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Neural heuristics for Problem Solving

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### Neural heuristics for Problem Solving

Siena

Outline

- Problem Solving
- Heuristic Search
- Optimal Search and Admissibility

- Neural heuristics for Problem Solving
- Architecture
- Asymmetric Regression
- Dataset Generation

- Likely-admissibility
- Multiple neural heuristics

Marco Ernandes - email: [email protected]

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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:[email protected]

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:[email protected]

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.
- Non-admissible search is polynomial.

Marco Ernandes - email:[email protected]

Our idea about admissible search

- 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:[email protected]

- Problem Solving
- Heuristic Search
- Optimal Search and Admissibility

- Neural heuristics for Problem Solving
- Architecture
- Asymmetric Regression
- Dataset Generation

- Likely-admissibility
- Multiple neural heuristics

Marco Ernandes - email:[email protected]

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Neural heuristicsWe used standard MLP networks.

ONLINE PHASE

OFFLINE PHASE

h(n)

Marco Ernandes - email:[email protected]

– 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:[email protected]

– 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:[email protected]

– 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:[email protected]

– 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:[email protected]

– Neural heuristics –Are sub-symbolic heuristics “online”?

- 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:[email protected]

- Problem Solving
- Heuristic Search
- Optimal Search and Admissibility

- Neural heuristics for Problem Solving
- Architecture
- Asymmetric Regression
- Dataset Generation

- Likely-admissibility
- Multiple neural heuristics

Marco Ernandes - email:[email protected]

Likely-Admissible Search

- 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:[email protected]

– 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 xX.
- ph shall be the probability of optimally solving a problem using h and A*.

- TO ESTIMATE OPTIMALITY FROM ADMISSIBILITY:

Marco Ernandes - email:[email protected]

– Likely-Admissible Search –Multiple Heuristics

- 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:[email protected]

– Likely-Admissible Search –Optimality prediction

- 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:[email protected]

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:[email protected]

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