The Phase Transition in Heuristic Search J. Christopher Beck - - PowerPoint PPT Presentation

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The Phase Transition in Heuristic Search J. Christopher Beck - - PowerPoint PPT Presentation

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The Phase Transition in Heuristic Search J. Christopher Beck Department of Mechanical & Industrial Engineering University of Toronto Canada jcb@mie.utoronto.ca PlanSOpt Workshop, ICAPS2017 University of Toronto June 19, 2017 Mechanical

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University of Toronto Mechanical & Industrial Engineering

The Phase Transition in Heuristic Search

  • J. Christopher Beck

Department of Mechanical & Industrial Engineering University of Toronto Canada jcb@mie.utoronto.ca

PlanSOpt Workshop, ICAPS2017 June 19, 2017

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University of Toronto Mechanical & Industrial Engineering 2

Nothing is as good as it used to be, and it never was. The “golden age of sports,” the golden age of anything, is the age of everyone’s childhood.

  • Ken Dryden, “The Game”

The lack of interest, the distain for history is what makes computing not-quite-a-field.

  • Alan Kay, Dr. Dobbs,

July 10, 2012 Corollary: The best papers are the ones we read during grad school.

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Outline

  • The Phase Transition

– aka Flashback to the 1990s

  • The Phase Transition in Heuristic Search

– An abstract model and benchmark problems

  • The Effect of Operator Cost Ratio
  • Next Steps

– Heavy-Tails and Local Minima?

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Where the Hard Problems Are

  • While NP problems are worst-case

exponential to solve, often typical instances are practically solvable

  • Q: What is the distribution of the

empirically hard instances?

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Graph Coloring

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[Cheeseman et al. 1991] IJCAI, 1991.

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Graph Coloring

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[Cheeseman et al. 1991] IJCAI, 1991.

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Conjectures

  • All NP-complete problems have an “order

parameter” (TSP, CSP, SAT, HC, ...)

  • A critical value of the order

parameter separates regions

  • f under-constrained and
  • ver-constrained problem instances
  • The hard problem instances are found

around this critical value

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[Cheeseman et al. 1991] IJCAI, 1991.

The Phase Transition

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Random 3-SAT

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[Crawford & Auton 1996] AIJ, 81, 31-57, 1996.

Clause/variable ratio % Solubility and Normalized difficulty

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Why Do We Care?

  • A lot of recent interest in understanding

the difficulty of heuristic search problems

– i.e., “A*-style” state-based search

  • The phase transition has not (yet) been

shown for heuristic search problems

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Does the phase transition phenomenon play a role in problem difficulty for heuristic search?

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Some more background ...

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State-Space Search (aka “Heuristic Search”)

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s * Possible transitions h = 10 h = 5 h = 8 Path from node to goal (estimate): h = 5 Greedy Best-First Search (GBFS): choose node with minimum h

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PT in Planning

  • Randomly generate planning problems

– operators, preconditions, effects, ...

  • Bylander [AIJ 1996]

– Bounds based on goals and atoms to

  • perators ratio
  • Rintanen [KR 2004]

– Gradual transition between soluble and insoluble based on operator/variable ratio – Hampered by lack of insolubility test

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Quantified SAT (2-QSAT)

  • Gent & Walsh [AAAI 1999]

– apply theory of “constrainedness” from NP to PSPACE – PT and easy-hard-easy observed for 2-QSAT

  • nce trivially insoluble instances removed

– More convincing evidence of abrupt PT than in the planning work

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Problem Difficulty for GBFS

  • Operator cost ratio

– higher ratio ≈ more effort

  • (but see Fan et al. ICAPS2017)
  • Uninformative Heuristic Regions (UHRs)

– plateaux and local minima ≈ more effort

  • Correlation between heuristic and distance

– lower correlation ≈ more effort

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Does the phase transition phenomenon play a role in problem difficulty for heuristic search? GBFS?

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University of Toronto Mechanical & Industrial Engineering

Outline

  • The Phase Transition

– aka Flashback to the 1990s

  • The Phase Transition in Heuristic Search

– An abstract model and benchmark problems

  • The Effect of Operator Cost Ratio
  • Next Steps

– Heavy-Tails and Local Minima?

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Abstract Model

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[Cohen & B. 2017] AAAI, 780-786, 2017.

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Control Parameter

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Solubility

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Is this surprising?

Solubility: 0.1% to 99.9%

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# Nodes Expanded

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Effect of the Heuristic

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True cost to goal

A new question: What is the impact of systematically stronger heuristics?

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Effect of the Heuristic

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Soluble instances only

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Abstract Model

  • Solubility phase transition
  • Easy-hard-easy pattern

associated with PT

  • New results on the impact
  • f heuristics across PT

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Standard PT work (CP, SAT) uses an abstract model on random problems analogous to ours. What about benchmark problems?

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Benchmarks

  • Given an existing benchmark problem, we

can generate relaxed/restricted instances by adding/removing transitions

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Benchmarks

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The Pancake Problem

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[Helmert 2010] SoCS, 109-110, 2010.

Action Fk: flip top k

Solution: F5, F6, F3, F4, F5

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The Pancake Problem

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The Grid Navigation Problem

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G S

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The Grid Navigation Problem

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Similar Results

  • TopSpin
  • Towers of Hanoi
  • Interesting differences

with 8 Sliding Tile Puzzle due to disconnected search space

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Effect of Heuristic (8-Pancake)

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So ...

  • Phase transition and easy-hard-easy

patterns exist in GBFS for both abstract model and benchmark problems

  • Heuristics of systematically increasing

strengths show radically different performance across the phase transition

– Lowest improvement on hardest problems

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What about existing ideas about problem difficulty in heuristic search?

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Outline

  • The Phase Transition

– aka Flashback to the 1990s

  • The Phase Transition in Heuristic Search

– An abstract model and benchmark problems

  • The Effect of Operator Cost Ratio
  • Next Steps

– Heavy-Tails and Local Minima?

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Operator Cost Ratio

  • [Wilt & Ruml 2011]

– Instances are far more difficult with non-unit costs despite the same connection structure

  • [Cushing et al. 2011]

– Cost variance fundamentally misleads heuristic search

  • [Fan et al. 2017]

– No Free Lunch Theorem for Dijkstra’s Alg.

  • Negative effects are balanced by positive effects in
  • ther cost functions

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Operator Cost Ratio and the PT

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[Cohen & B. 2017] SoCS, in press, 2017.

What is the impact of the operator cost ratio on problem difficulty across relaxed/ restricted benchmark problems?

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Grid Navigation

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Grid Navigation

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Pancake Problem

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[Helmert 2010] SoCS, 109-110, 2010.

Action Fk: flip top k

  • Cost = zm

– z: size of the bottom pancake in flipped sub-pile

  • For the 8-Pancake problem the operator

cost ratio is 8m

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Pancake Problem

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TopSpin

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[Wilt & Ruml 2014] for TopSpin, sometimes higher operator cost ratio is better

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Operator Cost Ratio and the PT

  • Impact of higher operator cost ratio follows

a low-high-low pattern, peaking in the PT

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Outline

  • The Phase Transition

– aka Flashback to the 1990s

  • The Phase Transition in Heuristic Search

– An abstract model and benchmark problems

  • The Effect of Operator Cost Ratio
  • Next Steps

– Heavy-Tails and Local Minima?

41

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The Pancake Problem

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Pancake Problem (Median)

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Pancake Problem

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“Exceptionally hard problems (ehps)” [Gent & Walsh 1994]

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Exceptionally Hard Problems

  • Very hard problems in underconstrained

regions of the PT

  • Not inherently hard problems

– Combination of problem structure and algorithm details

  • Heavy-tailed distributions

– Performance of randomized heuristic follows a heavy-tailed distribution

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[Smith & Grant 1997] CP, 182-195, 1997. [Gomes et al. 2005] Constraints, 10, 317-337, 2005.

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Heavy-Tailed Runtime Distributions

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log frequency

  • f a

solution log of a search effort

[Gomes et al. 1998] AAAI, 431–437, 1998.

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Failed Sub-trees and Local Minima

  • Failed sub-tree (CSP)

– A sub-tree with no solutions – If entered (e.g. by depth-first search) needs to be exhaustively searched

  • Local Minima (heuristic search)

– [Wilt & Ruml 2014] – A region that does not contain the goal but that the search will have to exhaust if it enters – Connected with difficulty due to higher

  • perator cost ratio

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Heavy-tails occurs when depths of failed sub-trees are exponentially distributed [Gomes et al. 2005]

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Problem Difficulty for GBFS

  • Operator cost ratio

– higher ratio ≈ more effort

  • (but see Fan et al. ICAPS2017)
  • Uninformative

Heuristic Regions (UHRs)

– plateaux and local minima ≈ more effort

  • Correlation between heuristic and distance

– lower correlation ≈ more effort

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Associated with size/extent of local minima [Wilt & Ruml 2014] Impacted by phase transition Connection with exceptionally hard problems and heavy tails? Connection between local minima and PT?

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So What Have We Done?

  • Showed that the phase transition

phenomenon from combinatorial search can be observed in heuristic search

  • Showed an (empirical) relation between

PT and problem hardness

– Both unit-cost problems and when varying operator cost ratio

  • Showed the existence of ehps for GBFS

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Conjectures

  • The size and extend of local minima is

effected by the phase transition

  • The analysis of problem difficulty based on

heavy-tailed distributions (in CSPs) can be imported into heuristic search

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Science requires a society because even people who are trying to be good thinkers love their own thoughts and theories – much

  • f the debugging has to be done by others.