Overfitting, Bias / Variance Analysis
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1 Overfitting, Bias / Variance Analysis Professor Ameet Talwalkar Professor Ameet Talwalkar CS260 Machine Learning Algorithms February 8, 207 / 40
2 Outline Administration 2 Review of last lecture 3 Basic ideas to overcome overfitting 4 Bias/Variance Analysis Professor Ameet Talwalkar CS260 Machine Learning Algorithms February 8, / 40
3 Announcements HW2 will be returned in section on Friday HW3 due in class next Monday Midterm is next Wednesday Professor Ameet Talwalkar CS260 Machine Learning Algorithms February 8, / 40
4 Midterm Next Wednesday in class from 0am - :50am Completely closed-book (no notes allowed) Will include roughly 6 short answer questions and 3 long questions Short questions should take 5 minutes on average Long questions should take 5 minutes each Covers all material through (and including) today s lecture Goal is to test conceptual understanding of the course material Suggestion: carefully review lecture notes and problem sets Office hours / Section (see timing on course website) Professor Ameet Talwalkar CS260 Machine Learning Algorithms February 8, / 40
5 Outline Administration 2 Review of last lecture Linear Regression Ridge Regression for Numerical Purposes Non-linear Basis 3 Basic ideas to overcome overfitting 4 Bias/Variance Analysis Professor Ameet Talwalkar CS260 Machine Learning Algorithms February 8, / 40
6 Linear regression Setup Input: x R D (covariates, predictors, features, etc) Output: y R (responses, targets, outcomes, outputs, etc) Model: f : x y, with f(x) = w 0 + d w dx d = w 0 + w T x w = [w w 2 w D ] T : weights, parameters, or parameter vector w 0 is called bias We also sometimes call w = [w 0 w w 2 w D ] T parameters too Training data: D = {(x n, y n ), n =, 2,..., N} Professor Ameet Talwalkar CS260 Machine Learning Algorithms February 8, / 40
7 Linear regression Setup Input: x R D (covariates, predictors, features, etc) Output: y R (responses, targets, outcomes, outputs, etc) Model: f : x y, with f(x) = w 0 + d w dx d = w 0 + w T x w = [w w 2 w D ] T : weights, parameters, or parameter vector w 0 is called bias We also sometimes call w = [w 0 w w 2 w D ] T parameters too Training data: D = {(x n, y n ), n =, 2,..., N} Least Mean Squares (LMS) Objective: Minimize squared difference on training data (or residual sum of squares) RSS( w) = n [y n f(x n )] 2 = n [y n (w 0 + d w d x nd )] 2 Professor Ameet Talwalkar CS260 Machine Learning Algorithms February 8, / 40
8 Linear regression Setup Input: x R D (covariates, predictors, features, etc) Output: y R (responses, targets, outcomes, outputs, etc) Model: f : x y, with f(x) = w 0 + d w dx d = w 0 + w T x w = [w w 2 w D ] T : weights, parameters, or parameter vector w 0 is called bias We also sometimes call w = [w 0 w w 2 w D ] T parameters too Training data: D = {(x n, y n ), n =, 2,..., N} Least Mean Squares (LMS) Objective: Minimize squared difference on training data (or residual sum of squares) RSS( w) = n [y n f(x n )] 2 = n [y n (w 0 + d w d x nd )] 2 D Solution: Identify stationary points by taking derivative with respect to parameters and setting to zero, yielding normal equations Professor Ameet Talwalkar CS260 Machine Learning Algorithms February 8, / 40
9 LMS when x is D-dimensional RSS( w) in matrix form RSS( w) = n [y n (w 0 + d w d x nd )] 2 Professor Ameet Talwalkar CS260 Machine Learning Algorithms February 8, / 40
10 LMS when x is D-dimensional RSS( w) in matrix form RSS( w) = n [y n (w 0 + d w d x nd )] 2 = n [y n w T x n ] 2 where we have redefined some variables (by augmenting) x [ x x 2... x D ] T, w [w 0 w w 2... w D ] T Professor Ameet Talwalkar CS260 Machine Learning Algorithms February 8, / 40
11 LMS when x is D-dimensional RSS( w) in matrix form RSS( w) = n [y n (w 0 + d w d x nd )] 2 = n [y n w T x n ] 2 where we have redefined some variables (by augmenting) x [ x x 2... x D ] T, w [w 0 w w 2... w D ] T which leads to RSS( w) = n (y n w T x n )(y n x T n w) Professor Ameet Talwalkar CS260 Machine Learning Algorithms February 8, / 40
12 LMS when x is D-dimensional RSS( w) in matrix form RSS( w) = n [y n (w 0 + d w d x nd )] 2 = n [y n w T x n ] 2 where we have redefined some variables (by augmenting) x [ x x 2... x D ] T, w [w 0 w w 2... w D ] T which leads to RSS( w) = n = n (y n w T x n )(y n x T n w) w T x n x T n w 2y n x T n w + const. Professor Ameet Talwalkar CS260 Machine Learning Algorithms February 8, / 40
13 LMS when x is D-dimensional RSS( w) in matrix form RSS( w) = n [y n (w 0 + d w d x nd )] 2 = n [y n w T x n ] 2 where we have redefined some variables (by augmenting) x [ x x 2... x D ] T, w [w 0 w w 2... w D ] T which leads to RSS( w) = n (y n w T x n )(y n x T n w) = w T x n x T n w 2y n x T n w + const. n { ( ) ( ) = w T x n x T n w 2 y n x T n n n w } + const. Professor Ameet Talwalkar CS260 Machine Learning Algorithms February 8, / 40
14 RSS( w) in new notations From previous slide { ) ( RSS( w) = w T x n x T n n ( ) w 2 y n x T n n w } + const. Professor Ameet Talwalkar CS260 Machine Learning Algorithms February 8, / 40
15 RSS( w) in new notations From previous slide { ) ( RSS( w) = w T x n x T n n ( ) w 2 y n x T n n w } + const. Design matrix and target vector x T x X T 2 =. RN (D+), y = x T N y y 2. y N Professor Ameet Talwalkar CS260 Machine Learning Algorithms February 8, / 40
16 RSS( w) in new notations From previous slide { ) ( RSS( w) = w T x n x T n n ( ) w 2 y n x T n n w } + const. Design matrix and target vector x T x X T 2 =. RN (D+), y = x T N Compact expression { RSS( w) = X w y 2 2 = y y 2. y N ( ) } T w T X T X w 2 X T y w + const Professor Ameet Talwalkar CS260 Machine Learning Algorithms February 8, / 40
17 Solution in matrix form Compact expression { RSS( w) = X w y 2 2 = ( ) } T w T X T X w 2 X T y w + const Professor Ameet Talwalkar CS260 Machine Learning Algorithms February 8, / 40
18 Solution in matrix form Compact expression { RSS( w) = X w y 2 2 = ( ) } T w T X T X w 2 X T y w + const Gradients of Linear and Quadratic Functions x b x = b x x Ax = 2Ax (symmetric A) Professor Ameet Talwalkar CS260 Machine Learning Algorithms February 8, / 40
19 Solution in matrix form Compact expression { RSS( w) = X w y 2 2 = ( ) } T w T X T X w 2 X T y w + const Gradients of Linear and Quadratic Functions x b x = b x x Ax = 2Ax (symmetric A) Normal equation w RSS( w) X T Xw X T y = 0 Professor Ameet Talwalkar CS260 Machine Learning Algorithms February 8, / 40
20 Solution in matrix form Compact expression { RSS( w) = X w y 2 2 = ( ) } T w T X T X w 2 X T y w + const Gradients of Linear and Quadratic Functions x b x = b x x Ax = 2Ax (symmetric A) Normal equation w RSS( w) X T Xw X T y = 0 This leads to the least-mean-square (LMS) solution w LMS = ( X T X) X T y Professor Ameet Talwalkar CS260 Machine Learning Algorithms February 8, / 40
21 Practical concerns Bottleneck of computing the LMS solution w = ( XT X) Xy Matrix multiply of X T X R (D+) (D+) Inverting the matrix X T X Scalable methods Batch gradient descent Stochastic gradient descent Professor Ameet Talwalkar CS260 Machine Learning Algorithms February 8, / 40
22 (Batch) Gradient Descent Initialize w to w (0) (e.g., randomly); set t = 0; choose η > 0 Loop until convergence Compute the gradient RSS( w) = X T X w (t) X T y 2 Update the parameters w (t+) = w (t) η RSS( w) 3 t t + What is the complexity of each iteration? Professor Ameet Talwalkar CS260 Machine Learning Algorithms February 8, 207 / 40
23 (Batch) Gradient Descent Initialize w to w (0) (e.g., randomly); set t = 0; choose η > 0 Loop until convergence Compute the gradient RSS( w) = X T X w (t) X T y 2 Update the parameters w (t+) = w (t) η RSS( w) 3 t t + What is the complexity of each iteration? O(ND) Why does this work? Professor Ameet Talwalkar CS260 Machine Learning Algorithms February 8, 207 / 40
24 (Batch) Gradient Descent Initialize w to w (0) (e.g., randomly); set t = 0; choose η > 0 Loop until convergence Compute the gradient RSS( w) = X T X w (t) X T y 2 Update the parameters w (t+) = w (t) η RSS( w) 3 t t + What is the complexity of each iteration? O(ND) Why does this work? RSS( w) is convex (Hessian is PSD) Professor Ameet Talwalkar CS260 Machine Learning Algorithms February 8, 207 / 40
25 Stochastic gradient descent Widrow-Hoff rule: update parameters using one example at a time Professor Ameet Talwalkar CS260 Machine Learning Algorithms February 8, / 40
26 Stochastic gradient descent Widrow-Hoff rule: update parameters using one example at a time Initialize w to some w (0) ; set t = 0; choose η > 0 Loop until convergence random choose a training a sample x t 2 Compute its contribution to the gradient g t = ( x T t w(t) y t ) x t 3 Update the parameters w (t+) = w (t) ηg t 4 t t + Professor Ameet Talwalkar CS260 Machine Learning Algorithms February 8, / 40
27 Stochastic gradient descent Widrow-Hoff rule: update parameters using one example at a time Initialize w to some w (0) ; set t = 0; choose η > 0 Loop until convergence random choose a training a sample x t 2 Compute its contribution to the gradient g t = ( x T t w(t) y t ) x t 3 Update the parameters w (t+) = w (t) ηg t 4 t t + How does the complexity per iteration compare with gradient descent? Professor Ameet Talwalkar CS260 Machine Learning Algorithms February 8, / 40
28 Stochastic gradient descent Widrow-Hoff rule: update parameters using one example at a time Initialize w to some w (0) ; set t = 0; choose η > 0 Loop until convergence random choose a training a sample x t 2 Compute its contribution to the gradient g t = ( x T t w(t) y t ) x t 3 Update the parameters w (t+) = w (t) ηg t 4 t t + How does the complexity per iteration compare with gradient descent? O(ND) for gradient descent versus O(D) for SGD Professor Ameet Talwalkar CS260 Machine Learning Algorithms February 8, / 40
29 What if X T X is not invertible Why might this happen? Professor Ameet Talwalkar CS260 Machine Learning Algorithms February 8, / 40
30 What if X T X is not invertible Why might this happen? Answer : N < D. Intuitively, not enough data to estimate all parameters. Professor Ameet Talwalkar CS260 Machine Learning Algorithms February 8, / 40
31 What if X T X is not invertible Why might this happen? Answer : N < D. Intuitively, not enough data to estimate all parameters. Answer 2: Columns of X are not linearly independent, e.g., some features are perfectly correlated. In this case, solution is not unique. Professor Ameet Talwalkar CS260 Machine Learning Algorithms February 8, / 40
32 Ridge regression What can we do when X T X is not invertible? Professor Ameet Talwalkar CS260 Machine Learning Algorithms February 8, / 40
33 Ridge regression What can we do when X T X is not invertible? Add regularizer so that all singular values are at least λ > 0! This is equivalent to adding an extra term to RSS( w) RSS( w) { { }}{ ( ) } T w T X T X w 2 X T y w λ w 2 2 }{{} regularization Professor Ameet Talwalkar CS260 Machine Learning Algorithms February 8, / 40
34 Ridge regression What can we do when X T X is not invertible? Solution Add regularizer so that all singular values are at least λ > 0! This is equivalent to adding an extra term to RSS( w) RSS( w) { { }}{ ( ) } T w T X T X w 2 X T y w λ w 2 2 }{{} regularization Can derive normal equations as before Solution is of the form: w = ( X T X + λi) XT y Professor Ameet Talwalkar CS260 Machine Learning Algorithms February 8, / 40
35 Is a linear modeling assumption always a good idea? Example of nonlinear classification Example of nonlinear regression t 0 0 x Professor Ameet Talwalkar CS260 Machine Learning Algorithms February 8, / 40
36 General nonlinear basis functions We can use a nonlinear mapping φ(x) : x R D z R M M is dimensionality of new features z (or φ(x)) M could be greater than, less than, or equal to D Professor Ameet Talwalkar CS260 Machine Learning Algorithms February 8, / 40
37 General nonlinear basis functions We can use a nonlinear mapping φ(x) : x R D z R M M is dimensionality of new features z (or φ(x)) M could be greater than, less than, or equal to D We can apply existing learning methods on the transformed data linear methods: prediction is based on w T φ(x) other methods: nearest neighbors, decision trees, etc Professor Ameet Talwalkar CS260 Machine Learning Algorithms February 8, / 40
38 Regression with nonlinear basis Residual sum squares [w T φ(x n ) y n ] 2 n where w R M, the same dimensionality as the transformed features φ(x). The LMS solution can be formulated with the new design matrix φ(x ) T φ(x 2 ) T Φ =. RN M, w lms = ( Φ T Φ ) Φ T y φ(x N ) T Professor Ameet Talwalkar CS260 Machine Learning Algorithms February 8, / 40
39 Example with regression Polynomial basis functions x φ(x) = x 2 f(x) = w 0 +. x M M w m x m m= Professor Ameet Talwalkar CS260 Machine Learning Algorithms February 8, / 40
40 Example with regression Polynomial basis functions x φ(x) = x 2 f(x) = w 0 +. x M M w m x m m= Fitting samples from a sine function: underfitting as f(x) is too simple M = 0 M = t t x 0 x Professor Ameet Talwalkar CS260 Machine Learning Algorithms February 8, / 40
41 Adding high-order terms M=3 t M = x Professor Ameet Talwalkar CS260 Machine Learning Algorithms February 8, / 40
42 Adding high-order terms M=3 M=9: overfitting t M = 3 t M = x 0 x More complex features lead to better results on the training data, but potentially worse results on new data, e.g., test data! Professor Ameet Talwalkar CS260 Machine Learning Algorithms February 8, / 40
43 Overfitting Parameters for higher-order polynomials are very large M = 0 M = M = 3 M = 9 w w w w w w w w w w Professor Ameet Talwalkar CS260 Machine Learning Algorithms February 8, / 40
44 Overfitting can be quite disastrous Fitting the housing price data with large M Predicted price goes to zero (and is ultimately negative) if you buy a big enough house! How might we prevent overfitting? Professor Ameet Talwalkar CS260 Machine Learning Algorithms February 8, / 40
45 Outline Administration 2 Review of last lecture 3 Basic ideas to overcome overfitting Use more training data Regularization methods 4 Bias/Variance Analysis Professor Ameet Talwalkar CS260 Machine Learning Algorithms February 8, / 40
46 Use more training data to prevent over fitting The more, the merrier t M = x Professor Ameet Talwalkar CS260 Machine Learning Algorithms February 8, / 40
47 Use more training data to prevent over fitting The more, the merrier M = 9 N = 5 N = 00 t t t x 0 x 0 x Professor Ameet Talwalkar CS260 Machine Learning Algorithms February 8, / 40
48 Use more training data to prevent over fitting The more, the merrier M = 9 N = 5 N = 00 t t t x 0 x What if we do not have a lot of data? 0 x Professor Ameet Talwalkar CS260 Machine Learning Algorithms February 8, / 40
49 Regularization methods Intuition: Give preference to simpler models How do we define a simple linear regression model w T x? Professor Ameet Talwalkar CS260 Machine Learning Algorithms February 8, / 40
50 Regularization methods Intuition: Give preference to simpler models How do we define a simple linear regression model w T x? Our Strategy: Place a prior on our weights Interpret w as a random variable Assume that each w d is centered around zero Use observed data D to update our prior belief on w Professor Ameet Talwalkar CS260 Machine Learning Algorithms February 8, / 40
51 Review: Probabilistic interpretation for LMS LMS model: Y = w X + η Professor Ameet Talwalkar CS260 Machine Learning Algorithms February 8, / 40
52 Review: Probabilistic interpretation for LMS LMS model: Y = w X + η η N(0, σ0) 2 is a Gaussian random variable Thus, Y N(w X, σ0) 2 Professor Ameet Talwalkar CS260 Machine Learning Algorithms February 8, / 40
53 Review: Probabilistic interpretation for LMS LMS model: Y = w X + η η N(0, σ0) 2 is a Gaussian random variable Thus, Y N(w X, σ0) 2 We assume that w is fixed (Frequentist interpretation) Professor Ameet Talwalkar CS260 Machine Learning Algorithms February 8, / 40
54 Review: Probabilistic interpretation for LMS LMS model: Y = w X + η η N(0, σ0) 2 is a Gaussian random variable Thus, Y N(w X, σ0) 2 We assume that w is fixed (Frequentist interpretation) The likelihood function maps parameters to probabilities L : w, σ0 2 p(d w, σ0) 2 Professor Ameet Talwalkar CS260 Machine Learning Algorithms February 8, / 40
55 Review: Probabilistic interpretation for LMS LMS model: Y = w X + η η N(0, σ0) 2 is a Gaussian random variable Thus, Y N(w X, σ0) 2 We assume that w is fixed (Frequentist interpretation) The likelihood function maps parameters to probabilities L : w, σ 2 0 p(d w, σ 2 0) = p(y X, w, σ 2 0) = n p(y n x n, w, σ 2 0) Professor Ameet Talwalkar CS260 Machine Learning Algorithms February 8, / 40
56 Review: Probabilistic interpretation for LMS LMS model: Y = w X + η η N(0, σ0) 2 is a Gaussian random variable Thus, Y N(w X, σ0) 2 We assume that w is fixed (Frequentist interpretation) The likelihood function maps parameters to probabilities L : w, σ 2 0 p(d w, σ 2 0) = p(y X, w, σ 2 0) = n p(y n x n, w, σ 2 0) Maximizing likelihood with respect to w minimizes RSS and yields the LMS solution: w LMS = w ML = arg max w L(w, σ 2 0) Professor Ameet Talwalkar CS260 Machine Learning Algorithms February 8, / 40
57 Probabilistic interpretation of Ridge Regression Ridge Regression model: Y = w X + η Professor Ameet Talwalkar CS260 Machine Learning Algorithms February 8, / 40
58 Probabilistic interpretation of Ridge Regression Ridge Regression model: Y = w X + η Y N(w X, σ0) 2 is a Gaussian random variable (as before) wd N(0, σ 2 ) are i.i.d. Gaussian random variables (unlike before) Note that all wd share the same variance σ 2 Professor Ameet Talwalkar CS260 Machine Learning Algorithms February 8, / 40
59 Probabilistic interpretation of Ridge Regression Ridge Regression model: Y = w X + η Y N(w X, σ0) 2 is a Gaussian random variable (as before) wd N(0, σ 2 ) are i.i.d. Gaussian random variables (unlike before) Note that all wd share the same variance σ 2 w is a random variable (Bayesian interpretation) Professor Ameet Talwalkar CS260 Machine Learning Algorithms February 8, / 40
60 Probabilistic interpretation of Ridge Regression Ridge Regression model: Y = w X + η Y N(w X, σ0) 2 is a Gaussian random variable (as before) wd N(0, σ 2 ) are i.i.d. Gaussian random variables (unlike before) Note that all wd share the same variance σ 2 w is a random variable (Bayesian interpretation) To find w given data D, we can compute posterior distribution of w: p(w D) = Professor Ameet Talwalkar CS260 Machine Learning Algorithms February 8, / 40
61 Probabilistic interpretation of Ridge Regression Ridge Regression model: Y = w X + η Y N(w X, σ0) 2 is a Gaussian random variable (as before) wd N(0, σ 2 ) are i.i.d. Gaussian random variables (unlike before) Note that all wd share the same variance σ 2 w is a random variable (Bayesian interpretation) To find w given data D, we can compute posterior distribution of w: p(w D) = p(d w)p(w) p(d) Professor Ameet Talwalkar CS260 Machine Learning Algorithms February 8, / 40
62 Probabilistic interpretation of Ridge Regression Ridge Regression model: Y = w X + η Y N(w X, σ0) 2 is a Gaussian random variable (as before) wd N(0, σ 2 ) are i.i.d. Gaussian random variables (unlike before) Note that all wd share the same variance σ 2 w is a random variable (Bayesian interpretation) To find w given data D, we can compute posterior distribution of w: p(w D) = p(d w)p(w) p(d) Maximum a posterior (MAP) estimate: w map = arg max w p(w D) = arg max w p(d w)p(w) Professor Ameet Talwalkar CS260 Machine Learning Algorithms February 8, / 40
63 Probabilistic interpretation of Ridge Regression Ridge Regression model: Y = w X + η Y N(w X, σ0) 2 is a Gaussian random variable (as before) wd N(0, σ 2 ) are i.i.d. Gaussian random variables (unlike before) Note that all wd share the same variance σ 2 w is a random variable (Bayesian interpretation) To find w given data D, we can compute posterior distribution of w: p(w D) = p(d w)p(w) p(d) Maximum a posterior (MAP) estimate: w map = arg max w p(w D) = arg max w p(d w)p(w) What s the relationship between MAP and MLE? Professor Ameet Talwalkar CS260 Machine Learning Algorithms February 8, / 40
64 Probabilistic interpretation of Ridge Regression Ridge Regression model: Y = w X + η Y N(w X, σ0) 2 is a Gaussian random variable (as before) wd N(0, σ 2 ) are i.i.d. Gaussian random variables (unlike before) Note that all wd share the same variance σ 2 w is a random variable (Bayesian interpretation) To find w given data D, we can compute posterior distribution of w: p(w D) = p(d w)p(w) p(d) Maximum a posterior (MAP) estimate: w map = arg max w p(w D) = arg max w p(d w)p(w) What s the relationship between MAP and MLE? MAP reduces to MLE if we assume uniform prior for p(w) Professor Ameet Talwalkar CS260 Machine Learning Algorithms February 8, / 40
65 Probabilistic interpretation of Ridge Regression Ridge Regression model: Y = w X + η Y N(w X, σ0) 2 is a Gaussian random variable (as before) wd N(0, σ 2 ) are i.i.d. Gaussian random variables (unlike before) Note that all wd share the same variance σ 2 w is a random variable (Bayesian interpretation) To find w given data D, we can compute posterior distribution of w: p(w D) = p(d w)p(w) p(d) Maximum a posterior (MAP) estimate: w map = arg max w p(w D) = arg max w p(d w)p(w) What s the relationship between MAP and MLE? MAP reduces to MLE if we assume uniform prior for p(w) Fully Bayesian treatment considers entire posterior, not just the mode Professor Ameet Talwalkar CS260 Machine Learning Algorithms February 8, / 40
66 Estimating w Let X,..., X N be IID with y w, x N(w x, σ0 2) Let w d be IID with w d N(0, σ 2 ) Joint likelihood of data and parameters (given σ 0, σ) p(d, w) = p(d w)p(w) = n p(y n x n, w) d p(w d ) Professor Ameet Talwalkar CS260 Machine Learning Algorithms February 8, / 40
67 Estimating w Let X,..., X N be IID with y w, x N(w x, σ0 2) Let w d be IID with w d N(0, σ 2 ) Joint likelihood of data and parameters (given σ 0, σ) p(d, w) = p(d w)p(w) = n p(y n x n, w) d p(w d ) Joint log likelihood Plugging in Gaussian PDF, we get: log p(d, w) = n log p(y n x n, w) + d log p(w d ) = n (wt x n y n ) 2 2σ 2 0 d 2σ 2 w2 d + const Professor Ameet Talwalkar CS260 Machine Learning Algorithms February 8, / 40
68 Estimating w Let X,..., X N be IID with y w, x N(w x, σ0 2) Let w d be IID with w d N(0, σ 2 ) Joint likelihood of data and parameters (given σ 0, σ) p(d, w) = p(d w)p(w) = n p(y n x n, w) d p(w d ) Joint log likelihood Plugging in Gaussian PDF, we get: log p(d, w) = n log p(y n x n, w) + d log p(w d ) = n (wt x n y n ) 2 2σ 2 0 d 2σ 2 w2 d + const MAP estimate: w map = arg max w log p(d, w) As with LMS, set gradient equal to zero and solve (for w) Professor Ameet Talwalkar CS260 Machine Learning Algorithms February 8, / 40
69 Maximum a posterior (MAP) estimate Regularized linear regression: a new error to minimize E(w) = n (w T x n y n ) 2 + λ w 2 2 where λ > 0 is used to denote σ0 2/σ2. This extra term w 2 2 regularization/regularizer and controls the model complexity. is called Professor Ameet Talwalkar CS260 Machine Learning Algorithms February 8, / 40
70 Maximum a posterior (MAP) estimate Regularized linear regression: a new error to minimize E(w) = n (w T x n y n ) 2 + λ w 2 2 where λ > 0 is used to denote σ0 2/σ2. This extra term w 2 2 regularization/regularizer and controls the model complexity. is called Intuitions Professor Ameet Talwalkar CS260 Machine Learning Algorithms February 8, / 40
71 Maximum a posterior (MAP) estimate Regularized linear regression: a new error to minimize E(w) = n (w T x n y n ) 2 + λ w 2 2 where λ > 0 is used to denote σ0 2/σ2. This extra term w 2 2 regularization/regularizer and controls the model complexity. is called Intuitions If λ +, then σ 2 0 σ 2. That is, the variance of noise is far greater than what our prior model can allow for w. In this case, our prior model on w would be more accurate than what data can tell us. Thus, we are getting a simple model. Numerically, w map 0 Professor Ameet Talwalkar CS260 Machine Learning Algorithms February 8, / 40
72 Maximum a posterior (MAP) estimate Regularized linear regression: a new error to minimize E(w) = n (w T x n y n ) 2 + λ w 2 2 where λ > 0 is used to denote σ0 2/σ2. This extra term w 2 2 regularization/regularizer and controls the model complexity. is called Intuitions If λ +, then σ 2 0 σ 2. That is, the variance of noise is far greater than what our prior model can allow for w. In this case, our prior model on w would be more accurate than what data can tell us. Thus, we are getting a simple model. Numerically, w map 0 If λ 0, then we trust our data more. Numerically, w map w lms = arg min n (w T x n y n ) 2 Professor Ameet Talwalkar CS260 Machine Learning Algorithms February 8, / 40
73 Closed-form solution For regularized linear regression: the solution changes very little (in form) from the LMS solution arg min n (w T x n y n ) 2 + λ w 2 2 w map = ( X T X + λi ) X T y and reduces to the LMS solution when λ = 0, as expected. Professor Ameet Talwalkar CS260 Machine Learning Algorithms February 8, / 40
74 Closed-form solution For regularized linear regression: the solution changes very little (in form) from the LMS solution arg min n (w T x n y n ) 2 + λ w 2 2 w map = ( X T X + λi ) X T y and reduces to the LMS solution when λ = 0, as expected. Gradients and Hessian change nominally too E(w) = 2(X T Xw X T y + λw), H = 2(X T X + λi) As long as λ 0, the optimization is convex. Professor Ameet Talwalkar CS260 Machine Learning Algorithms February 8, / 40
75 Example: fitting data with polynomials Our regression model y = M w m x m m= Regularization would discourage large parameter values as we saw with the LMS solution, thus potentially preventing overfitting. M = 0 M = M = 3 M = 9 w w w w w w w w w w Professor Ameet Talwalkar CS260 Machine Learning Algorithms February 8, / 40
76 Overfitting in terms of λ Overfitting is reduced from complex model to simpler one with the help of increasing regularizers t M = x Professor Ameet Talwalkar CS260 Machine Learning Algorithms February 8, / 40
77 Overfitting in terms of λ Overfitting is reduced from complex model to simpler one with the help of increasing regularizers M = 9 ln λ = 8 t t x 0 x Professor Ameet Talwalkar CS260 Machine Learning Algorithms February 8, / 40
78 Overfitting in terms of λ Overfitting is reduced from complex model to simpler one with the help of increasing regularizers M = 9 ln λ = 8 ln λ = 0 t t t x 0 x 0 x Professor Ameet Talwalkar CS260 Machine Learning Algorithms February 8, / 40
79 Overfitting in terms of λ Overfitting is reduced from complex model to simpler one with the help of increasing regularizers M = 9 ln λ = 8 ln λ = 0 t t t x 0 x 0 x λ vs. residual error shows the difference of the model performance on training and testing dataset Training Test ERMS ln λ Professor Ameet Talwalkar CS260 Machine Learning Algorithms February 8, / 40
80 The effect of λ Large λ attenuates parameters towards 0 ln λ = ln λ = 8 ln λ = 0 w w w w w w w w w w Professor Ameet Talwalkar CS260 Machine Learning Algorithms February 8, / 40
81 Regularized methods for classification Adding regularizer to the cross-entropy functions used for binary and multinomial logistic regression E(w) = n E(w, w 2,..., w K ) = n {y n log σ(w T x n ) + ( y n ) log[ σ(w T x n )]} + λ w 2 2 log P (C k x n ) + λ w k 2 2 k k Professor Ameet Talwalkar CS260 Machine Learning Algorithms February 8, / 40
82 Regularized methods for classification Adding regularizer to the cross-entropy functions used for binary and multinomial logistic regression E(w) = n E(w, w 2,..., w K ) = n Numerical optimization {y n log σ(w T x n ) + ( y n ) log[ σ(w T x n )]} + λ w 2 2 log P (C k x n ) + λ w k 2 2 k k Objective functions remain to be convex as long as λ 0. Gradients and Hessians change marginally and can be easily derived. Professor Ameet Talwalkar CS260 Machine Learning Algorithms February 8, / 40
83 How to choose the right amount of regularization? Can we tune λ on the training dataset? Professor Ameet Talwalkar CS260 Machine Learning Algorithms February 8, / 40
84 How to choose the right amount of regularization? Can we tune λ on the training dataset? No: as this will always set λ to zero, i.e., no regularization, defeating our intention of controlling model complexity λ is thus a hyperparemeter. To tune it, We can use a validation set or do cross validation (as previously discussed) Professor Ameet Talwalkar CS260 Machine Learning Algorithms February 8, / 40
85 Outline Administration 2 Review of last lecture 3 Basic ideas to overcome overfitting 4 Bias/Variance Analysis Professor Ameet Talwalkar CS260 Machine Learning Algorithms February 8, / 40
86 Basic and important machine learning concepts Supervised learning We aim to build a function h(x) to predict the true value y associated with x. If we make a mistake, we incur a loss l(h(x), y) Professor Ameet Talwalkar CS260 Machine Learning Algorithms February 8, / 40
87 Basic and important machine learning concepts Supervised learning We aim to build a function h(x) to predict the true value y associated with x. If we make a mistake, we incur a loss l(h(x), y) Example: quadratic loss function for regression when y is continuous l(h(x), y) = [h(x) y] 2 Ex: when y = 0 h(x) Professor Ameet Talwalkar CS260 Machine Learning Algorithms February 8, / 40
88 Other types of loss functions For classification: cross-entropy loss (also called logistic loss) l(h(x), y) = y log h(x) ( y) log[ h(x)] Ex: when y = h(x) Professor Ameet Talwalkar CS260 Machine Learning Algorithms February 8, / 40
89 Measure how good our predictor is Risk: Given the true distribution of data p(x, y), the risk is R[h(x)] = l(h(x), y)p(x, y)dxd y x,y Professor Ameet Talwalkar CS260 Machine Learning Algorithms February 8, / 40
90 Measure how good our predictor is Risk: Given the true distribution of data p(x, y), the risk is R[h(x)] = l(h(x), y)p(x, y)dxd y x,y However, we cannot compute R[h(x)], so we use empirical risk, given a training dataset D R emp [h(x)] = l(h(x n ), y n ) N n Professor Ameet Talwalkar CS260 Machine Learning Algorithms February 8, / 40
91 Measure how good our predictor is Risk: Given the true distribution of data p(x, y), the risk is R[h(x)] = l(h(x), y)p(x, y)dxd y x,y However, we cannot compute R[h(x)], so we use empirical risk, given a training dataset D Intuitively, as N +, R emp [h(x)] = l(h(x n ), y n ) N R emp [h(x)] R[h(x)] n Professor Ameet Talwalkar CS260 Machine Learning Algorithms February 8, / 40
92 How this relates to what we have learned? So far, we have been doing empirical risk minimization (ERM) For linear regression, h(x) = w T x, and we use squared loss For logistic regression, h(x) = σ(w T x), and we use cross-entropy loss Professor Ameet Talwalkar CS260 Machine Learning Algorithms February 8, / 40
93 How this relates to what we have learned? So far, we have been doing empirical risk minimization (ERM) For linear regression, h(x) = w T x, and we use squared loss For logistic regression, h(x) = σ(w T x), and we use cross-entropy loss ERM might be problematic Professor Ameet Talwalkar CS260 Machine Learning Algorithms February 8, / 40
94 How this relates to what we have learned? So far, we have been doing empirical risk minimization (ERM) For linear regression, h(x) = w T x, and we use squared loss For logistic regression, h(x) = σ(w T x), and we use cross-entropy loss ERM might be problematic If h(x) is complicated enough, R emp [h(x)] 0 But then h(x) is unlikely to do well in predicting things out of the training dataset D This is called poor generalization or overfitting. We have just discussed approaches to address this issue. We ll explore why regularization might work from the context of the bias-variance tradeoff, focusing on regression / squared loss Professor Ameet Talwalkar CS260 Machine Learning Algorithms February 8, / 40
95 Bias/variance tradeoff (Looking ahead) Error decomposes into 3 terms E D R[h D (x)] = variance + bias 2 + noise We will prove this result, and interpret what it means... Professor Ameet Talwalkar CS260 Machine Learning Algorithms February 8, / 40
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