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 Rd for 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 Rd and 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., KDD 2007

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 minimizer exercise
Compare SVM to regularized logistic regression
Software:
Version 2.1 executables for OSL machines available on course website