Logistic Regression. Vibhav Gogate The University of Texas at Dallas. Some Slides from Carlos Guestrin, Luke Zettlemoyer and Dan Weld.

Transcription

1 Logistic Regression Vibhav Gogate The University of Texas at Dallas Some Slides from Carlos Guestrin, Luke Zettlemoyer and Dan Weld.

2 Generative vs. Discriminative Classifiers Want to Learn: h:x Y X features Y target classes Generative classifier, e.g., Naïve Bayes: Assume some functional form for P(X Y), P(Y) Estimate parameters of P(X Y), P(Y) directly from training data Use Bayes rule to calculate P(Y X= x) This is a generative model Indirect computation of P(Y X) through Bayes rule As a result, can also generate a sample of the data, P(X) = y P(y) P(X y) Discriminative classifiers, e.g., Logistic Regression: Assume some functional form for P(Y X) Estimate parameters of P(Y X) directly from training data This is the discriminative model Directly learn P(Y X) But cannot obtain a sample of the data, because P(X) is not available P(Y X) P(X Y) P(Y) 2

3 Univariate Linear Regression h w (x) = w 1 x + w 0 n Loss(h w ) = L 2(y j, h w (x j )) = (y j - h w (x j )) 2 = j=1 n j=1 n (y j (w 1 x j +w 0 )) 2 j=1 3

4 Understanding Weight Space h w (x) = w 1 x + w 0 Loss(h w ) = n (y j (w 1 x j +w 0 )) 2 j=1 4

5 Understanding Weight Space h w (x) = w 1 x + w 0 Loss(h w ) = n (y j (w 1 x j +w 0 )) 2 j=1 5

6 Finding Minimum Loss Argmin w Loss(h w ) h w (x) = w 1 x + w 0 w 0 Loss(h w ) = Loss(h w ) = 0 n (y j (w 1 x j +w 0 )) 2 j=1 w 1 Loss(h w ) = 0 6

7 Unique Solution! Argmin w Loss(h w ) h w (x) = w 1 x + w 0 w 1 = N (x j y j ) ( x j )( y j ) N (x j2 ) ( x j ) 2 w 0 = ( (y j ) w 1 ( x j )/N

8 Could also Solve Iteratively Argmin w Loss(h w ) w = any point in weight space Loop until convergence For each w i in w do w i := w i - Loss(w) w i

9 Multivariate Linear Regression h w (x j ) = w 0 + w i x j,i = w i x j,i =w T x j Argmin w Loss(h w ) Unique Solution = (x T x) -1 x T y Problem. Overfitting: Possible that some dimension that is actually irrelevant appears by chance to be useful. 9

10 Overfitting Regularize!! Penalize high weights (complex hypothesis) Minimize cost: Loss + Complexity n Cost h w = y j w i x j 2 + λ w i p j=1 i i p=1: L1 regularization (Lasso) p=2: L2 regularization 10

11 Regularization L1 L2 11

12 Back to Classification P(edible X)=1 Decision Boundary P(edible X)=0 12

13 Logistic Regression Learn P(Y X) directly! Assume a particular functional form Not differentiable P(Y)=0 P(Y)=1 13

14 Logistic Regression Learn P(Y X) directly! Assume a particular functional form Logistic Function Aka Sigmoid 14

15 Logistic Function in n Dimensions Sigmoid applied to a linear function of the data: Features can be discrete or continuous! 15

16 Understanding Sigmoids w 0 =-2, w 1 = w 0 =0, w 1 = w 0 =0, w 1 = -0.5

17 Very convenient! implies implies implies linear classification rule! Y=0 if the RHS>0 17 Carlos Guestrin

18 Likelihood vs. Conditional Likelihood Generative (Naïve Bayes) maximizes Data likelihood Discriminative (Logistic Regr.) maximizes Conditional Data Likelihood Discriminative models can t compute P(x j w)! Or, They don t waste effort learning P(X) Focus only on P(Y X) - all that matters for classification 18 Carlos Guestrin

19 Maximizing Conditional Log Likelihood Bad news: no closed-form solution to maximize l(w) Good news: l(w) is concave function of w! No local minima Concave functions easy to optimize Carlos Guestrin

20 Optimizing concave function Gradient ascent Conditional likelihood for Logistic Regression is concave! Gradient: Update rule: Learning rate, >0 Gradient ascent is simplest of optimization approaches

21 Maximize Conditional Log Likelihood: Gradient ascent

22 Gradient Ascent for LR Gradient ascent algorithm: (learning rate η > 0) do: For i=1 to n: (iterate over weights) until change < Loop over training examples!

23 Large parameters a=1 a=5 a=10 Maximum likelihood solution: prefers higher weights higher likelihood of (properly classified) examples close to decision boundary larger influence of corresponding features on decision can cause overfitting!!! Regularization: penalize high weights

24 That s all MCLE. How about MCAP? One common approach is to define priors on w Normal distribution, zero mean, identity covariance Pushes parameters towards zero Regularization Helps avoid very large weights and overfitting MAP estimate:

25 M C AP as Regularization Add log p(w) to objective: Quadratic penalty: drives weights towards zero Adds a negative linear term to the gradients Penalizes high weights, like we did in linear regression

26 MCLE vs. MCAP Maximum conditional likelihood estimate Maximum conditional a posteriori estimate

27 Naïve Bayes Learning: h:x Y vs. Logistic Regression X features Y target classes Generative Assume functional form for P(X Y) assume cond indep P(Y) Est params from train data Gaussian NB for cont features Bayes rule to calc. P(Y X= x) P(Y X) P(X Y) P(Y) Indirect computation Can also generate a sample of the data 27 Discriminative Assume functional form for P(Y X) no assumptions Est params from training data Handles discrete & cont features Directly calculate P(Y X=x) Can t generate data sample

28 Remember Gaussian Naïve Bayes? Sometimes Assume Variance is independent of Y (i.e., i ), or independent of X i (i.e., k ) or both (i.e., ) P(X i =x Y=y k ) = N( ik, ik ) P(Y X) P(X Y) P(Y) N( ik, ik ) =

29 Gaussian Naïve Bayes vs. Logistic Regression Learning: h:x Y X Real-valued features Y target classes Generative What can we say about the Assume functional form for form of P(Y=1 X i )? P(X Y) assume X i cond indep given Y P(Y) Est params from train data Gaussian NB for continuous features model P(X i Y = y k ) as Gaussian N( ik, i ) model P(Y) as Bernoulli (q,1-q) Bayes rule to calc. P(Y X= x) P(Y X) P(X Y) P(Y) 29 Cool!!!!

30 Derive form for P(Y X) for continuous X i up to now, all arithmetic only for Naïve Bayes models Looks like a setting for w 0? Can we solve for w i? Yes, but only in Gaussian case

31 Ratio of class-conditional probabilities Linear function! Coefficients expressed with original Gaussian parameters!

32 Derive form for P(Y X) for continuous Xi

33 Gaussian Naïve Bayes vs. Logistic Regression Set of Gaussian Naïve Bayes parameters (feature variance independent of class label) Can go both ways, we only did one way Set of Logistic Regression parameters Representation equivalence But only in a special case!!! (GNB with class-independent variances) But what s the difference??? LR makes no assumptions about P(X Y) in learning!!! Loss function!!! Optimize different functions! Obtain different solutions

34 Naïve Bayes vs. Logistic Regression Consider Y boolean, X=<X 1... X n > continuous Number of parameters: Naïve Bayes: 4n +1 Logistic Regression: n+1

35 Naïve Bayes vs. Logistic Regression Generative vs. Discriminative classifiers Asymptotic comparison (# training examples infinity) when model correct GNB (with class independent variances) and LR produce identical classifiers when model incorrect LR is less biased does not assume conditional independence therefore LR expected to outperform GNB [Ng & Jordan, 2002]

36 Naïve Bayes vs. Logistic Regression [Ng & Jordan, 2002] Generative vs. Discriminative classifiers Non-asymptotic analysis convergence rate of parameter estimates, (n = # of attributes in X) Size of training data to get close to infinite data solution Naïve Bayes needs O(log n) samples Logistic Regression needs O(n) samples GNB converges more quickly to its (perhaps less helpful) asymptotic estimates

37 Naïve bayes Logistic Regression Some experiments from UCI data sets Carlos Guestrin

38 What you should know about Logistic Regression (LR) Gaussian Naïve Bayes with class-independent variances representationally equivalent to LR Solution differs because of objective (loss) function In general, NB and LR make different assumptions NB: Features independent given class! assumption on P(X Y) LR: Functional form of P(Y X), no assumption on P(X Y) LR is a linear classifier decision rule is a hyperplane LR optimized by conditional likelihood no closed-form solution concave! global optimum with gradient ascent Maximum conditional a posteriori corresponds to regularization Convergence rates GNB (usually) needs less data LR (usually) gets to better solutions in the limit

39 LR: MCAP Algorithm Given Data matrix of size m x (n+2) (n attributes and m examples) Data[i][n+1] gives the class for example i Data[i][0] is the dummy threshold attribute always set to 1. Arrays Pr[0..m-1] and w[0..n] initialized to random values Until convergence do For each example i Compute Pr[i]=Pr(class=1 Data[i],w) Array dw[0..n] initialized to zero For i=0 to n // Go over all the weights For j=0 to m-1 // Go over all the training examples dw[i]=dw[i]+data[j][i]*(data[j][n+1]-pr[j]) For i=0 to n w[i]=w[i]+ (dw[i]- w[i]) 39