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Randomized Variable EliminationPowerPoint Presentation

Randomized Variable Elimination

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Presentation Transcript

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.

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

else

replace the selected k variables

fail ← fail + 1

success ← 0

RVErS (contd…) ‘r’ (

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)

Results ‘r’ (

Variable Selection results using naïve algorithmsBayes and C4.5 algorithms

My implementation algorithms

- 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

RECAP algorithms

Questions algorithms

References algorithms

- 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…) algorithms

- 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

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