Randomized Variable Elimination

1. Randomized Variable Elimination David J. Stracuzzi Paul E. Utgoff

2. 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

3. 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.

4. 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.

5. 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)

6. 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.

7. The Cost Function

8. 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

9. 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.

10. 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.

11. Optimal cost of removing irrelevant variables • The optimal cost of removing irrelevant variables from n remaining and r relevant is given by

12. 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.

13. Algorithms

14. 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

15. 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

16. 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.

17. 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

18. 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)

19. Comparison of RVE and RVErS

20. Results

21. Variable Selection results using naïve Bayes and C4.5 algorithms

22. Variable Selection results using naïve Bayes and C4.5 algorithms

23. 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

24. RECAP

25. Questions

26. 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.

27. 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