Randomized Variable Elimination

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Randomized Variable Elimination. David J. Stracuzzi Paul E. Utgoff. Agenda. Background Filter and wrapper methods Randomized Variable Elimination Cost Function RVE algorithm when r is known (RVE) RVE algorithm when r is not known ( RVErS ) Results Questions.

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### Randomized Variable Elimination

David J. Stracuzzi

Paul E. Utgoff

Agenda
• Background
• Filter and wrapper methods
• Randomized Variable Elimination
• Cost Function
• RVE algorithm when r is known (RVE)
• RVE algorithm when r is not known (RVErS)
• Results
• Questions
Variable Selection Problem
• Choosing relevant attributes from set of attributes.
• Producing a subset of variables from large set of input variables that best predicts target function.
• Forward selection algorithm starts with an empty set and searches for variables to add.
• Backward selection algorithm starts with entire set of variables and go on removing irrelevant variable(s).
• In some cases, forward selection algorithm also removes variables in order to recover from previous poor selections.
• Caruna and Freitag (1994) experimented with greedy search methods and found that allowing search to add or remove variables outperform simple forward and backward searches
• Filter and wrapper methods for variable selection.
Filter methods
• Uses statistical measures to evaluate the quality of variable subsets.
• Subset of variables are evaluated with respect to specific quality measure.
• Statistical evaluation of variables require very little computational cost as compared to running the learning algorithm.
• FOCUS (Almuallim and Dietterich, 1991) searches for smallest subset that completely discriminates between target classes.
• Relief (Kira and Rendell, 1992) ranks variables as per distance.
• In filter methods, variables are evaluated independently and not in context of learning problem.
Wrapper methods
• Uses performance of the learning algorithm to evaluate the quality of subset of input variables.
• The learning algorithm is executed on the candidate variable set and then tested for the accuracy of resulting hypothesis.
• Advantage: Since wrapper methods evaluate variables in the context of learning problem, they outperform filter methods.
• Disadvantage: Cost of repeatedly executing the learning algorithm can become problematic.
• John, Kohavi, and Pfleger (1994) coined the term “wrapper” but the technique was used before that (Devijver and Kittler, 1982)
Randomized Variable Elimination
• Falls under the category of wrapper methods.
• First, a hypothesis is produced for entire set of ‘n’ variables.
• A subset if formed by randomly selecting ‘k’ variables.
• A hypothesis is then produced for remaining (n-k) variables.
• Accuracy of the two hypotheses are compared.
• Removal of any relevant variable should cause an immediate decline in performance
• Uses a cost function to achieve a balance between successive failures and cost of running the learning algorithm several times.
Probability of selecting ‘k’ variables
• The probability of successfully selecting ‘k’ irrelevant variables at random is given by

where,

n … remaining variables

r … relevant variables

Expected number of failures
• The expected number of consecutive failures before a success at selecting k irrelevant variables is given by
• Number of consecutive trials in which at least one of the r relevant variables will be randomly selected along with irrelevant variables.
Cost of removing k variables
• The expected cost of successfully removing k variables from n remaining given r relevant variables is given by

where, M(L, n) represents an upper bound on the cost of running algorithm ‘L’ on n inputs.

Optimal cost of removing irrelevant variables
• The optimal cost of removing irrelevant variables from n remaining and r relevant is given by
Optimal value for ‘k’
• The optimal value is computed as
• It is the value of k for which the cost of removing variables is optimal.
Algorithm for computing k and cost values
• Given: L, N, r
• Isum[r+1…N] ← 0

kopt[r+1…N] ← 0

fori ← r+1 to Ndo

bestCost ← ∞

for k ← 1 to i-r do

temp ← I(i,r,k) + Isum[i-k]

if (temp < bestCost) then

bestCost ← temp

bestK ← k

Isum[i] ← bestCost

kopt[i] ← bestK

Randomized Variable Elimination (RVE) when r is known
• Given: L,n,r, tolerance
• Compute tables for Isum(i,r) and kopt(i,r)

h ← hypothesis produced by L on ‘n’ inputs

• whilen > rdo

k ← kopt(n,r)

select k variables at random and remove them

h’ ← hypothesis produced by L on n-k inputs

ife(h’) – e(h) ≤ tolerancethen

n ← n-k

h ← h’

else

replace the selected k variables

RVE example
• Plot of expected cost of running RVE(Isum(N,r = 10)) along with cost of removing inputs individually, and the estimated number of updates M(L,n).
• L is function that learns a boolean function using perceptron unit.
• Given: L, c1, c2, n, rmax , rmin , tolerance
• Compute tables Isum(i,r) and kopt(i,r) for rmin ≤ r ≤ rmax

r ← (rmax + rmin) / 2

success, fail ← 0

h ← hypothesis produced by L on ‘n’ inputs

• repeat

k ← kopt(n,r)

select k variables at random and remove them

h’ ← hypothesis produced by L on (n-k) inputs

ife(h’) – e(h) ≤ tolerance then

n ← n – k

h ← h’

success ← success + 1

fail ← 0

else

replace the selected k variables

fail ← fail + 1

success ← 0

RVErS (contd…)

ifn ≤ rminthen

r, rmax, rmin ← n

elseiffail ≥ c1E⁻(n,r,k)then

rmin ← r

r ← (rmax + rmin) / 2

success, fail ← 0

elseifsuccess ≥ c2(r – E⁻(n,r,k)) then

rmax ← r

r ← (rmax + rmin) / 2

success, fail ← 0

until rmin < rmaxandfail ≤ c1E⁻(n,r,k)

My implementation
• Integrate with Weka
• Extend the NaiveBayes and J48 algorithms
• Obtain results for some UCI datasets used
• Compare results with those reported by authors
• Work in progress
References
• H. Almuallim and T.G Dietterich. Leraning with many irrelevant features. In Proceedings of the Ninth National Conference on Artificial Intelligence, Anaheim, CA, 1991. MIT Press.
• R. Caruna and D. Freitag. Greedy attribute selection. In Machine Learning: Proceedings of Eleventh International Conference, Amherst, MA, 1993. Morgan Kaufmann.
• K. Kira and L. Rendell. A practical approach to feature selection. In D. Sleeman and P. Edwards, editors, Machine Learning: Proceedings of Ninth International Conference, San Mateo, CA, 1992. Morgan Kaufmann.
References (contd…)
• G. H. John, R. Kohavi, and K. Pfleger. Irrelevant features and subset selection problem. In Machine Learning: Proceedings of Eleventh Internaltional Conference, pages 121-129, New Brunswick, NJ, 1994. Morgan Kauffmann.
• P.A. Devijver and J. Kittler. Pattern Recognition: A statistical approach. Prentice Hall/International, 1982