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Regularized risk minimization Usman Roshan

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

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

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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.

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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.

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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 p emp (x,y) The delta function δ x (y)=1 if x=y and 0 otherwise.

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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.

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

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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.

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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

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Empirical risk We pose the empirical risk in terms of a loss function and go about to solve it. Input: n training samples x i each of dimension d along with labels y i Output: a linear function f(x)=w T x+w 0 that minimizes the empirical risk

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Empirical risk examples Linear regression How about logistic regression?

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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

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Logistic regression We find our parameters w and w 0 by maximizing the likelihood or minimizing the -log(likelihood). The -log of the likelihood is

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Logistic regression loss function

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

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Different loss functions Linear regression Logistic regression SVM

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

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Other loss functions From “A Scalable Modular Convex Solver for Regularized Risk Minimization”, Teo et. al., KDD 2007

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

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

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