1. Searching by Constraint (Continued)
CMSC 25000
Artificial Intelligence
January 29, 2008

2. Incremental Repair Start with initial complete assignment
Use greedy approach
Probably invalid - I.e. violates some constraints
Incrementally convert to valid solution
Use heuristic to replace value that violates
“min-conflict” strategy:
Change value to result in fewest constraint violations
Break ties randomly
Incorporate in local or backtracking hill-climber

3. Incremental Repair Q2 Q4 5 conflicts Q1 Q3 Q1 Q2 Q3 Q4 Q1 Q2 Q3 Q4

4. Question
How would we apply iterative repair to Traveling Salesman Problem?

5. Iterative Improvement
Alternate formulation of CSP
Rather than DFS through partial assignments
Start with some complete, valid assignment
Search for optimal assignment wrt some criterion
Example: Traveling Salesman Problem
Minimum length tour through cities, visiting each one once

6. Iterative Improvement Example
TSP
Start with some valid tour
E.g. find greedy solution
Make incremental change to tour
E.g. hill-climbing - take change that produces greatest improvement
Problem: Local minima
Solution: Randomize to search other parts of space
Other methods: Simulated annealing, Genetic alg’s

7. Min-Conflict Effectiveness
N-queens: Given initial random assignment, can solve in ~ O(n)
For n < 10^7
GSAT (satisfiability)
Best (near linear in practice) solution uses min-conflict-type hill-climbing strategy
Adds randomization to escape local min
~Linear seems true for most CSPs
Except for some range of ratios of constraints to variables
Avoids storage of assignment history (for BT)

8. Artificial Intelligence CMSC 25000 January 29, 2008
Evolutionary Search
Artificial Intelligence
CMSC 25000
January 29, 2008

9. Agenda Motivation: Genetic Algorithms Conclusions Evolving a solution
Modelling search as evolution
Mutation
Crossover
Survival of the fittest
Survival of the most diverse
Conclusions

10. Motivation: Evolution
Evolution through natural selection
Individuals pass on traits to offspring
Individuals have different traits
Fittest individuals survive to produce more offspring
Over time, variation can accumulate
Leading to new species

11. Simulated Evolution Evolving a solution
Begin with population of individuals
Individuals = candidate solutions ~chromosomes
Produce offspring with variation
Mutation: change features
Crossover: exchange features between individuals
Apply natural selection
Select “best” individuals to go on to next generation
Continue until satisfied with solution

12. Genetic Algorithms Applications
Search parameter space for optimal assignment
Not guaranteed to find optimal, but can approach
Classic optimization problems:
E.g. Travelling Salesman Problem
Program design (“Genetic Programming”)
Aircraft carrier landings

13. Genetic Algorithm Example
Cookie recipes (Winston, AI, 1993)
As evolving populations
Individual = batch of cookies
Quality: 0-9
Chromosomes = 2 genes: 1 chromosome each
Flour Quantity, Sugar Quantity: 1-9
Mutation:
Randomly select Flour/Sugar: +/- 1 [1-9]
Crossover:
Split 2 chromosomes & rejoin; keeping both

14. Fitness Natural selection: Most fit survive
Fitness= Probability of survival to next gen
Question: How do we measure fitness?
“Standard method”: Relate fitness to quality
:0-1; :1-9:
Chromosome Quality Fitness
1 4
3 1
1 2
1 1 4 3 2 1
0.4
0.3
0.2
0.1

15. GA Design Issues Genetic design: Population: How many chromosomes?
Identify sets of features = genes; Constraints?
Population: How many chromosomes?
Too few => inbreeding; Too many=>too slow
Mutation: How frequent?
Too few=>slow change; Too many=> wild
Crossover: Allowed? How selected?
Duplicates?

16. GA Design: Basic Cookie GA
Genetic design:
Identify sets of features: 2 genes: flour+sugar;1-9
Population: How many chromosomes?
1 initial, 4 max
Mutation: How frequent?
1 gene randomly selected, randomly mutated
Crossover: Allowed? No
Duplicates? No
Survival: Standard method

17. Basic Cookie GA Results
Results are for 1000 random trials
Initial state: 1 1-1, quality 1 chromosome
On average, reaches max quality (9) in 16 generations
Best: max quality in 8 generations
Conclusion:
Low dimensionality search
Successful even without crossover

18. Basic Cookie GA+Crossover Results
Results are for 1000 random trials
Initial state: 1 1-1, quality 1 chromosome
On average, reaches max quality (9) in 14 generations
Conclusion:
Faster with crossover: combine good in each gene
Key: Global max achievable by maximizing each dimension independently - reduce dimensionality

19. Solving the Moat Problem
No single step mutation can reach optimal values using standard fitness (quality=0 => probability=0)
Solution A:
Crossover can combine fit parents in EACH gene
However, still slow: 155 generations on average 1 2 3 4 5 4 3 2 1 2 2 3 3 4 7 8 7 4 5 8 9 8 5 4 7 8 7 4 3 3 2 2 1 2 3 4 5 4 3 2 1

20. Questions How can we avoid the 0 quality problem?
How can we avoid local maxima?

21. Rethinking Fitness Goal: Explicit bias to best Solution: Rank method
Remove implicit biases based on quality scale
Solution: Rank method
Ignore actual quality values except for ranking
Step 1: Rank candidates by quality
Step 2: Probability of selecting ith candidate, given that i-1 candidate not selected, is constant p.
Step 2b: Last candidate is selected if no other has been
Step 3: Select candidates using the probabilities

22. Rank Method Chromosome Quality Rank Std. Fitness Rank Fitness 1 4 1 3
1 2
5 2
7 5 4 3 2 1 1 2 3 4 5
0.4
0.3
0.2
0.1
0.0
0.667
0.222
0.074
0.025
0.012
Results: Average over 1000 random runs on Moat problem
- 75 Generations (vs 155 for standard method)
No 0 probability entries: Based on rank not absolute quality

23. Diversity
Diversity:
Degree to which chromosomes exhibit different genes
Rank & Standard methods look only at quality
Need diversity: escape local min, variety for crossover
“As good to be different as to be fit”

24. Rank-Space Method Combines diversity and quality in fitness
Diversity measure:
Sum of inverse squared distances in genes
Diversity rank: Avoids inadvertent bias
Rank-space:
Sort on sum of diversity AND quality ranks
Best: lower left: high diversity & quality

25. Rank-Space Method W.r.t. highest ranked 5-1
Chromosome Q D D Rank Q Rank Comb Rank R-S Fitness
1 4
3 1
1 2
1 1
7 5 4 3 2 1
0.04
0.25
0.059
0.062
0.05 1 5 3 4 2 1 2 3 4 5 1 4 2 5 3
0.667
0.025
0.222
0.012
0.074
Diversity rank breaks ties
After select others, sum distances to both
Results: Average (Moat) 15 generations

26. GA’s and Local Maxima Quality metrics only: Quality + Diversity:
Susceptible to local max problems
Quality + Diversity:
Can populate all local maxima
Including global max
Key: Population must be large enough

27. GA Discussion Similar to stochastic local beam search Why crossover?
Beam: Population size
Stochastic: selection & mutation
Local: Each generation from single previous
Key difference: Crossover – 2 sources!
Why crossover?
Schema: Partial local subsolutions
E.g. 2 halves of TSP tour

28. Question Traveling Salesman Problem N-Queens
CSP-style Iterative refinement
Genetic Algorithm
N-Queens

29. Iterative Improvement Example
TSP
Start with some valid tour
E.g. find greedy solution
Make incremental change to tour
E.g. hill-climbing - take change that produces greatest improvement
Problem: Local minima
Solution: Randomize to search other parts of space
Other methods: Simulated annealing, Genetic alg’s

30. Machine Learning: Nearest Neighbor & Information Retrieval Search
Artificial Intelligence
CMSC 25000
January 29, 2008

31. Agenda Machine learning: Introduction Nearest neighbor techniques
Applications:
Credit rating
Text Classification
K-nn
Issues:
Distance, dimensions, & irrelevant attributes
Efficiency:
k-d trees, parallelism

32. Machine Learning
Learning: Acquiring a function, based on past inputs and values, from new inputs to values.
Learn concepts, classifications, values
Identify regularities in data

33. Machine Learning Examples
Pronunciation:
Spelling of word => sounds
Speech recognition:
Acoustic signals => sentences
Robot arm manipulation:
Target => torques
Credit rating:
Financial data => loan qualification

34. Complexity & Generalization
Goal: Predict values accurately on new inputs
Problem:
Train on sample data
Can make arbitrarily complex model to fit
BUT, will probably perform badly on NEW data
Strategy:
Limit complexity of model (e.g. degree of equ’n)
Split training and validation sets
Hold out data to check for overfitting

35. Nearest Neighbor Memory- or case- based learning
Supervised method: Training
Record labeled instances and feature-value vectors
For each new, unlabeled instance
Identify “nearest” labeled instance
Assign same label
Consistency heuristic: Assume that a property is the same as that of the nearest reference case.

36. Nearest Neighbor Example
Credit Rating:
Classifier: Good / Poor
Features:
L = # late payments/yr;
R = Income/Expenses
Name L R G/P
A G
B P
C G
D P
E P
F G
G G
H P

37. Nearest Neighbor Example
Name L R G/P
A G A F
B P 1 G R
C G E H D C
D P
E P B
F G
G G 10 20 30 L
H P

38. Nearest Neighbor Example
Name L R G/P I G A F J K P 1 I G K ?? E R H D C J B
Distance Measure:
Sqrt ((L1-L2)^2 + [sqrt(10)*(R1-R2)]^2))
- Scaled distance 10 20 30 L

39. Nearest Neighbor Analysis
Problem:
Ambiguous labeling, Training Noise
Solution:
K-nearest neighbors
Not just single nearest instance
Compare to K nearest neighbors
Label according to majority of K
What should K be?
Often 3, can train as well

40. Text Classification

41. Matching Topics and Documents
Two main perspectives:
Pre-defined, fixed, finite topics:
“Text Classification”
Arbitrary topics, typically defined by statement of information need (aka query)
“Information Retrieval”

42. Vector Space Information Retrieval
Task:
Document collection
Query specifies information need: free text
Relevance judgments: 0/1 for all docs
Word evidence: Bag of words
No ordering information

43. Vector Space Model Tv Program Computer
Two documents: computer program, tv program
Query: computer program : matches 1 st doc: exact: distance=2 vs 0
educational program: matches both equally: distance=1

44. Vector Space Model Represent documents and queries as
Vectors of term-based features
Features: tied to occurrence of terms in collection
E.g.
Solution 1: Binary features: t=1 if present, 0 otherwise
Similiarity: number of terms in common
Dot product

45. Vector Space Model II Problem: Not all terms equally interesting
E.g. the vs dog vs Levow
Solution: Replace binary term features with weights
Document collection: term-by-document matrix
View as vector in multidimensional space
Nearby vectors are related
Normalize for vector length

46. Vector Similarity Computation
Similarity = Dot product
Normalization:
Normalize weights in advance
Normalize post-hoc

47. Term Weighting “Aboutness” “Specificity”
To what degree is this term what document is about?
Within document measure
Term frequency (tf): # occurrences of t in doc j
“Specificity”
How surprised are you to see this term?
Collection frequency
Inverse document frequency (idf):

48. Term Selection & Formation
Some terms are truly useless
Too frequent, no content
E.g. the, a, and,…
Stop words: ignore such terms altogether
Creation:
Too many surface forms for same concepts
E.g. inflections of words: verb conjugations, plural
Stem terms: treat all forms as same underlying

49. Efficient Implementations
Classification cost:
Find nearest neighbor: O(n)
Compute distance between unknown and all instances
Compare distances
Problematic for large data sets
Alternative:
Use binary search to reduce to O(log n)

50. Efficient Implementation: K-D Trees
Divide instances into sets based on features
Binary branching: E.g. > value
2^d leaves with d split path = n
d= O(log n)
To split cases into sets,
If there is one element in the set, stop
Otherwise pick a feature to split on
Find average position of two middle objects on that dimension
Split remaining objects based on average position
Recursively split subsets

51. K-D Trees: Classification
Yes No
L > 17.5?
L > 9 ? No
Yes
Yes No
R > 0.6?
R > 0.75?
R > ?
R > ? No
Yes No
Yes No No
Yes
Yes
Poor
Good
Good
Poor
Good
Good
Poor
Good

52. Efficient Implementation: Parallel Hardware
Classification cost:
# distance computations
Const time if O(n) processors
Cost of finding closest
Compute pairwise minimum, successively
O(log n) time

53. Nearest Neighbor: Issues
Prediction can be expensive if many features
Affected by classification, feature noise
One entry can change prediction
Definition of distance metric
How to combine different features
Different types, ranges of values
Sensitive to feature selection

54. Nearest Neighbor: Analysis
Issue:
What is a good distance metric?
How should features be combined?
Strategy:
(Typically weighted) Euclidean distance
Feature scaling: Normalization
Good starting point:
(Feature - Feature_mean)/Feature_standard_deviation
Rescales all values - Centered on 0 with std_dev 1

55. Nearest Neighbor: Analysis
Issue:
What features should we use?
E.g. Credit rating: Many possible features
Tax bracket, debt burden, retirement savings, etc..
Nearest neighbor uses ALL
Irrelevant feature(s) could mislead
Fundamental problem with nearest neighbor

56. Nearest Neighbor: Advantages
Fast training:
Just record feature vector - output value set
Can model wide variety of functions
Complex decision boundaries
Weak inductive bias
Very generally applicable

57. Summary Machine learning:
Acquire function from input features to value
Based on prior training instances
Supervised vs Unsupervised learning
Classification and Regression
Inductive bias:
Representation of function to learn
Complexity, Generalization, & Validation