Supervised Learning. George Konidaris

Transcription

1 Supervised Learning George Konidaris Fall 2017

2 Machine Learning Subfield of AI concerned with learning from data. Broadly, using: Experience To Improve Performance On Some Task (Tom Mitchell, 1997)

3 A breakthrough in machine learning would be worth ten Microsofts. - Bill Gates

4 Supervised Learning Input: X = {x1,, xn} Y = {y1,, yn} inputs labels training data Learn to predict new labels. Given x: y?

5 Classification vs. Regression If the set of labels Y is discrete: Classification Minimize number of errors If Y is real-valued: Regression Minimize sum squared error Today we focus on classification.

6 Supervised Learning Formal definition: Given training data: X = {x1,, xn} Y = {y1,, yn} inputs labels Produce: Decision function f : X! Y That minimizes error: X err(f(x i ),y i ) i

7 Test/Train Split Minimize error measured on what? Don t get to see future data. Could use test data but! may not generalize. General principle: Do not measure error on the data you train on! Methodology: Split data into training set and test set. Fit f using training set. Measure error on test set. Always do this.

8 Test/Train Split What if you choose unluckily? And aren t we wasting data? k-fold Cross Validation: Common alternative Repeat k times: Partition data into train (n - n/k) and test (n/k) data sets Train on training set, test on test set Average results across k choices of test set.

9 Key Idea: Hypothesis Space Typically Fixed representation of classifier. Learning algorithm constructed to match. Representation induces class of functions F, from which to find f. F is known as the hypothesis space. Tradeoff: power vs. expressibility vs. data efficiency. Not every F can represent every function. F = {f1, f2,, fn} Set of possible functions that can be returned Typically infinite set (not always) Learning is finding f i 2 F that minimizes error.

10 Key Idea: Decision Boundary Boundary at which label changes

11 Decision Trees Let s assume: Two classes (true and false). Input: vector of discrete values. What s the simplest thing we could do? How about some if-then rules? Relatively simple classifier: Tree of tests. Evaluate test for for each xi, follow branch. Leaves are class labels.

12 Decision Trees xi = [a, b, c] a? each boolean true false b? c? true false true false y=1 y=2 b? y=1 true false y=2 y=1

13 Decision Trees How to make one? Given X = {x1,, xn} Y = {y1,, yn} repeat: if all the labels are the same, we have a leaf node. pick an attribute and split data bases on its value. recurse on each half. If we run out of splits, and data not perfectly in one class, then take a max.

14 Decision Trees a? A B C L T F T 1 T T F 1 T F F 1 F T F 2 F T T 2 F T F 2 F F T 1 F F F 1

15 Decision Trees true y=1 a? A B C L T F T 1 T T F 1 T F F 1 F T F 2 F T T 2 F T F 2 F F T 1 F F F 1

16 Decision Trees true y=1 a? false b? A B C L T F T 1 T T F 1 T F F 1 F T F 2 F T T 2 F T F 2 F F T 1 F F F 1

17 Decision Trees true y=1 a? false b? true y=2 A B C L T F T 1 T T F 1 T F F 1 F T F 2 F T T 2 F T F 2 F F T 1 F F F 1

18 Decision Trees true y=1 a? false b? true y=2 false y=1 A B C L T F T 1 T T F 1 T F F 1 F T F 2 F T T 2 F T F 2 F F T 1 F F F 1

19 Attribute Picking Key question: Which attribute to split over? Information contained in a data set: I(A) = f 1 log 2 f 1 f 2 log 2 f 2 How many bits of information do we need to determine the label in a dataset? Pick the attribute with the max information gain: X Gain(B) =I(A) f i I(B i ) i

20 Example X Gain(B) =I(A) f i I(B i ) I(A) = f 1 log 2 f 1 f 2 log 2 f 2 i A B C L T F T 1 T T F 1 T F F 1 F T F 2 F T T 2 F T F 2 F F T 1 F F F 1

21 Decision Trees What if the inputs are real-valued? Have inequalities rather than equalities. Can repeat variables. a > 3.1 true false y=1 b < 0.6? true false y=2 y=1

22 Hypothesis Class What is the hypothesis class for a decision tree? Discrete inputs? Real-valued inputs?

23 The Perceptron If your input (xi) is real-valued explicit decision boundary?

24 The Perceptron If x = [x(1),, x(n)]: Create an n-d line Slope for each x(i) Constant offset f(x) =sign(w x + c) y = wx + c gradient offset

25 The Perceptron Which side of a line are you on? w x = w x cos( ) x w

26 The Perceptron How do you reduce error? - e =(y i (w x i + j = 2(y i (w i x i + c))x i (j) descend this gradient to reduce error

27 The Perceptron Algorithm Assume you have a batch of data: X = {x1,, xn} Y = {y1,, yn} set w, c to 0. for each xi: predict zi = sign(w.xi + c) if zi!= yi: w = w + a(yi - zi)xi converges if data is linearly separate learning rate

28 Probabilities What if you want a probabilistic classifier? Instead of sign, squash output of linear sum down to [0, 1]: (w x + c) Resulting algorithm: logistic regression.

29 Frank Rosenblatt Built the Mark I in 1960.

30 Perceptrons What can t you do?

31 Perceptrons 1969

32 Neural Networks (w x + c) logistic regression

33 Neurons

34 Neural Networks output layer o1 o2 hidden layer h1 h2 h3 input layer x1 x2

35 Neural Networks (w o 1 1 h 1 + w o 1 2 h 2 + w o 1 3 h 3 + w o 1 4 ) value computed o1 (w o 2 1 h 1 + w o 2 2 h 2 + w o 2 3 h 3 + w o 2 4 ) o2 value computed h 1 = (w h 1 1 x 1 + w h 1 2 x 2 + w h 1 h1 1 x 1 + w h 2 2h2 x 2 + w h 2 3 ) (wh 3 h3 3 ) (w h 2 x1 x2 1 x 1 + w h 3 2 x 2 + w h 3 3 ) feed forward input layer x 1,x 2 2 [0, 1]

36 Neural Networks probability of class 1 (w 1 h 1 + w 2 h 2 + w 3 h 3 + w 4 ) o1 o2 (w 1 h 1 + w 2 h 2 + w 3 h 3 + w 4 ) probability of class 2 (w 1 x 1 + w 2 x 2 + w 3 ) (w 1 x 1 + w 2 x 2 + w 3 ) (w 1 x 1 + w 2 x 2 + w 3 ) h1 h2 h3 weights (parameters) x1 x2 input data x 1,x 2 2 [0, 1]

37 Neural Classification A neural network is just a parametrized function: y = f(x, w) How to train it? Write down an error function: Minimize it! (w.r.t. w) (y i f(x i,w)) 2

38 Neural Classification Recall that the squashing function is defined as: (t) = 1 1+e = (t)(1 (t))

39 Neural Classification OK, so we can minimize error using gradient descent. To do so, we i for each wi. How to do so? Easy for output i o i ) 2 = 2(y i o i i o i ) = 2(o i y i )o i (1 o i i chain rule Interior weights: repeat chain rule application.

40 Backpropagation This algorithm is called backpropagation. Bryson and Ho, 1969 Rumelhart, Hinton, and Williams, 1986.

41 Deep Neural Networks o1 o2 hn1 hn2 hn3. h11 h12 h13 x1 x2

42 Applications Fraud detection Internet advertising Friend or link prediction Sentiment analysis Face recognition Spam filtering

43 Applications MNIST Data Set Training set: 60k digits Test set: 10k digits