithms:

 wAOBF  wR-AOBF  BRAOBB

 3 datasets of hardest instances:

 Pedigrees  Grids  WCSP

 time limit - 2 hour s  memory limit - 120 Gb

MPE task – higher values are better! Two best weight schedule piecewise() revsqrt() (=sqrt())

Experiments 5: large memory limit: pedigrees

Experiments 5: large memory limit: pedigrees

Experiments 5: large memory limit: WCSPs

Experiments 5: large memory limit: WCSPs

Experiments 5: large memory limit: grids

Experiments 5: large memory limit: grids

ANA-RAGGED

 Main idea:

 Maintain best partial solution tree (as usual for AOBF) + feasible

partial solution tree (constructed based on the e(n)), e(n)=(C- g(n))/h(n)

 For each node:

 q(n) – lower bound on the cost of the best solution below n  u(n) – upper bound; u(s) – upper bound on the overall optimal solution

 Expand tip nodes of the feasible tree. If it has no tip nodes, expand tip

node of best partial solution tree

 Each time a new node is expanded, best partial solution tree is

recalculated

 If a new leaf found – recalculate u(n) for best partial solution tree  If an improved solution is found, it is outputted and the feasible

partial solution tree is recalculated

ANA-RAGGED

ANA-RAGGED

 Problem: After feasible partial solution tree can

no longer be expanded and before finding a new improved solution ANA-RAGGED performs BF search, which can take a lot of time, because we

• nly update u(n) when a new leaf is found -> until

then we can’t find new solution -> can’t re- compute feasible partial solution tree

 -> long period between finding new solutions =

not so good anytime performance

ANA-SMOOTH

 Main idea:

Run similar to ANA-RAGGED After each new node expansion:

 Update u(n) values -> possible to find a new solution  e-compute feasible partial solution tree -> possible to

• btain new tip nodes and coninue depth-first dive

 Drawback:

More updates at every step

 Benefit:

More chances to find a new solution -> smoother

anytime behaviour

Experiments 6: nonparametric hybrid schemes

 Algorithms:

 wAOBF  wR-AOBF  BRAOBB

 3 datasets of hardest instances (no determinism):

 Pedigrees  Grids  WCSP

 time limit - 2 hour s  memory limit - 120 Gb

Experiments 6: nonparametric hybrid schemes

Experiments 6: nonparametric hybrid schemes

Experiments 6: nonparametric hybrid schemes

Iterative weighted AOBB (iAOBB)

 Main idea:

Very similar to wAOBF, but at each iteration run

not weighted AOBF, but weighted AOBB

 Implementation issue:

The code was solving max-sum and not min-sum

problem (unlike all other code), thus the weight should be w<1, not w>1 -> discrepancy in the weighting schemes used

Preliminary experiments: iAOBB

 Algorithms:

 iAOBB  BRAOBB

 Datasets:

 Pedigrees  Grids  WCSP  proteins

 time limit - 1 hour s  memory limit - 2 Gb

Two weight schedule piecewise() revsqrt() However, weight used is 1/w

Preliminary experiments: iAOBB, i=4

Preliminary experiments: iAOBB, i=18, i=10