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UVA CS 6316: Machine Learning Lecture 9: K-nearest-neighbor Dr. Yanjun Qi University of Virginia Department of Computer Science 9/30/19 Dr. Yanjun Qi / UVA CS 1

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Page 1: UVA CS 6316: Machine Learning Lecture 9: K-nearest-neighbor · e.g., support vector machine, decision tree, logistic regression, e.g. neural networks (NN), deep NN 2. Generative:

UVA CS 6316:Machine Learning

Lecture 9: K-nearest-neighbor

Dr. Yanjun Qi

University of Virginia Department of Computer Science

9/30/19 Dr. Yanjun Qi / UVA CS 1

Page 2: UVA CS 6316: Machine Learning Lecture 9: K-nearest-neighbor · e.g., support vector machine, decision tree, logistic regression, e.g. neural networks (NN), deep NN 2. Generative:

Course Content Plan èSix major sections of this course

q Regression (supervised)q Classification (supervised)q Unsupervised models q Learning theory

q Graphical models

qReinforcement Learning

9/30/19 Yanjun Qi / UVA CS 2

Y is a continuous

Y is a discrete

NO Y

About f()

About interactions among X1,… Xp

Learn program to Interact with its environment

Page 3: UVA CS 6316: Machine Learning Lecture 9: K-nearest-neighbor · e.g., support vector machine, decision tree, logistic regression, e.g. neural networks (NN), deep NN 2. Generative:

Last Recap: Supervised Classifiers

Classification

Separation Boundary / Tree Local Smooth / Generative

?

?

ACC / F1 / Many metric

Task

Representation

Score Function

Search/Optimization

Models, Parameters

9/30/19 Dr. Yanjun Qi / UVA CS 3

Output Y is categorical

Page 4: UVA CS 6316: Machine Learning Lecture 9: K-nearest-neighbor · e.g., support vector machine, decision tree, logistic regression, e.g. neural networks (NN), deep NN 2. Generative:

Three major sections for classification

• We can divide the large variety of classification approaches into roughly three major types

1. Discriminativedirectly estimate a decision rule/boundarye.g., support vector machine, decision tree, logistic regression,e.g. neural networks (NN), deep NN

2. Generative:build a generative statistical modele.g., Bayesian networks, Naïve Bayes classifier

3. Instance based classifiers- Use observation directly (no models)- e.g. K nearest neighbors

9/30/19 Dr. Yanjun Qi / UVA CS 4

Page 5: UVA CS 6316: Machine Learning Lecture 9: K-nearest-neighbor · e.g., support vector machine, decision tree, logistic regression, e.g. neural networks (NN), deep NN 2. Generative:

(1) K-Nearest Neighbor

Classification / Regression

Local Smoothness

EPE with L2 loss èconditional mean (Extra)

NA

Training Samples

Task

Representation

Score Function

Search/Optimization

Models, Parameters

9/30/19 5Dr. Yanjun Qi / UVA CS

Page 6: UVA CS 6316: Machine Learning Lecture 9: K-nearest-neighbor · e.g., support vector machine, decision tree, logistic regression, e.g. neural networks (NN), deep NN 2. Generative:

© Eric Xing @ CMU, 2006-2008

For example: Vector Space Representation of Text

• Each document is a vector, one component for each term (= word).

• High-dimensional vector space:• Terms are axes, 10,000+ dimensions, or even 100,000+• Docs are vectors in this space• Normally Normalize to unit length.

Doc 1 Doc 2 Doc 3 ...Word 1 3 0 0 ...Word 2 0 8 1 ...Word 3 12 1 10 ...

... 0 1 3 ...

... 0 0 0 ...

Page 7: UVA CS 6316: Machine Learning Lecture 9: K-nearest-neighbor · e.g., support vector machine, decision tree, logistic regression, e.g. neural networks (NN), deep NN 2. Generative:

© Eric Xing @ CMU, 2006-2008

Multiple Classes in a Vector Space

Sports

Science

Arts

Page 8: UVA CS 6316: Machine Learning Lecture 9: K-nearest-neighbor · e.g., support vector machine, decision tree, logistic regression, e.g. neural networks (NN), deep NN 2. Generative:

© Eric Xing @ CMU, 2006-2008

Test Document = ?

Sports

Science

Arts

?

Page 9: UVA CS 6316: Machine Learning Lecture 9: K-nearest-neighbor · e.g., support vector machine, decision tree, logistic regression, e.g. neural networks (NN), deep NN 2. Generative:

© Eric Xing @ CMU, 2006-2008

K-Nearest Neighbor (kNN) classifier

Sports

Science

Arts

?

Page 10: UVA CS 6316: Machine Learning Lecture 9: K-nearest-neighbor · e.g., support vector machine, decision tree, logistic regression, e.g. neural networks (NN), deep NN 2. Generative:

10

Instance-based Learning

• Simplest form of learning:• Training instances are searched for those that most closely

resembles new instance• The instances themselves represent the knowledge

• Similarity function defines what’s “learned”• Instance-based learning is lazy learning

From witten&eibe

Page 11: UVA CS 6316: Machine Learning Lecture 9: K-nearest-neighbor · e.g., support vector machine, decision tree, logistic regression, e.g. neural networks (NN), deep NN 2. Generative:

Instance-based Learning

• K-Nearest Neighbor Algorithm• Weighted Regression• Case-based reasoning

Page 12: UVA CS 6316: Machine Learning Lecture 9: K-nearest-neighbor · e.g., support vector machine, decision tree, logistic regression, e.g. neural networks (NN), deep NN 2. Generative:

© Eric Xing @ CMU, 2006-2008

What makes an Instance-Based Learner?

•A distance metric

•How many nearby neighbors to look at?

•A weighting function (optional)

•How to relate to the local points?

Page 13: UVA CS 6316: Machine Learning Lecture 9: K-nearest-neighbor · e.g., support vector machine, decision tree, logistic regression, e.g. neural networks (NN), deep NN 2. Generative:

© Eric Xing @ CMU, 2006-2008

Popular Distance Metric

• Euclidean

• Or equivalently,

• Other metrics:• L1 norm: |x-x'|• L∞ norm: max |x-x'| (elementwise …)• Mahalanobis: where S is full, and symmetric • Correlation• Angle• Hamming distance, Manhattan distance• …

å -=i

iii xxxxD 22 )'()',( s

)'()'()',( xxxxxxD T -S-=

Page 14: UVA CS 6316: Machine Learning Lecture 9: K-nearest-neighbor · e.g., support vector machine, decision tree, logistic regression, e.g. neural networks (NN), deep NN 2. Generative:

Feature Scaling in Nearest neighbor method

• Scaling issues• Attributes may have to be scaled to prevent distance measures from

being dominated by one of the attributes• Example:

• height of a person may vary from 1.5 m to 1.8 m• weight of a person may vary from 90 lb to 300 lb• income of a person may vary from $10K to $1M

9/30/19 Dr. Yanjun Qi / UVA CS 14

The relative scalings in the distance metric affect region shapes.

Page 15: UVA CS 6316: Machine Learning Lecture 9: K-nearest-neighbor · e.g., support vector machine, decision tree, logistic regression, e.g. neural networks (NN), deep NN 2. Generative:

© Eric Xing @ CMU, 2006-2008

What makes an Instance-Based Learner?

•A distance metric

•How many nearby neighbors to look at?

•A weighting function (optional)

•How to relate to the local points?

Page 16: UVA CS 6316: Machine Learning Lecture 9: K-nearest-neighbor · e.g., support vector machine, decision tree, logistic regression, e.g. neural networks (NN), deep NN 2. Generative:

Nearest neighbor is instance-based method

9/30/19 Dr. Yanjun Qi / UVA CS 16

Requires three inputs:1. The set of stored

training samples2. Distance metric to

compute distance between samples

3. The value of k, i.e., the number of nearest neighbors to retrieve

Page 17: UVA CS 6316: Machine Learning Lecture 9: K-nearest-neighbor · e.g., support vector machine, decision tree, logistic regression, e.g. neural networks (NN), deep NN 2. Generative:

Unknown record

Nearest neighbor classifiers

9/30/19 Dr. Yanjun Qi / UVA CS 17

Requires three inputs:1. The set of stored

training samples2. Distance metric to

compute distance between samples

3. The value of k, i.e., the number of nearest neighbors to retrieve

Page 18: UVA CS 6316: Machine Learning Lecture 9: K-nearest-neighbor · e.g., support vector machine, decision tree, logistic regression, e.g. neural networks (NN), deep NN 2. Generative:

Nearest neighbor classifiers

9/30/19 Dr. Yanjun Qi / UVA CS

To classify unknown sample:1. Compute distance to

training records2. Identify k nearest

neighbors 3. Use class labels of nearest

neighbors to determine the class label of unknown record (e.g., by taking majority vote)

Unknown record

Page 19: UVA CS 6316: Machine Learning Lecture 9: K-nearest-neighbor · e.g., support vector machine, decision tree, logistic regression, e.g. neural networks (NN), deep NN 2. Generative:

1-Nearest Neighbor

Page 20: UVA CS 6316: Machine Learning Lecture 9: K-nearest-neighbor · e.g., support vector machine, decision tree, logistic regression, e.g. neural networks (NN), deep NN 2. Generative:

3-Nearest Neighbor

Page 21: UVA CS 6316: Machine Learning Lecture 9: K-nearest-neighbor · e.g., support vector machine, decision tree, logistic regression, e.g. neural networks (NN), deep NN 2. Generative:

9/30/19 Dr. Yanjun Qi / UVA CS 21

Page 22: UVA CS 6316: Machine Learning Lecture 9: K-nearest-neighbor · e.g., support vector machine, decision tree, logistic regression, e.g. neural networks (NN), deep NN 2. Generative:

K-Nearest Neighbor: How to decide:

• Decision of output value is delayed till a new instance arrives

• Target variable may be discrete or real-valued• When target is discrete, the naïve prediction is the majority vote

Page 23: UVA CS 6316: Machine Learning Lecture 9: K-nearest-neighbor · e.g., support vector machine, decision tree, logistic regression, e.g. neural networks (NN), deep NN 2. Generative:

e.g., 1-nearest neighbor

9/30/19 Dr. Yanjun Qi / UVA CS 23

Voronoi diagram: partitioning of a

plane into regions based on distance to points in a specific subset of the plane.

Page 24: UVA CS 6316: Machine Learning Lecture 9: K-nearest-neighbor · e.g., support vector machine, decision tree, logistic regression, e.g. neural networks (NN), deep NN 2. Generative:

e.g. Decision boundary implemented by 3NNThe boundary is always the perpendicular bisector of the line between two points (Vornoi tesselation)

k-nearest neighbors of a sample x are datapointsthat have the k smallest distances to x

Page 25: UVA CS 6316: Machine Learning Lecture 9: K-nearest-neighbor · e.g., support vector machine, decision tree, logistic regression, e.g. neural networks (NN), deep NN 2. Generative:

K-Nearest Neighbor: How to decide:

•Decision of output value is delayed till a new instance arrives

•Target variable may be discrete or real-valued• When target is discrete, the naïve prediction is the majority vote• When target is continuous, the naïve prediction is the mean

value of the k nearest training examples

Page 26: UVA CS 6316: Machine Learning Lecture 9: K-nearest-neighbor · e.g., support vector machine, decision tree, logistic regression, e.g. neural networks (NN), deep NN 2. Generative:

Nearest Neighbor (1D input) for Regression

9/30/19 Dr. Yanjun Qi / UVA CS 26

Page 27: UVA CS 6316: Machine Learning Lecture 9: K-nearest-neighbor · e.g., support vector machine, decision tree, logistic regression, e.g. neural networks (NN), deep NN 2. Generative:

K=5-Nearest Neighbor (1D input) for Regression

9/30/19 Dr. Yanjun Qi / UVA CS 27

Page 28: UVA CS 6316: Machine Learning Lecture 9: K-nearest-neighbor · e.g., support vector machine, decision tree, logistic regression, e.g. neural networks (NN), deep NN 2. Generative:

Probabilistic Interpretation of KNN

•Estimate conditional probability Pr(y|x)•Count of data points in class y in the neighborhood of x

Page 29: UVA CS 6316: Machine Learning Lecture 9: K-nearest-neighbor · e.g., support vector machine, decision tree, logistic regression, e.g. neural networks (NN), deep NN 2. Generative:

Summary of Nearest neighbor methods

•For regression, average the predictions of the K nearest neighbors.

•For classification, pick the class with the most votes.

• How should we break ties?• E.g., Let the k’th nearest neighbor contribute a count

that falls off with k. For example, k2

1 1+

Page 30: UVA CS 6316: Machine Learning Lecture 9: K-nearest-neighbor · e.g., support vector machine, decision tree, logistic regression, e.g. neural networks (NN), deep NN 2. Generative:

© Eric Xing @ CMU, 2006-2008

What makes an Instance-Based Learner?

•A distance metric

•How many nearby neighbors to look at?

•A weighting function (optional)

•How to relate to the local points?

Page 31: UVA CS 6316: Machine Learning Lecture 9: K-nearest-neighbor · e.g., support vector machine, decision tree, logistic regression, e.g. neural networks (NN), deep NN 2. Generative:

© Eric Xing @ CMU, 2006-2008

1-Nearest Neighbor (kNN) classifier

Sports

Science

Arts

Page 32: UVA CS 6316: Machine Learning Lecture 9: K-nearest-neighbor · e.g., support vector machine, decision tree, logistic regression, e.g. neural networks (NN), deep NN 2. Generative:

© Eric Xing @ CMU, 2006-2008

2-Nearest Neighbor (kNN) classifier

Sports

Science

Arts

Page 33: UVA CS 6316: Machine Learning Lecture 9: K-nearest-neighbor · e.g., support vector machine, decision tree, logistic regression, e.g. neural networks (NN), deep NN 2. Generative:

© Eric Xing @ CMU, 2006-2008

3-Nearest Neighbor (kNN) classifier

Sports

Science

Arts

Page 34: UVA CS 6316: Machine Learning Lecture 9: K-nearest-neighbor · e.g., support vector machine, decision tree, logistic regression, e.g. neural networks (NN), deep NN 2. Generative:

© Eric Xing @ CMU, 2006-2008

5-Nearest Neighbor (kNN) classifier

Sports

Science

Arts

Page 35: UVA CS 6316: Machine Learning Lecture 9: K-nearest-neighbor · e.g., support vector machine, decision tree, logistic regression, e.g. neural networks (NN), deep NN 2. Generative:

9/30/19 Dr. Yanjun Qi / UVA CS 35

Page 36: UVA CS 6316: Machine Learning Lecture 9: K-nearest-neighbor · e.g., support vector machine, decision tree, logistic regression, e.g. neural networks (NN), deep NN 2. Generative:

9/30/19 Dr. Yanjun Qi / UVA CS 36

Page 37: UVA CS 6316: Machine Learning Lecture 9: K-nearest-neighbor · e.g., support vector machine, decision tree, logistic regression, e.g. neural networks (NN), deep NN 2. Generative:

Decision boundaries in Linear vs. kNN models (Later)

9/30/19 Dr. Yanjun Qi / UVA CS 37

linear regression• global• stable• can be inaccurate

15-nearest neighbor 1-nearest neighbor

KNN• local• accurate• unstable

What ultimately matters: GENERALIZATION

Low Bias / High Variance

Low Variance / High Bias

Page 38: UVA CS 6316: Machine Learning Lecture 9: K-nearest-neighbor · e.g., support vector machine, decision tree, logistic regression, e.g. neural networks (NN), deep NN 2. Generative:

Model Selection for Nearest neighbor classification

•Choosing the value of k:• If k is too small, sensitive to noise points• If k is too large, neighborhood may include points

from other classes

9/30/19 Dr. Yanjun Qi / UVA CS 38

•Bias and variance tradeoff•A small neighborhood à large variance à unreliable estimation •A large neighborhood à large bias à inaccurate estimation

Page 39: UVA CS 6316: Machine Learning Lecture 9: K-nearest-neighbor · e.g., support vector machine, decision tree, logistic regression, e.g. neural networks (NN), deep NN 2. Generative:

© Eric Xing @ CMU, 2006-2008

What makes an Instance-Based Learner?

•A distance metric

•How many nearby neighbors to look at?

•A weighting function (optional)

•How to relate to the local points?

Page 40: UVA CS 6316: Machine Learning Lecture 9: K-nearest-neighbor · e.g., support vector machine, decision tree, logistic regression, e.g. neural networks (NN), deep NN 2. Generative:

Nearest neighbor Variations

• Options for determining the class from nearest neighbor list• 1. majority vote of class labels among the k-nearest neighbors• 2. Weight the votes according to distance

• example: weight factor w = 1 / d2

9/30/19 Dr. Yanjun Qi / UVA CS 40

Spurious or less relevant points need to be downweighted

Page 41: UVA CS 6316: Machine Learning Lecture 9: K-nearest-neighbor · e.g., support vector machine, decision tree, logistic regression, e.g. neural networks (NN), deep NN 2. Generative:

Another Weighted kNN

• Weight the contribution of each close neighbor based on their distances

• Weight function

• Prediction

Page 42: UVA CS 6316: Machine Learning Lecture 9: K-nearest-neighbor · e.g., support vector machine, decision tree, logistic regression, e.g. neural networks (NN), deep NN 2. Generative:

Variants: Distance-Weighted k-Nearest Neighbor Algorithm

• Assign weights to the neighbors based on their “distance” from the query point

• Weight “may” be inverse square of the distances w = 1 / d2

• Extreme Option: All training points may influence a particular instance

• E.g., Shepard’s method/ Modified Shepard, … by Geospatial Analysis

Page 43: UVA CS 6316: Machine Learning Lecture 9: K-nearest-neighbor · e.g., support vector machine, decision tree, logistic regression, e.g. neural networks (NN), deep NN 2. Generative:

K-Nearest Neighbor

Regression/ Classification

Local Smoothness / Training Samples

NA

NA

Training Samples

Task

Representation

Score Function

Search/Optimization

Models, Parameters

9/30/19 Dr. Yanjun Qi / UVA CS 43

Computational Scalable?

Page 44: UVA CS 6316: Machine Learning Lecture 9: K-nearest-neighbor · e.g., support vector machine, decision tree, logistic regression, e.g. neural networks (NN), deep NN 2. Generative:

Computational Time Cost

9/30/19 Dr. Yanjun Qi / UVA CS 44

Page 45: UVA CS 6316: Machine Learning Lecture 9: K-nearest-neighbor · e.g., support vector machine, decision tree, logistic regression, e.g. neural networks (NN), deep NN 2. Generative:

K-Nearest Neighbor

Regression/ Classification

Local Smoothness / Training Samples

NA

NA

Training Samples

Task

Representation

Score Function

Search/Optimization

Models, Parameters

9/30/19 Dr. Yanjun Qi / UVA CS 45

Asymptotically Sound?

Page 46: UVA CS 6316: Machine Learning Lecture 9: K-nearest-neighbor · e.g., support vector machine, decision tree, logistic regression, e.g. neural networks (NN), deep NN 2. Generative:

© Eric Xing @ CMU, 2006-2008

Is kNN ideal? … See Extra

Page 47: UVA CS 6316: Machine Learning Lecture 9: K-nearest-neighbor · e.g., support vector machine, decision tree, logistic regression, e.g. neural networks (NN), deep NN 2. Generative:

References

q Prof. Tan, Steinbach, Kumar’s “Introduction to Data Mining” slideq Prof. Andrew Moore’s slidesq Prof. Eric Xing’s slidesq Hastie, Trevor, et al. The elements of statistical learning. Vol. 2. No.

1. New York: Springer, 2009.

9/30/19 Dr. Yanjun Qi / UVA CS 47


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