CMU-Q Lecture 24:

Transcription

1 CMU-Q Lecture 24: Supervised Learning 2 Teacher: Gianni A. Di Caro

2 SUPERVISED LEARNING Hypotheses space Hypothesis function Labeled Given Errors Performance criteria Given a collection of input features and outputs h θ x (i), y (i), i = 1,, m and a hypothesis function h θ, find parameters values θ that minimize the average empirical error: minimize θ 1 m m i=1 l h θ x (i), y (i) We need to specify: 1. The hypothesis class H, h θ H 2. The loss function l 3. The algorithm for solving the optimization problem (often approximately) 4. A complete ML design: from data processing to learning to validation and testing 2

3 CLASSIFICATION AND REGRESSION Features: Width, Lightness Classification: (Width, Lightness) {Salmon, Sea bass} (discrete) Regression: (Width, Lightness) Weight (continuous) h θ x : X R 2 Y = {0,1} Which hypothesis class H? h θ x : X R 2 Y R Complex boundaries, relations 3

4 PROBABILISTIC MODELS: DISCRIMINATIVE VS. GENERATIVE Regression and classification problems can be stated in probabilistic terms (later) The mapping y = h θ x that we are learning can be naturally interpreted as the probability of the output being y given the input data x (under the selected hypothesis h and the learned parameter vector θ) Discriminative models: Directly learn p y x) Parametric hypothesis Allow to discriminate between classes / predicted outputs Generative models / Probability distributions: Learn p(x, y), the probabilistic model that describes the data, then use Bayes rule Allow to generate data any relevant data = sea bass = salmon x 1 x 2 x 1 x 2 p y x) = p x y)p(y) p(x) = p(x, y) p(x) 4

5 GENERATIVE MODELS A discriminative model, that learn learns p y x; θ), can be used to label the data, to discriminate the data, but not to generate the data o E.g., a discriminative approach tries to find out which (linear, in this case) decision boundary allows for the best classification based on the training data, and takes decisions accordingly o Direct learning of the mapping from X to Y A generative approach would proceed as follows: 1. By looking at the feature data about salmons, build a model of a salmon 2. By looking at the feature data about sea basses, build a model of a sea bass 3. To classify a new fish based on its features x, we can match it against the salmon and the sea bass models, to see whether it looks more like the salmons or more like the sea basses we had seen in the training set 1,2,3 is equivalent to model p x y), where y = {ω 1, ω 2 }: the conditional probability that the observed features x are those of a salmon or a sea bass 5

6 GENERATIVE MODELS p x p x y = ω 1 ) models the distribution of salmon s features y = ω 2 ) models the distribution of sea bass features p(y) can be derived from the dataset or from other sources o E.g., p(ω 1 ) = ratio of salmons in the dataset, p(ω 2 ) = ratio of sea basses Bayes rule: p y x) = p x y)p(y) p(x) = p(x, y) p(x) posterior = likelihood prior evidence p x = p x y = ω 1 )p y = ω 1 + p x y = ω 2 )p(y = ω 2 ) To make a prediction: arg max y p y x) = arg max y p x y)p(y) p(x) = arg max y p x y)p(y) Equivalent to: decide ω 1 if p ω 1 x) > p ω 2 x), otherwise decide ω 2 6

7 GENERATIVE MODELS AND BAYES DECISION RULE Decide ω 1 if p x ω 1 )p ω 1 > p x ω 2 )p(ω 2 ) otherwise decide ω 2 Decide ω 1 if p x ω 1) > p(ω 2) p x ω 2 ) p(ω 1 ) otherwise decide ω 2 Likelihood ratio Two disconnected regions for class 2 7

8 GENERATIVE MODELS Given the joint distribution we can generate any conditional or marginal probability Sample from p(x, y) to obtain labeled data points Given the priors p(y), sample a class or a predictor value Given the class y, sample instance data p x predictor variable sample an expected output y) of that class, or, given a Downside: higher complexity, more parameters to learn Density estimation problem: Parametric (e.g., Gaussian densities) Non-parametric (full density estimation) 8

9 LET S GO BACK TO LINEAR REGRESSION Linear model as hypothesis: y = h x; w = w 0 + w 1 x 1 + w 2 x w d x d = w T x x = (1, x 1, x 2,, x d ) h Find w that minimizes the deviation from the desired answers: y (i) h x i, i in dataset Loss function: Mean squared error (MSE) m l = 1 m i=1 y i h x i 2 The model does not try to explain variation in observed ys for the data 9

10 STATISTICAL MODEL FOR LINEAR REGRESSION A statistical model of linear regression: y = w T x + ε ε ~ N(0, σ 2 ) y ~ N(w T x, σ 2 ) The model does explain variation in observed ys for the data in terms of a white Gaussian noise The conditional distribution of y given x: p y x; w, σ) = 1 σ 2π exp 1 2σ 2 y wt x 2 Probability of the output being y given the predictor x E y x = w T x 10

11 STATISTICAL MODEL FOR LINEAR REGRESSION Let s consider the entire data set D, and let s assume that all samples are independent and identically distributed (i.i.d.) random variables What is the joint probability of all training data? That is, the probability of observing all the outputs y in D given w and σ? By iid: p y (1), y (2),, y (m) p y (1), y (2),, y (m) x (1), x (2),, x (m) ; w, σ) x (1), x (2),, x (m) ; w, σ) = p y i m i=1 x (i) ; w, σ) L(D, w, σ) = ς m i=1 p y i x (i) ; w, σ) Likelihood function of predictions, the probability of observing the outputs y in D given w and σ Maximum likelihood estimation of the parameters w: parameter values maximizing the likelihood of the predictions, the value of the parameters such that the probability of observing the data in D is maximized w = arg max w L(D, w, σ) 11

12 STATISTICAL MODEL FOR LINEAR REGRESSION Log-Likelihood: l D, w, σ = log(l(d, w, σ)) = log ς m i=1 p y i x (i) ; w, σ) m = log p y i i=1 x (i) ; w, σ) Using the conditional density: p y x; w, σ) = 1 σ 2π exp 1 2σ 2 y wt x 2 m l D, w, σ = log i=1 = 1 m 2σ 2 i=1 1 σ 2π exp 1 2σ 2 y i w T x (i) 2 = y i w T x (i) 2 + c(σ) m i=1 1 2σ 2 y i w T x (i) 2 c(σ) Maximizing the predictive log-likelihood with regard to w, is equivalent to minimizing the MSE loss function Does it look familiar? max w l D, w, σ ~ min MSE w More in general, least squares linear fit under Gaussian noise corresponds to the maximum likelihood estimator of the data 12

13 NON-LINEAR, ADDITIVE REGRESSION MODELS 13

14 NON-LINEAR PROBLEMS? Design a non-linear regressor / classifier Modify the input data to make the problem linear 14

15 MAP DATA IN HIGHER DIMENSIONALITY FEATURE SPACES 15

16 MAP DATA IN HIGHER DIMENSIONALITY FEATURE SPACES The hyperplane is found in z-space, then projected back in x-space, where is an ellipsis The property of the solution of SVMs (that are in terms of dot products between feature vectors) allows to easily define a kernel function that implicitly perform the desired transformation, allowing keeping using linear classifiers. 16

17 NON-LINEAR, ADDITIVE REGRESSION MODELS Main idea to model nonlinearities: Replace inputs to linear units with b feature (basis) functions φ j x, j = 1,, b, where φ j x is an arbitrary function of x y = h x; w = w 0 + w 1 φ 1 x + w 2 φ 2 x + + w b φ b x = w T φ(x) h b b Original feature input New input Linear model 17

18 EXAMPLES OF FEATURE FUNCTIONS Higher order polynomial with one-dimensional input, x = (x) φ 1 x = x, φ 2 x = x 2, φ 3 x = x 3, Quadratic polynomial with two-dimensional inputs, x = (x 1, x 2 ) φ 1 x = x 1, φ 2 x = x 2 1, φ 3 x = x 2, φ 4 x = x 2 2, φ 3 x = x 1 x 2 Transcendent functions: φ 1 x = sin(x), φ 2 x = cos(x) 18

19 SOLUTION USING FEATURE FUNCTIONS The same techniques (analytical gradient + system of equations, or gradient descent) used for the plain linear case with MSE as loss function h x i ; w = w 0 + w 1 φ 1 x i + w 2 φ 2 x i + + w b φ b x i = w T φ(x i ) φ x i = (1, φ 1 x i, φ 2 x i,, φ b (x i )) m l = 1 m i=1 y i h x i 2 To find min w l we have to look where w l = 0 m w l = 2 m i=1 y i h x i φ x i = 0 Results in a system of b linear equations: m m m m w 0 1φ j x i + w 1 φ 1 x i φ j x i + + w k φ k x i φ j x i + w b φ b x i i=1 m i=1 i=1 i=1 = y i φ j x i j = 1,, b i=1 φ j x i 19

20 EXAMPLE OF SDG WITH FEATURE FUNCTIONS One dimensional feature vectors and high-order polynomial: x = x, φ i x = x i h x; w = w 0 + w 1 φ 1 x + w 2 φ 2 x + + w b φ b x = w 0 + w i x i On-line, single sample, (x i, y i ), gradient update, j = 1,, b w j = w j + α w l h x i ; w, y i = w j + α y i h x i φ j x i b i=1 Same form as in the linear regression model, with x j (i) φj x i 20

21 ELECTRICITY EXAMPLE New data: it doesn t look linear anymore 21

22 NEW HYPOTHESIS The complexity of the model grows: one parameter for each feature transformed according to a polynomial of order 2 (at least 3 parameters vs. 2 of original hypothesis) 22

23 NEW HYPOTHESIS At least 5 parameters (if we had multiple predicting features, all their order d products should be considered, resulting into a number of additional parameters) 23

24 NEW HYPOTHESIS The number of parameters is now larger than the data points, such that the polynomial can almost precisely fit the data Overfitting 24

25 SELECTING MODEL COMPLEXITY Dataset with 10 points, 1D features: which hypothesis class should we use? Linear regression: y = h x; w = w 0 + w 1 x Polynomial regression, cubic: y = h x; w = w 0 + w 1 x + w 2 x 2 + w 3 x 3 MSE for the loss functions Which model would give the smaller error in terms of MSE / least squares fit? 25

26 SELECTING MODEL COMPLEXITY Cubic regression provides a better fit to the data, and a smaller MSE Should we stick with the hypothesis h x; w = w 0 + w 1 x + w 2 x 2 + w 3 x 3? Since a higher order polynomial seems to provide a better fit, why don t we use a polynomial of order higher than 3? What is the highest order that makes sense for the given problem? 26

27 SELECTING MODEL COMPLEXITY For 10 data points, a degree 9 polynomial gives a perfect fit (Lagrange interpolation). Error is zero. Is it always good to minimize (even reduce to zero) the training error? Related (and more important) question: How do we (will) perform on new, unseen data? 27

28 OVERFITTING The 9-polynomial model totally fails the prediction for the new point! Overfitting: Situation when the training error is low and the generalization error is high. Causes of the phenomenon: Highly complex hypothesis model, with a large number of parameters (degrees of freedom) Small data size (as compared to the complexity of the model) The learned function has enough degrees of freedom to (over)fit all data perfectly 28

29 OVERFITTING Empirical loss vs. Generalization loss 29

30 TRAINING AND VALIDATION LOSS 30

31 SPLITTING DATASET IN TWO 31

32 PERFORMANCE ON VALIDATION SET 32

33 PERFORMANCE ON VALIDATION SET 33

34 INCREASING MODEL COMPLEXITY In this case, the small size of the dataset favors an easy overfitting by increasing the degree of the polynomial (i.e., hypothesis complexity). For a large multi-dimensional dataset this effect is less strong / evident 34

35 TRAINING VS. VALIDATION LOSS 35

36 MODEL SELECTION AND EVALUATION PROCESS 1. Break all available data into training and testing sets (e.g., 70% / 30%) 2. Break training set into training and validation sets (e.g., 70% / 30%) 3. Loop: i. Set a hyperparameter value (e.g., degree of polynomial model complexity) ii. iii. iv. Train the model using training sets Validate the model using validation sets Exit loop if (validation errors keep growing && training errors go to zero) 4. Choose hyperparameters using validation set results: hyperparameter values corresponding to lowest validation errors 5. (Optional) With the selected hyperparameters, retrain the model using all training data sets 6. Evaluate (generalization) performance on the testing sets (more on this next time) 36

37 MODEL SELECTION AND EVALUATION PROCESS Dataset Training set Testing set Internal training set Validation set Model 1 Model 2 Learn 1 Learn 2 Validate 1 Validate 2 Select best model Learn Model Model n Learn n Validate n 37