Instance-based Learning
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1 Instance-based Learning Machine Learning 10701/15781 Carlos Guestrin Carnegie Mellon University October 15 th, Carlos Guestrin 1 1-Nearest Neighbor Four things make a memory based learner: 1. A distance metric Euclidian (and many more) 2. How many nearby neighbors to look at? One 3. A weighting function (optional) Unused 4. How to fit with the local points? Just predict the same output as the nearest neighbor Carlos Guestrin 2 1
2 Consistency of 1-NN Consider an estimator f n trained on n examples e.g., 1-NN, neural nets, regression,... Estimator is consistent if true error goes to zero as amount of data increases e.g., for no noise data, consistent if: Regression is not consistent! Representation bias 1-NN is consistent (under some mild fineprint) What about variance??? Carlos Guestrin 3 1-NN overfits? Carlos Guestrin 4 2
3 k-nearest Neighbor Four things make a memory based learner: 1. A distance metric Euclidian (and many more) 2. How many nearby neighbors to look at? k 1. A weighting function (optional) Unused 2. How to fit with the local points? Just predict the average output among the k nearest neighbors Carlos Guestrin 5 k-nearest Neighbor (here k=9) K-nearest neighbor for function fitting smoothes away noise, but there are clear deficiencies. What can we do about all the discontinuities that k-nn gives us? Carlos Guestrin 6 3
4 Weighted k-nns Neighbors are not all the same Carlos Guestrin 7 Kernel regression Four things make a memory based learner: 1. A distance metric Euclidian (and many more) 2. How many nearby neighbors to look at? All of them 3. A weighting function (optional) w i = exp(-d(x i, query) 2 / K w2 ) Nearby points to the query are weighted strongly, far points weakly. The K W parameter is the Kernel Width. Very important. 4. How to fit with the local points? Predict the weighted average of the outputs: predict = Σw i y i / Σw i Carlos Guestrin 8 4
5 Weighting functions w i = exp(-d(x i, query) 2 / K w2 ) Typically optimize K w using gradient descent (Our examples use Gaussian) Carlos Guestrin 9 Kernel regression predictions K W =10 K W =20 K W =80 Increasing the kernel width K w means further away points get an opportunity to influence you. As K w 1, the prediction tends to the global average Carlos Guestrin 10 5
6 Kernel regression on our test cases KW=1/32 of x-axis width. KW=1/32 of x-axis width. KW=1/16 axis width. Choosing a good K w is important. Not just for Kernel Regression, but for all the locally weighted learners we re about to see Carlos Guestrin 11 Kernel regression can look bad KW = Best. KW = Best. KW = Best. Time to try something more powerful Carlos Guestrin 12 6
7 Locally weighted regression Kernel regression: Take a very very conservative function approximator called AVERAGING. Locally weight it. Locally weighted regression: Take a conservative function approximator called LINEAR REGRESSION. Locally weight it Carlos Guestrin 13 Locally weighted regression Four things make a memory based learner: A distance metric Any How many nearby neighbors to look at? All of them A weighting function (optional) Kernels wi = exp(-d(xi, query) 2 / Kw 2 ) How to fit with the local points? General weighted regression: ∠= argmin! â N 2 w k k= 1 ( ) 2 T y " â x k k Carlos Guestrin 14 7
8 How LWR works Query Linear regression Same parameters for all queries ∠= T!1 T ( X X) X Y Locally weighted regression Solve weighted linear regression for each query (( ) T WX) "1 ( WX) T WY? ˆ = WX & w1 $ $ 0 W = $ 0 $ 0 #! 0! 0!! % w n " Carlos Guestrin 15 0 w O Another view of LWR Image from Cohn, D.A., Ghahramani, Z., and Jordan, M.I. (1996) Carlos "Active GuestrinLearning with Statistical Models", JAIR Volume 4, pages
9 LWR on our test cases KW = 1/16 of x-axis width. KW = 1/32 of x-axis width. KW = 1/8 of x-axis width Carlos Guestrin 17 Locally weighted polynomial regression Kernel Regression Kernel width K W at optimal level. KW = 1/100 x-axis LW Linear Regression Kernel width K W at optimal level. KW = 1/40 x-axis LW Quadratic Regression Kernel width K W at optimal level. KW = 1/15 x-axis Local quadratic regression is easy: just add quadratic terms to the WXTWX matrix. As the regression degree increases, the kernel width can increase without introducing bias Carlos Guestrin 18 9
10 Curse of dimensionality for instance-based learning Must store and retreve all data! Most real work done during testing For every test sample, must search through all dataset very slow! We ll see fast methods for dealing with large datasets Instance-based learning often poor with noisy or irrelevant features Carlos Guestrin 19 Curse of the irrelevant feature Carlos Guestrin 20 10
11 What you need to know about instance-based learning k-nn Simplest learning algorithm With sufficient data, very hard to beat strawman approach Picking k? Kernel regression Set k to n (number of data points) and optimize weights by gradient descent Smoother than k-nn Locally weighted regression Generalizes kernel regression, not just local average Curse of dimensionality Must remember (very large) dataset for prediction Irrelevant features often killers for instance-based approaches Carlos Guestrin 21 Acknowledgment This lecture contains some material from Andrew Moore s excellent collection of ML tutorials: Carlos Guestrin 22 11
12 Announcements Recitation this week: Neural networks Project proposals due next Wednesday Exciting data: Swivel.com - user generated graphs Recognizing Captchas Election contributions Activity recognition Carlos Guestrin 23 Support Vector Machines Machine Learning 10701/15781 Carlos Guestrin Carnegie Mellon University October 15 th, Carlos Guestrin 24 12
13 Linear classifiers Which line is better? Data: Example i: w.x = j w (j) x (j) Carlos Guestrin 25 Pick the one with the largest margin! w.x + b = 0 w.x = j w (j) x (j) Carlos Guestrin 26 13
14 Maximize the margin w.x + b = Carlos Guestrin 27 But there are a many planes w.x + b = Carlos Guestrin 28 14
15 Review: Normal to a plane w.x + b = Carlos Guestrin 29 Normalized margin Canonical hyperplanes w.x + b = +1 w.x + b = 0 w.x + b = -1 x + x - margin 2γ Carlos Guestrin 30 15
16 Normalized margin Canonical hyperplanes w.x + b = +1 w.x + b = 0 w.x + b = -1 x + x - margin 2γ Carlos Guestrin 31 Margin maximization using canonical hyperplanes w.x + b = +1 w.x + b = 0 w.x + b = -1 margin 2γ Carlos Guestrin 32 16
17 Support vector machines (SVMs) w.x + b = +1 w.x + b = 0 w.x + b = -1 Solve efficiently by quadratic programming (QP) Well-studied solution algorithms Hyperplane defined by support vectors margin 2γ Carlos Guestrin 33 What if the data is not linearly separable? Use features of features of features of features Carlos Guestrin 34 17
18 What if the data is still not linearly separable? Minimize w.w and number of training mistakes Tradeoff two criteria? Tradeoff #(mistakes) and w.w 0/1 loss Slack penalty C Not QP anymore Also doesn t distinguish near misses and really bad mistakes Carlos Guestrin 35 Slack variables Hinge loss If margin 1, don t care If margin < 1, pay linear penalty Carlos Guestrin 36 18
19 Side note: What s the difference between SVMs and logistic regression? SVM: Logistic regression: Log loss: Carlos Guestrin 37 What about multiple classes? Carlos Guestrin 38 19
20 One against All Learn 3 classifiers: Carlos Guestrin 39 Learn 1 classifier: Multiclass SVM Simultaneously learn 3 sets of weights Carlos Guestrin 40 20
21 Learn 1 classifier: Multiclass SVM Carlos Guestrin 41 What you need to know Maximizing margin Derivation of SVM formulation Slack variables and hinge loss Relationship between SVMs and logistic regression 0/1 loss Hinge loss Log loss Tackling multiple class One against All Multiclass SVMs Carlos Guestrin 42 21
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