Neural heuristics for Problem Solving

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10-09-2004 Siena Neural heuristics for Problem Solving Outline Problem Solving Heuristic Search Optimal Search and Admissibility Neural heuristics for Problem Solving Architecture Asymmetric Regression Dataset Generation Likely-admissibility Multiple neural heuristics

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10-09-2004

Siena

### Neural heuristics for Problem Solving

Outline

• Problem Solving
• Heuristic Search
• Neural heuristics for Problem Solving
• Architecture
• Asymmetric Regression
• Dataset Generation
• Multiple neural heuristics

Marco Ernandes - email: ernandes@dii.unisi.it

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Problem Solving
• PS is a decision-making process that aims to find a sequence of operators that takes the agent from a given state to a goal state.
• We talk about: states, successor function (defines the operators available at each state), initial state, goal state, problem space.

Marco Ernandes - email:ernandes@dii.unisi.it

Heuristic Search
• Search algorithms (best-first, greedy, …): define a strategy to investigate the search-tree.
• Heuristic information h(n): typically the distance from node n (of the search-tree)to goal
• Heuristic usage policy : how to combine h(n) and g(n) to obtain f(n).

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Marco Ernandes - email:ernandes@dii.unisi.it

Optimal Heuristic Search
• GOAL: retrieve the solution that minimizes the path cost (sum of all the operator costs) C=C*
• Requires:
• an optimal search algorithm (A*, IDA*, BS*)
• an admissible heuristic, h(n)h*(n) (i.e. Manhattan)
• an admissible heuristic usage (f(n) =h(n) + g(n) , )
• Complexity:
• Optimal solving: any puzzle is NP-Hard.

Marco Ernandes - email:ernandes@dii.unisi.it

• The best performance in literature: memory-based heuristics (Disjoint Pattern DBs, <Korf&Taylor, 2002>)
• Offline phase: resolution of all possibile subproblems and storage of all the results.
• Online phase: decomposition of a node in subproblems and database querying.
• Our idea:
• memory-based heuristics have little future because they only shift NP-completeness from time to space.
• ANN (as universal approx.) can provide effective non-memory-based, “nearly” admissible heuristics.

Marco Ernandes - email:ernandes@dii.unisi.it

Outline:

• Problem Solving
• Heuristic Search
• Neural heuristics for Problem Solving
• Architecture
• Asymmetric Regression
• Dataset Generation
• Multiple neural heuristics

Marco Ernandes - email:ernandes@dii.unisi.it

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Neural heuristics

We used standard MLP networks.

ONLINE PHASE

OFFLINE PHASE

h(n)

Marco Ernandes - email:ernandes@dii.unisi.it

– Neural heuristics –Outputs, Targets & Entrances
• It’s a regression problem, hence we used 1 linear output neuron (modified a posteriori exploiting information from Manhattan-like heuristics).
• 2 possible targets:
• A) “direct” target function  o(x) = h*(x)
• B) “gap” target  o(x) = h*(x)-hM(x)

(which takes advantage of Manhattan too)

• Entrances coding:
• we tried 3 different vector-valued codings
• future work: represent configurations as graphs, in order to have non-dimension-dependent learning. (i.e. exploit learning from 15-puzzle, in the 24-puzzle).

Marco Ernandes - email:ernandes@dii.unisi.it

– Neural heuristics –Learning Algorithm
• Normal backpropagation algorithm, but …
• Introducing a coefficient of asymmetry in the error function. This stresses admissibility:
• Ed = (1-w) (od –td) if (od –td) < 0
• Ed = (1+w) (od –td) if (od –td) > 0
• We used a dynamic decreasing w, in order to stress underestimations when learning is simple and to ease it successively. Momentum a=0,8 helped smoothness.

with 0 < w < 1

Marco Ernandes - email:ernandes@dii.unisi.it

– Neural heuristics –Asymmetric Regression
• This is a general idea for backprop learning.
• It can suit any regression problem where overestimations harm more than underestimations (or contrary).
• Heuristic machine learning is an ideal application field.
• We believe that totally admissible neural heuristics are theoretically impossible, or at least impracticable.

Symmetric error

Asymmetric error

Marco Ernandes - email:ernandes@dii.unisi.it

– Neural heuristics –Dataset Generation
• Examples are previously optimally solved configurations. It seems a big problem, but …
• Few examples are sufficient for good learning. A few hundreds to have faster search than Manhattan. We used a training set of 25000 to (500 million times smaller than the problem space).
• These examples are mainly “easy” ones, over 60% of 15-puzzle examples have d < 30, whereas only 0,1% of random cases have d < 30 [see 15-puzzle search tree distribution].
• All the process is fully parallelizable.
• Further works: auto-feed learning.

Marco Ernandes - email:ernandes@dii.unisi.it

• We believe so. Even that there is an offline learning phase. For 2 reasons:
• 1. Nodes visited during search are generally UNSEEN.
• Exactly like often humans do with learned heuristics: we don’t recover a heuristic value from a database, we compute it employing the inner rules that the heuristic provides.
• 2. The learned heuristic should be dimension-independent: learning over small problems could be used for bigger problems (i.e. 8-puzzle  15-puzzle). This is not possible with mem-based heuristics.

Marco Ernandes - email:ernandes@dii.unisi.it

Outline:

• Problem Solving
• Heuristic Search
• Neural heuristics for Problem Solving
• Architecture
• Asymmetric Regression
• Dataset Generation
• Multiple neural heuristics

Marco Ernandes - email:ernandes@dii.unisi.it

• We relax the optimality requirement in a probabilistic sense (not qualitatively like e-admissible search).
• Why is it a better approach than e-admissibility?
• It allows to retrieve TRULY OPTIMAL solutions.
• It still allows to change the nature of search complexity.
• Because search can rely on any heuristic, unlike e-admissible search that works only on already-proven-admissible ones.
• Because we can better combine search with statistical machine learning techniques. Using universal approximators we can automatically generate heuristics.

Marco Ernandes - email:ernandes@dii.unisi.it

– Likely-Admissible Search –A statistical framework
• One requisite: to have a previous statistical analysis of overestimation frequencies of our h.
• P(h\$) shall be the probability that heuristic h underestimates h* for any given state xX.
• ph shall be the probability of optimally solving a problem using h and A*.
• TO ESTIMATE OPTIMALITY FROM ADMISSIBILITY:

Marco Ernandes - email:ernandes@dii.unisi.it

• To enrich the heuristic information we can generate many heuristics and use them simultaneously, as:
• Thus:
• If we will consider for simplicity that all j heuristics have the same given P(h+2):

j grows logarithmically with d and pH

Marco Ernandes - email:ernandes@dii.unisi.it

• Unfortunately the last equation is very optimistic since it assumes a total error independency among neural heuristics.
• For predicions we have to use which is:
• Extremely precise for optimality over 80%.
• Imprecise for low predictions.
• Predictions are much more accurate than e-admissible search predictions.

Marco Ernandes - email:ernandes@dii.unisi.it

Experimental Results & Demo
• Compared to Manhattan:
• IDA* with 1 ANN (optimality  30%): 1/1000 execution time, 1/15000 nodes visited
• IDA* with 2 ANN (opt.  50%): 1/500 time, 1/13000 nodes.
• IDA* with 4 ANN-1 (opt.  90%): 1/70 time, 1/2800 nodes.
• Compared to DPDBs:
• IDA* with 1 ANN (optimality  30%): between -17% and +13% nodes visited, between 1,4 and 3,5 times slower
• IDA* with 2 ANN (opt.  50%): -5% nodes visited, 5 times slower (but this could be parallelized completely!)

Try the “demo” at: http://www.dii.unisi.it/ ~ernandes/samloyd/

Marco Ernandes - email:ernandes@dii.unisi.it