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SatPlan Restart Strategies

SatPlan Restart Strategies. Henry Kautz University of Rochester. Planning. Input Description of initial state of world Description of goal state(s) Description of available actions Optional: Cost for each action Output Sequence of actions that converts the initial state into a goal state

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SatPlan Restart Strategies

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  1. SatPlanRestart Strategies Henry KautzUniversity of Rochester

  2. Planning Input • Description of initial state of world • Description of goal state(s) • Description of available actions • Optional: Cost for each action Output • Sequence of actions that converts the initial state into a goal state • May wish to minimizelength or cost of plan

  3. Classical Planning • Atomic time • Deterministic actions • Complete knowledge • No numeric reward (just goals) • Only planner changes world

  4. Route Planning State = intersection Operators = block between intersections Operator cost = length of block

  5. Blocks World Control a robot arm that can pick up and stack blocks. • Arm can hold exactly one block • Blocks can either be on the table, or on top of exactly one other block State = configuration of blocks • { (on-table G), (on B G), (clear B), (holding R) } Operator = pick up or put down a block • (put-down R) put on table • (stack R B) put on another block

  6. State Space Planning = Finding (shortest) paths in state graph put-down(R) stack(R,B) pick-up(R) pick-up(G) stack(G,R)

  7. STRIPS Representation (define (domain prodigy-bw) (:requirements :strips) (:predicates (on ?x ?y) (on-table ?x) (clear ?x) (arm-empty) (holding ?x))

  8. Problem Instance (define (problem bw-sussman) (:domain prodigy-bw) (:objects A B C) (:init (on-table a) (on-table b) (on c a) (clear b) (clear c) (arm-empty)) (:goal (and (on a b) (on b c)))) goal may be a partial description

  9. Operator Schemas (:action stack :parameters (?obj ?under_obj) :precondition (and (holding ?obj) (clear ?under_obj)) :effect (and (not (holding ?obj)) (not (clear ?under_obj)) (clear ?obj) (arm-empty) (on ?obj ?under_obj))) add effects – make true delete effects – make false

  10. Blocks World Blackbox Planner Demo

  11. Satplan • Model planning as Boolean satisfiability • (Kautz & Selman 1992): Hard structured benchmarks for SAT solvers • Pushing the envelope: planning, propositional logic, and stochastic search (1996) • Can outperform best current planning systems

  12. Satplan in 15 Seconds • Time = bounded sequence of integers • Translate planning operators to propositional schemas that assert:

  13. SAT Solvers Stochastic local search solvers (walksat) when they work, scale well cannot show unsat fail on certain domains must use very simple (fast) heuristics Systematic solvers (Davis Putnam Loveland style) complete fail on (often different) domains might use more sophisticated (costly) heuristics often to scale badly Can we combine best features of each approach?

  14. Background Combinatorial search methods often exhibit a remarkable variability in performance. It is common to observe significant differences between: - different heuristics - same heuristic on different instances - different runs of same heuristic with different seeds (stochastic methods)

  15. Preview of Strategy We’ll put variability / unpredictability to our advantage via randomization / averaging.

  16. Cost Distributions Backtrack-style search (e.g. Davis-Putnam) characterized by: I Erratic behavior of mean. II Distributions have “heavy tails”.

  17. Heavy-Tailed Distributions … infinite variance … infinite mean Introduced by Pareto in the 1920’s --- “probabilistic curiosity.” Mandelbrot established the use of heavy-tailed distributions to model real-world fractal phenomena. Examples: stock-market, earth-quakes, weather,...

  18. Decay of Distributions Standard --- Exponential Decay e.g. Normal: Heavy-Tailed --- Power Law Decay e.g. Pareto-Levy:

  19. Power Law Decay Exponential Decay Standard Distribution (finite mean & variance)

  20. How to Check for “Heavy Tails”? Log-Log plot of tail of distribution should be approximately linear. Slope gives value of infinite mean and infinite variance infinite variance

  21. Heavy Tails Bad scaling of systematic solvers can be caused by heavy tailed distributions Deterministic algorithms get stuck on particular instances but that same instance might be easy for a different deterministic algorithm! Expected (mean) solution time increases without limit over large distributions

  22. Randomized Restarts Solution: randomize the systematic solver Add noise to the heuristic branching (variable choice) function Cutoff and restart search after a fixed number of backtracks Provably Eliminates heavy tails In practice: rapid restarts with low cutoff can dramatically improve performance (Gomes, Kautz, and Selman 1997, 1998)

  23. Run-Time Distribution of Restart Strategies Independent trials:

  24. Rapid Restart on LOG.D Note Log Scale: Exponential speedup!

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