Supervised Learning: Nearest Neighbors
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1 CS 2750: Machine Learning Supervised Learning: Nearest Neighbors Prof. Adriana Kovashka University of Pittsburgh February 1, 2016
2 Today: Supervised Learning Part I Basic formulation of the simplest classifier: K Nearest Neighbors Example uses Generalizing the distance metric and weighting neighbors differently Problems: The curse of dimensionality Picking K Approximation strategies
3 Key Idea Supervised learning: We want to learn to predict, for a new data point x, its label y (e.g. spam / not spam) Don t learn an explicit function F: X Y Keep all training data {X, Y} For a test example x, find the training example x i closest to it (e.g. using Euclidean distance) Then copy the target label y i as the label for x
4 Related Methods Instance-based methods Exemplar methods Memory-based methods Non-parametric methods
5 1-Nearest Neighbor Classifier Training examples from class 1 Test example Training examples from class 2 f(x) = label of the training example nearest to x All we need is a distance function for our inputs No training required! Slide credit: Lana Lazebnik
6 K-Nearest Neighbors Classifier For a new point, find the k closest points from training data (e.g. k=5) Labels of the k points vote to classify Black = negative Red = positive If query lands here, the 5 NN consist of 3 negatives and 2 positives, so we classify it as negative. Slide credit: David Lowe
7 1-nearest neighbor x x o o x + x o o o x o + x x x x2 o x1 Slide credit: Derek Hoiem
8 3-nearest neighbor x x o o x + x o o o x o + x x x x2 o x1 Slide credit: Derek Hoiem
9 5-nearest neighbor x x o o x + x o o o x o + x x x x2 o x1 What are the tradeoffs of having a too large k? Too small k? Slide credit: Derek Hoiem
10 Instance/Memory-based Learning Four things make a memory based learner: A distance metric How many nearby neighbors to look at? A weighting function (optional) How to fit with the local points? Slide credit: Carlos Guestrin
11 1-Nearest Neighbor Four things make a memory based learner: A distance metric Euclidean (and others) How many nearby neighbors to look at? 1 A weighting function (optional) Not used How to fit with the local points? Just predict the same output as the nearest neighbor Slide credit: Carlos Guestrin
12 k-nearest Neighbor Four things make a memory based learner: A distance metric Euclidean (and others) How many nearby neighbors to look at? k A weighting function (optional) Not used How to fit with the local points? Just predict the average output among the nearest neighbors Slide credit: Carlos Guestrin
13 Formal Definition Classification: Regression:
14 Example: Predict where this picture was taken Hays and Efros, IM2GPS: Estimating Geographic Information from a Single Image, CVPR 2008
15 Example: Predict where this picture was taken Hays and Efros, IM2GPS: Estimating Geographic Information from a Single Image, CVPR 2008
16 Example: Predict where this picture was taken Hays and Efros, IM2GPS: Estimating Geographic Information from a Single Image, CVPR 2008
17 6+ million geotagged photos by 109,788 photographers Hays and Efros, IM2GPS: Estimating Geographic Information from a Single Image, CVPR 2008
18 Scene Matches Hays and Efros, IM2GPS: Estimating Geographic Information from a Single Image, CVPR 2008
19 Hays and Efros, IM2GPS: Estimating Geographic Information from a Single Image, CVPR 2008
20 Scene Matches Hays and Efros, IM2GPS: Estimating Geographic Information from a Single Image, CVPR 2008
21 Hays and Efros, IM2GPS: Estimating Geographic Information from a Single Image, CVPR 2008
22 Scene Matches Hays and Efros, IM2GPS: Estimating Geographic Information from a Single Image, CVPR 2008
23 Hays and Efros, IM2GPS: Estimating Geographic Information from a Single Image, CVPR 2008
24 The Importance of Data Hays and Efros, IM2GPS: Estimating Geographic Information from a Single Image, CVPR 2008
25 4 + 9 = = = = = 0 Slide credit: Alexander Ihler
26 Slide credit: Alexander Ihler
27 Slide credit: Alexander Ihler
28 Slide credit: Alexander Ihler Latent Semantic Indexing (LSI) worked-out example:
29 k-nearest Neighbor Four things make a memory based learner: A distance metric Euclidean (and others) How many nearby neighbors to look at? k A weighting function (optional) Not used How to fit with the local points? Just predict the average output among the nearest neighbors Slide credit: Carlos Guestrin
30 Distances Suppose I want to charge my overall distance more for differences in x 2 direction as opposed to x 1 direction Setup A: equal weighing on all directions Setup B: more weight on x 2 direction Will my neighborhoods be longer in the x 1 or x 2 direction?
31 Voronoi partitioning Nearest neighbor regions All points in a region are closer to the seed in that region than to any other seed (black dots = seeds) Figure from Wikipedia
32 Multivariate distance metrics Suppose the input vectors x 1, x 2, x N are two dimensional: x 1 = ( x 11, x 12 ), x 2 = ( x 21, x 22 ),, x N = ( x N1, x N2 ). Dist(x i,x j ) = (x i1 x j1 ) 2 + (x i2 x j2 ) 2 Dist(x i,x j ) =(x i1 x j1 ) 2 +(3x i2 3x j2 ) 2 The relative scalings in the distance metric affect region shapes Slide credit: Carlos Guestrin
33 Distance metrics Euclidean: Mahalanobis: Minkowski:
34 Distance metrics Euclidean Manhattan Figures from Wikipedia
35 Another generalization: Weighted K-NNs Neighbors weighted differently: Extremes Bandwidth = infinity: prediction is dataset average Bandwidth = zero: prediction becomes 1-NN
36 Kernel Regression/Classification Four things make a memory based learner: A distance metric Euclidean (and others) How many nearby neighbors to look at? All of them A weighting function (optional) w i = exp(-d(x i, query) 2 / σ 2 ) Nearby points to the query are weighted strongly, far points weakly. The σ parameter is the Kernel Width. Very important. How to fit with the local points? (Regression) Predict the weighted average of the outputs, i.e. Σw i y i / Σw i Slide credit: Carlos Guestrin
37 Problems with Instance-Based Learning Too many features? Doesn t work well if large number of irrelevant features, distances overwhelmed by noisy features Distances become meaningless in high dimensions (the curse of dimensionality) What is the impact of the value of K? Expensive No learning: most real work done during testing For every test sample, must search through all dataset very slow! Must use tricks like approximate nearest neighbor search Need to store all training data Adapted from Dhruv Batra
38 Curse of Dimensionality
39 Curse of Dimensionality Consider: Sphere of radius 1 in d-dims Consider: an outer ε-shell in this sphere shell volume What is? sphere volume Slide credit: Dhruv Batra
40 Figure 1.22 from Bishop Curse of Dimensionality
41 Curse of Dimensionality Problem: In very high dimensions, all points are equally close This problem applies to all types of classifiers, not just K-NN
42 Problems with Instance-Based Learning Too many features? Doesn t work well if large number of irrelevant features, distances overwhelmed by noisy features Distances become meaningless in high dimensions (the curse of dimensionality) What is the impact of the value of K? Expensive No learning: most real work done during testing For every test sample, must search through all dataset very slow! Must use tricks like approximate nearest neighbor search Need to store all training data Adapted from Dhruv Batra
43 Slide credit: Alexander Ihler simplifies / complicates
44 Slide credit: Alexander Ihler
45 Slide credit: Alexander Ihler
46 Slide credit: Alexander Ihler
47 Use a validation set to pick K Too complex Slide credit: Alexander Ihler
48 Problems with Instance-Based Learning Too many features? Doesn t work well if large number of irrelevant features, distances overwhelmed by noisy features Distances become meaningless in high dimensions (the curse of dimensionality) What is the impact of the value of K? Expensive No learning: most real work done during testing For every test sample, must search through all dataset very slow! Must use tricks like approximate nearest neighbor search Need to store all training data Adapted from Dhruv Batra
49 Approximate distance methods Build a balanced tree of data points (kd-trees), splitting along different dimensions Go down tree (starting from root) and at each point compute current best guess of nearest neighbor Intelligently eliminate parts of the search space if they cannot contain a better current best Only search for neighbors up until some budget exhausted
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56 Figure from Szeliski
57 Summary K-Nearest Neighbor is the most basic and simplest to implement classifier Cheap at training time, expensive at test time Unlike other methods we ll see later, naturally works for any number of classes Pick K through a validation set, use approximate methods for finding neighbors Success of classification depends on the amount of data and the meaningfulness of the distance function
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