Accuracy of Admissible Heuristic Functions in Selected Planning Domains Malte Helmert Robert Mattm¨ uller Albert-Ludwigs-Universit¨ at Freiburg, Germany AAAI 2008
Introduction Analyses Summary and Conclusion Outline Introduction 1 Analyses 2 Summary and Conclusion 3
Introduction Analyses Summary and Conclusion Motivation Goal: Develop efficient optimal planning algorithms Subgoal: Find accurate admissible heuristics How to assess the accuracy of an admissible heuristic? Most common approach Run planners on benchmarks and count node expansions. Drawback: Only comparative statements Alternative approach Analytical comparison to optimal heuristic on benchmark domains Advantage: Absolute statements, theoretical limitations
Introduction Analyses Summary and Conclusion Scope of our analysis Considered heuristics h + : optimal plan length for delete relaxation h k : cost of most costly size- k goal subset (roughly) h PDB : pattern database heuristics h PDB add : additive pattern database heuristics Reference point: optimal plan length h ∗ Considered planning domains Gripper , Logistics , Blocksworld , Miconic-Strips , Miconic-Simple-Adl , Schedule , Satellite
Introduction Analyses Summary and Conclusion Domains: Gripper initial state goal state
Introduction Analyses Summary and Conclusion Domains: Blocksworld initial state goal state
Introduction Analyses Summary and Conclusion Domains: Miconic-Strips , Miconic-Simple-Adl initial state goal state
Introduction Analyses Summary and Conclusion Asymptotic accuracy Definition Let D be a planning domain (family of planning tasks). A heuristic h has asymptotic accuracy α ∈ [0 , 1] on D iff h ( s ) ≥ αh ∗ ( s ) + o ( h ∗ ( s )) for all initial states s of tasks in D , and h ( s ) ≤ αh ∗ ( s ) + o ( h ∗ ( s )) for all initial states s of an infinite subfamily of D with unbounded h ∗ ( s ) If solution lengths in D are unbounded, there is exactly one such α for a given heuristic and domain. We write it as α ( h, D ) .
Introduction Analyses Summary and Conclusion Outline Introduction 1 Analyses 2 Summary and Conclusion 3
Introduction Analyses Summary and Conclusion Delete relaxation Considered heuristics h + : optimal plan length for delete relaxation h k : cost of most costly size- k goal subset (roughly) h PDB : pattern database heuristics h PDB add : additive pattern database heuristics
Introduction Analyses Summary and Conclusion Delete relaxation: Blocksworld Example ( Blocksworld ) Lower bound: m = number of blocks touched in optimal plan h ∗ ( s ) ≤ 4 m, h + ( s ) ≥ m ⇒ α ( h + , Blocksworld ) ≥ 1 / 4 Upper bound: B 1 B 1 B n B n B n +1 B n +2 B n +1 B n +2 h ∗ ( s n ) = 4 n − 2 , h + ( s n ) = n + 1 ⇒ α ( h + , Blocksworld ) ≤ 1 / 4 � α ( h + , Blocksworld ) = 1 / 4
Introduction Analyses Summary and Conclusion Delete relaxation: Blocksworld Example ( Blocksworld ) Lower bound: m = number of blocks touched in optimal plan h ∗ ( s ) ≤ 4 m, h + ( s ) ≥ m ⇒ α ( h + , Blocksworld ) ≥ 1 / 4 Upper bound: B 1 B 1 B n B n B n +1 B n +2 B n +1 B n +2 h ∗ ( s n ) = 4 n − 2 , h + ( s n ) = n + 1 ⇒ α ( h + , Blocksworld ) ≤ 1 / 4 � α ( h + , Blocksworld ) = 1 / 4
Introduction Analyses Summary and Conclusion Delete relaxation: Blocksworld Example ( Blocksworld ) Lower bound: m = number of blocks touched in optimal plan h ∗ ( s ) ≤ 4 m, h + ( s ) ≥ m ⇒ α ( h + , Blocksworld ) ≥ 1 / 4 Upper bound: B 1 B 1 B n B n B n +1 B n +2 B n +1 B n +2 h ∗ ( s n ) = 4 n − 2 , h + ( s n ) = n + 1 ⇒ α ( h + , Blocksworld ) ≤ 1 / 4 � α ( h + , Blocksworld ) = 1 / 4
Introduction Analyses Summary and Conclusion The h k heuristic family Considered heuristics h + : optimal plan length for delete relaxation h k : cost of most costly size- k goal subset (roughly) h PDB : pattern database heuristics h PDB add : additive pattern database heuristics
Introduction Analyses Summary and Conclusion The h k heuristic family α ( h k , D ) = 0 for all considered domains Proof idea. There are families of states ( s n ) n ∈ N with h ∗ ( s n ) ∈ Ω( n ) and h k ( s n ) ∈ O ( k ) .
Introduction Analyses Summary and Conclusion The h k heuristic family Example ( Blocksworld ) B 1 B 2 B 3 . . . . . . B 1 B 2 B 3 B n B n h ∗ ( s n ) = 2 n − 2 , h k ( s n ) ≤ 2 k � α ( h k , Blocksworld ) = 0
Introduction Analyses Summary and Conclusion Non-additive pattern database heuristics Considered heuristics h + : optimal plan length for delete relaxation h k : cost of most costly size- k goal subset (roughly) h PDB : pattern database heuristics h PDB add : additive pattern database heuristics Let n be the problem size. Bounded memory: database size limit O ( n k ) entries Consequently: pattern size limit O (log n ) variables
Introduction Analyses Summary and Conclusion Non-additive pattern database heuristics α ( h PDB , D ) = 0 for all considered domains Proof idea. At most O (log n ) variables in pattern ⇒ at most O (log n ) goals represented in abstraction There are families of states ( s n ) n ∈ N with h ∗ ( s n ) ∈ Ω( n ) and h PDB ( s n ) ∈ O (log n ) .
Introduction Analyses Summary and Conclusion Additive pattern database heuristics Considered heuristics h + : optimal plan length for delete relaxation h k : cost of most costly size- k goal subset (roughly) h PDB : pattern database heuristics h PDB add : additive pattern database heuristics Let n be the problem size. Bounded memory: overall database size limit O ( n k ) entries Consequently: size limit O (log n ) variables for each pattern
Introduction Analyses Summary and Conclusion Additive pattern database heuristics: Miconic-Strips Example ( Miconic-Strips ) Lower bound: f 2 n m passengers, singleton pattern for each passenger: h ∗ ( s ) ≤ 4 m, h PDB add ( s n ) = 2 m f 2 n − 1 ⇒ α ( h PDB add , Miconic-Strips ) ≥ 1 / 2 Upper bound: Optimal additive PDB: f 4 { elev , pass 1 , . . . , pass K } ( K ∈ O (log n ) ) f 3 { pass K +1 } , . . . , { pass n } f 2 � h ∗ ( s n ) = 4 n, h PDB add ( s n ) = 2 n + 2 K f 1 ⇒ α ( h PDB add , Miconic-Strips ) ≤ 1 / 2 f 0 init α ( � h PDB add , Miconic-Strips ) = 1 / 2
Introduction Analyses Summary and Conclusion Additive pattern database heuristics: Miconic-Strips Example ( Miconic-Strips ) Lower bound: f 2 n m passengers, singleton pattern for each passenger: h ∗ ( s ) ≤ 4 m, h PDB add ( s n ) = 2 m f 2 n − 1 ⇒ α ( h PDB add , Miconic-Strips ) ≥ 1 / 2 Upper bound: Optimal additive PDB: f 4 { elev , pass 1 , . . . , pass K } ( K ∈ O (log n ) ) f 3 { pass K +1 } , . . . , { pass n } f 2 � h ∗ ( s n ) = 4 n, h PDB add ( s n ) = 2 n + 2 K f 1 ⇒ α ( h PDB add , Miconic-Strips ) ≤ 1 / 2 f 0 init α ( � h PDB add , Miconic-Strips ) = 1 / 2
Introduction Analyses Summary and Conclusion Additive pattern database heuristics: Miconic-Strips Example ( Miconic-Strips ) Lower bound: f 2 n m passengers, singleton pattern for each passenger: h ∗ ( s ) ≤ 4 m, h PDB add ( s n ) = 2 m f 2 n − 1 ⇒ α ( h PDB add , Miconic-Strips ) ≥ 1 / 2 Upper bound: Optimal additive PDB: f 4 { elev , pass 1 , . . . , pass K } ( K ∈ O (log n ) ) f 3 { pass K +1 } , . . . , { pass n } f 2 � h ∗ ( s n ) = 4 n, h PDB add ( s n ) = 2 n + 2 K f 1 ⇒ α ( h PDB add , Miconic-Strips ) ≤ 1 / 2 f 0 init α ( � h PDB add , Miconic-Strips ) = 1 / 2
Introduction Analyses Summary and Conclusion Outline Introduction 1 Analyses 2 Summary and Conclusion 3
Introduction Analyses Summary and Conclusion Summary of results Asymptotic accuracy h + h k h PDB h PDB Domain add 2 / 3 0 0 2 / 3 Gripper Logistics 3 / 4 0 0 1 / 2 1 / 4 0 0 0 Blocksworld Miconic-Strips 6 / 7 0 0 1 / 2 3 / 4 0 0 0 Miconic-Simple-Adl Schedule 1 / 4 0 0 1 / 2 Satellite 1 / 2 0 0 1 / 6
Introduction Analyses Summary and Conclusion Summary and conclusion Method: Analytical comparison of domain-specific accuracy of the heuristics h + , h k , h PDB , h PDB add Results: h + : usually most accurate (but NP-hard to compute in general) h k , h PDB : arbitrarily inaccurate h PDB add : good accuracy/effort trade-off (but how to determine a good pattern collection?) Future work: additive h k explicit-state abstraction heuristics
Introduction Analyses Summary and Conclusion The end Thank you for your attention!
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