Machine Learning Basics Lecture 7: Multiclass Classification. Princeton University COS 495 Instructor: Yingyu Liang
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1 Machine Learning Basics Lecture 7: Multiclass Classification Princeton University COS 495 Instructor: Yingyu Liang
2 Example: image classification indoor Indoor outdoor
3 Example: image classification (multiclass) ImageNet figure borrowed from vision.standford.edu
4 Multiclass classification Given training data x i R d, y i {1,2,, K} x i, y i : 1 i n i.i.d. from distribution D Find f x : R d {1,2,, K} that outputs correct labels What kind of f?
5 Approaches for multiclass classification
6 Approach 1: reduce to regression Given training data Find f w x = w T x that minimizes L f w x i, y i : 1 i n i.i.d. from distribution D = 1 σ n n i=1 w T x i y i 2 Bad idea even for binary classification Reduce to linear regression; ignore the fact y {1,2..., K}
7 Approach 1: reduce to regression Bad idea even for binary classification Figure from Pattern Recognition and Machine Learning, Bishop
8 Approach 2: one-versus-the-rest Find K 1 classifiers f 1, f 2,, f K 1 f 1 classifies 1 vs {2,3,, K} f 2 classifies 2 vs {1,3,, K} f K 1 classifies K 1 vs {1,2,, K 2} Points not classified to classes {1,2,, K 1} are put to class K Problem of ambiguous region: some points may be classified to more than one classes
9 Approach 2: one-versus-the-rest Figure from Pattern Recognition and Machine Learning, Bishop
10 Approach 3: one-versus-one Find K 1 K/2 classifiers f (1,2), f (1,3),, f (K 1,K) f (1,2) classifies 1 vs 2 f (1,3) classifies 1 vs 3 f (K 1,K) classifies K 1 vs K Computationally expensive: think of K = 1000 Problem of ambiguous region
11 Approach 3: one-versus-one Figure from Pattern Recognition and Machine Learning, Bishop
12 Approach 4: discriminant functions Find K scoring functions s 1, s 2,, s K Classify x to class y = argmax i s i (x) Computationally cheap No ambiguous regions
13 Linear discriminant functions Find K discriminant functions s 1, s 2,, s K Classify x to class y = argmax i s i (x) Linear discriminant: s i (x) = w i T x, with w i R d
14 Linear discriminant functions Linear discriminant: s i (x) = w i T x, with w i R d Lead to convex region for each class: by y = argmax i w i T x Figure from Pattern Recognition and Machine Learning, Bishop
15 Conditional distribution as discriminant Find K discriminant functions s 1, s 2,, s K Classify x to class y = argmax i s i (x) Conditional distributions: s i (x) = p(y = i x) Parametrize by w i : s i (x) = p w i(y = i x)
16 Multiclass logistic regression
17 Review: binary logistic regression Sigmoid σ w T 1 x + b = 1 + exp( (w T x + b)) Interpret as conditional probability p w y = 1 x = σ w T x + b p w y = 0 x = 1 p w y = 1 x = 1 σ w T x + b How to extend to multiclass?
18 Review: binary logistic regression Suppose we model the class-conditional densities p x y = i and class probabilities p y = i Conditional probability by Bayesian rule: p y = 1 x = p x y = 1 p(y = 1) p x y = 1 p y = 1 + p x y = 2 p(y = 2) = exp( a) = σ(a) where we define a ln p x y = 1 p(y = 1) p x y = 2 p(y = 2) = ln p y = 1 x p y = 2 x
19 Review: binary logistic regression Suppose we model the class-conditional densities p x y = i and class probabilities p y = i p y = 1 x = σ a = σ(w T x + b) is equivalent to setting log odds a = ln p y = 1 x p y = 2 x = wt x + b Why linear log odds?
20 Review: binary logistic regression Suppose the class-conditional densities p x y = i is normal log odd is p x y = i = N x μ i, I = 1 2π d/2 exp{ 1 2 x μ i p x y = 1 p(y = 1) a = ln p x y = 2 p(y = 2) = wt x + b where w = μ 1 μ 2, b = 1 2 μ 1 T μ μ 2 T p(y = 1) μ 2 + ln p(y = 2) 2 }
21 Multiclass logistic regression Suppose we model the class-conditional densities p x y = i and class probabilities p y = i Conditional probability by Bayesian rule: p y = i x = p x y = i p(y = i) σ j p x y = j p(y = j) = exp(a i) σ j exp(a j ) where we define a i ln [p x y = i p y = i ]
22 Multiclass logistic regression Suppose the class-conditional densities p x y = i is normal 1 p x y = i = N x μ i, I = 2π d/2 exp{ 1 2 x μ i Then a i ln p x y = i p y = i = 1 2 xt x + w i Tx + b i 2 } where w i = μ i, b i = 1 2 μ i T μ i + ln p y = i + ln 1 2π d/2
23 Multiclass logistic regression Suppose the class-conditional densities p x y = i is normal p x y = i = N x μ i, I = 1 2π d/2 exp{ 1 2 x μ i Cancel out 1 2 xt x, we have p y = i x = exp(a i) σ j exp(a j ), a i w i T x + b i where w i = μ i, b i = 1 2 μ i T μ i + ln p y = i + ln 1 2 } 2π d/2
24 Multiclass logistic regression: conclusion Suppose the class-conditional densities p x y = i is normal 1 p x y = i = N x μ i, I = 2π d/2 exp{ 1 2 x μ i 2 } Then p y = i x = exp( wi T x + b i ) σ j exp( w j T x + b j ) which is the hypothesis class for multiclass logistic regression It is softmax on linear transformation; it can be used to derive the negative loglikelihood loss (cross entropy)
25 Softmax A way to squash a = (a 1, a 2,, a i, ) into probability vector p softmax a = exp(a 1) σ j exp(a j ), exp(a 2 ) σ j exp(a j ),, exp a i, σ j exp a j Behave like max: when a i a j j i, p i 1, p j 0
26 Cross entropy for conditional distribution Let p data (y x) denote the empirical distribution of the data Negative log-likelihood 1 σ n i=1 n log p y = y i x i = E pdata (y x) log p(y x) is the cross entropy between p data and the model output p Information theory viewpoint: KL divergence D(p data p = E pdata [log p data ] = E p p data [log p data ] E pdata [log p] Entropy; constant Cross entropy
27 Cross entropy for full distribution Let p data (x, y) denote the empirical distribution of the data Negative log-likelihood 1 σ n i=1 n log p(x i, y i ) = E pdata (x,y) log p(x, y) is the cross entropy between p data and the model output p
28 Multiclass logistic regression: summary Last hidden layer h Label y i (w j ) T h + b j softmax p j Cross entropy Linear Convert to probability Loss
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