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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

Randomized Variable Elimination

David J. Stracuzzi

Paul E. Utgoff

  • 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
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
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
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 elimination1
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
Probability of selecting ‘k’ variables
  • The probability of successfully selecting ‘k’ irrelevant variables at random is given by


n … remaining variables

r … relevant variables

expected number of failures
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
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
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
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
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
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’


replace the selected k variables

rve example
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.
randomized variable elimination including a search for r rvers
Randomized Variable Elimination including a search for ‘r’ (RVErS)
  • 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


replace the selected k variables

fail ← fail + 1

success ← 0

rvers contd
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
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
  • 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
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