Regularized risk minimization

1. Regularized risk minimization Usman Roshan

2. Supervised learning for two classes • We are given n training samples (xi,yi) for i=1..n drawn i.i.d from a probability distribution P(x,y). • Each xi is a d-dimensional vector (xi in Rd) and yi is +1 or -1 • Our problem is to learn a function f(x) for predicting the labels of test samples xi’ in Rdfor i=1..n’ also drawn i.i.d from P(x,y)

3. Loss function • Loss function: c(x,y,f(x)) • Maps to [0,inf] • Examples:

4. Test error • We quantify the test error as the expected error on the test set (in other words the average test error). In the case of two classes: • We’d like to find f that minimizes this but we need P(y|x) which we don’t have access to.

5. Expected risk • Suppose we didn’t have test data (x’). Then we average the test error over all possible data points x • We want to find f that minimizes this but we don’t have all data points. We only have training data.

6. Empirical risk • Since we only have training data we can’t calculate the expected risk (we don’t even know P(x,y)). • Solution: we approximate P(x,y) with the empirical distribution pemp(x,y) • The delta function δx(y)=1 if x=y and 0 otherwise.

7. Empirical risk • We can now define the empirical risk as • Once the loss function is defined and training data is given we can then find f that minimizes this.

8. Example of minimizing empirical risk (least squares) • Suppose we are given n data points (xi,yi) where each xi in Rdand yi in R. We want to determine a linear function f(x)=ax+b for predicting test points. • Loss function c(xi,yi,f(xi))=(yi-f(xi))2 • What is the empirical risk?

9. Empirical risk for least squares Now finding f has reduced to finding a and b. Since this function is convex in a and b we know there is a global optimum which is easy to find by setting first derivatives to 0.

10. Maximum likelihood and empirical risk • Maximizing the likelihood P(D|M) is the same as maximizing log(P(D|M)) which is the same as minimizing -log(P(D|M)) • Set the loss function to • Now minimizing the empirical risk is the same as maximizing the likelihood

11. Empirical risk • We pose the empirical risk in terms of a loss function and go about to solve it. • Input: n training samples xi each of dimension d along with labels yi • Output: a linear function f(x)=wTx+w0 that minimizes the empirical risk

12. Empirical risk examples • Linear regression • How about logistic regression?

13. Logistic regression • Recall the logistic regression model: • Let y=+1 be case and y=-1 be control. • The sample likelihood of the training data is given by

14. Logistic regression • We find our parameters w and w0 by maximizing the likelihood or minimizing the -log(likelihood). • The -log of the likelihood is

15. Logistic regression loss function

16. SVM loss function • Recall the SVM optimization problem: • The loss function (second term) can be written as

17. Different loss functions • Linear regression • Logistic regression • SVM

18. Regularized risk minimization • Minimize • Note the additional term added to the empirical risk.

19. Other loss functions • From “A Scalable Modular Convex Solver for Regularized Risk Minimization”, Teo et. al., KDD2007

20. Regularizer • L1 norm: • L1 gives sparse solution (many entries will be zero) • Logistic loss with L1 also known as “lasso” • L2 norm:

21. Regularized risk minimizerexercise • Compare SVM to regularized logistic regression • Software: http://users.cecs.anu.edu.au/~chteo/BMRM.html • Version 2.1 executables for OSL machines available on course website