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CS 446: Machine Learning

Gerald [email protected] SC

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INTRODUCTION CS446-Fall 06 2

Office hours: after most classes and Thur @ 3 Text: Mitchell’s Machine Learning Midterm: Oct. 4 Final: Dec. 12 each a third Homeworks / projects

Submit at the beginning of class Late penalty: 20% / day up to 3 days Programming, some in-class assignments

Class web site soon

Cheating: none allowed! We adopt dept. policy


Please answer these and hand in now

Name Department Where (If?*) you had Intro AI course Who taught it (esp. if not here)

1) Why interested in Machine Learning?2) Any topics you would like to see

covered?

* may require significant additional effort


Approx. Course Overview / Topics Introduction: Basic problems and questions A detailed examples: Linear threshold units Basic Paradigms:

PAC (Risk Minimization); Bayesian Theory; SRM (Structural Risk Minimization); Compression; Maximum Entropy;…

Generative/Discriminative; Classification/Skill;… Learning Protocols

Online/Batch; Supervised/Unsupervised/Semi-supervised; Delayed supervision

Algorithms: Decision Trees (C4.5) [Rules and ILP (Ripper, Foil)] Linear Threshold Units (Winnow, Perceptron; Boosting; SVMs;

Kernels) Probabilistic Representations (naïve Bayes, Bayesian trees;

density estimation) Delayed supervision: RL Unsupervised/Semi-supervised: EM

Clustering, Dimensionality Reduction, or others of student interest


What to Learn Classifiers: Learn a hidden function

Concept Learning: chair ? face ? game ? Diagnosis: medical; risk assessment

Models: Learn a map (and use it to navigate)

Learn a distribution (and use it to answer queries) Learn a language model; Learn an Automaton

Skills: Learn to play games; Learn a Plan / Policy Learn to Reason; Learn to Plan

Clusterings: Shapes of objects; Functionality; Segmentation Abstraction

Focus on classification (importance, theoretical richness, generality,…)


What to Learn? Direct Learning: (discriminative, model-free[bad

name]) Learn a function that maps an input instance to

the sought after property. Model Learning: (indirect, generative)

Learning a model of the domain; then use it to answer various questions about the domain

In both cases, several protocols can be used – Supervised – learner is given examples and

answers Unsupervised – examples, but no answers Semi-supervised – some examples w/answers,

others w/o Delayed supervision


Supervised Learning

Given: Examples (x,f ((x)) of some unknown function f

Find: A good approximation to f

x provides some representation of the input The process of mapping a domain element into a

representation is called Feature Extraction. (Hard; ill-understood; important)

x 2 {0,1}n or x 2 <n The target function (label)

f(x) 2 {-1,+1} Binary Classification f(x) 2 {1,2,3,.,k-1} Multi-class classification f(x) 2 < Regression


Example and Hypothesis Spaces

--

+

+

+

-

X H

--

+

X: Example Space – set of all well-formed inputs [w/a distribution]

H: Hypothesis Space – set of all well-formed outputs


Supervised Learning: Examples

Disease diagnosis x: Properties of patient (symptoms, lab tests) f : Disease (or maybe: recommended therapy)

Part-of-Speech tagging x: An English sentence (e.g., The can will rust) f : The part of speech of a word in the sentence

Face recognition x: Bitmap picture of person’s face f : Name the person (or maybe: a property of)

Automatic Steering x: Bitmap picture of road surface in front of car f : Degrees to turn the steering wheel


y = f (x1, x2, x3, x4)Unknownfunction

x1

x2

x3

x4

A Learning Problem

X H

? ?

(Boolean: x1, x2, x3, x4, f)


y = f (x1, x2, x3, x4)Unknownfunction

x1

x2

x3

x4

Example x1 x2 x3 x4 y 1 0 0 1 0 0

3 0 0 1 1 1

4 1 0 0 1 1

5 0 1 1 0 0

6 1 1 0 0 0

7 0 1 0 1 0

2 0 1 0 0 0

Training Set


Hypothesis SpaceComplete Ignorance: How many possible functions?216 = 56536 over four input

features.

After seven examples how many possibilities for f?

29 possibilities remain for f

How many examples until we figure out which is correct?

We need to see labels for all 16 examples!

Is Learning Possible?

Example x1 x2 x3 x4 y

1 1 1 1 ?

0 0 0 0 ?

1 0 0 0 ?

1 0 1 1 ? 1 1 0 0 0 1 1 0 1 ?

1 0 1 0 ? 1 0 0 1 1

0 1 0 0 0 0 1 0 1 0 0 1 1 0 0 0 1 1 1 ?

0 0 1 1 1 0 0 1 0 0 0 0 0 1 ?

1 1 1 0 ?


Another Hypothesis Space

Simple Rules: There are only 16 simple conjunctive rules of the form y=xi Æ xj Æ xk...

No simple rule explains the data. The same is true for simple clauses

1 0 0 1 0 0

3 0 0 1 1 1 4 1 0 0 1 1 5 0 1 1 0 0 6 1 1 0 0 0 7 0 1 0 1 0

2 0 1 0 0 0

y=c x1 1100 0

x2 0100 0

x3 0110 0

x4 0101 1

x1 x2 1100 0

x1 x3 0011 1

x1 x4 0011 1

Rule Counterexamplex2 x3 0011 1

x2 x4 0011 1

x3 x4 1001 1

x1 x2 x3 0011 1

x1 x2 x4 0011 1

x1 x3 x4 0011 1

x2 x3 x4 0011 1

x1 x2 x3 x4 0011 1

Rule Counterexample


Third Hypothesis Spacem-of-n rules: There are 29 possible rules of the form ”y = 1 if and only if at least m of the following n variables are 1”

Found a consistent hypothesis.

1 0 0 1 0 0

3 0 0 1 1 1 4 1 0 0 1 1 5 0 1 1 0 0 6 1 1 0 0 0 7 0 1 0 1 0

2 0 1 0 0 0

x1 3 - - -

x2 2 - - -

x3 1 - - -

x4 7 - - -

x1,x2 2 3 - -

x1, x3 1 3 - -

x1, x4 6 3 - -

x2,x3 2 3 - -

variables 1-of 2-of 3-of 4-ofx2, x4 2 3 - -

x3, x4 4 4 - -

x1,x2, x3 1 3 3 -

x1,x2, x4 2 3 3 -

x1,x3,x4 1 3 -

x2, x3,x4 1 5 3 -

x1, x2, x3,x4 1 5 3 3

variables 1-of 2-of 3-of 4-of


Views of Learning Learning is the removal of our remaining

uncertainty: Suppose we knew that the unknown function was an m-of-n

Boolean function, then we could use the training data to infer which function it is.

Learning requires guessing a good, small hypothesis class: We can start with a very small class and enlarge it until it

contains an hypothesis that fits the data.

We could be wrong ! Our prior knowledge might be wrong: y=x4 one-of (x1,

x3) is also consistent Our guess of the hypothesis class could be wrong

If this is the unknown function, then we will make errors when we are given new examples, and are asked to predict the value of the function


General strategy for Machine Learning H should respect our prior understanding:

Excess expressivity makes learning difficult Expressivity of H should match our ignorance

Understand flexibility of std. hypothesis spaces: Decision trees, neural networks, rule grammars,

stochastic models Hypothesis spaces of flexible size; Nested collections of

hypotheses. ML succeeds when these interrelate

Develop algorithms for finding a hypothesis h that fits the data

h will likely perform well when the richness of H is less than the information in the training set


Terminology Training example: An pair of the form (x, f (x))

Target function (concept): The true function f (?) Hypothesis: A proposed function h, believed to be similar to

f. Concept: Boolean function. Example for which f (x)= 1 are

positive examples; those for which f (x)= 0 are negative examples (instances) (sometimes used interchangeably w/ “Hypothesis”)

Classifier: A discrete valued function. The possible value of f: {1,2,…K} are the classes or class labels.

Hypothesis space: The space of all hypotheses that can, in principle, be output by the learning algorithm.

Version Space: The space of all hypothesis in the hypothesis space that have not yet been ruled out.


Key Issues in Machine Learning Modeling

How to formulate application problems as machine learning problems ?

Learning Protocols (where is the data coming from, how?)

Project examples: [complete products] EMAIL

Given a seminar announcement, place the relevant information in my outlook

Given a message, place it in the appropriate folder Image processing:

Given a folder with pictures; automatically rotate all those that need it.

My office: have my office greet me in the morning and unlock the door

(but do it only for me!) Context Sensitive Spelling: Incorporate into Word


Key Issues in Machine Learning Modeling

How to formulate application problems as machine learning problems ?

Learning Protocols (where is the data coming from, how?)

Representation: What are good hypothesis spaces ? Any rigorous way to find these? Any general approach?

Algorithms: What are good algorithms? How do we define success? Generalization Vs. over fitting The computational problem


Example: Generalization vs Overfitting

What is a Tree ?

A botanist Her brother

A tree is something with A tree is a green thing

leaves I’ve seen before

Neither will generalize well


Self-organize into Groups of 4 or 5 Assignment 1 The Badges Game……

Prediction or Modeling? Representation Background Knowledge When did learning take place? Learning Protocol? What is the problem? Algorithms

http://l2r.cs.uiuc.edu/~danr/Teaching/CS446-05/game.html


Linear DiscriminatorsI don’t know {whether, weather} to laugh or cry

How can we make this a learning problem?

We will look for a function F: Sentences {whether, weather} We need to define the domain of this function better.

An option: For each word w in English define a Boolean feature xw :

[xw =1] iff w is in the sentence This maps a sentence to a point in {0,1}50,000

In this space: some points are whether points some are weather points

Learning Protocol?

Supervised? Unsupervised?


What’s Good? Learning problem:

Find a function that best separates the data

What function? What’s best? How to find it?

A possibility: Define the learning problem to be:Find a (linear) function that best separates the data


Exclusive-OR (XOR) (x1 Æ x2) Ç (:{x1} Æ :{x2}) In general: a parity function.

xi 2 {0,1}

f(x1, x2,…, xn) = 1

iff xi is even

This function is not linearly separable.

x1

x2


Sometimes Functions Can be Made Linearx1 x2 x4 Ç x2 x4 x5 Ç x1 x3 x7

Space: X= x1, x2,…, xn

input Transformation New Space: Y = {y1,y2,…} =

{xi,xi xj, xi xj xj}

Weather

Whether

y3 Ç y4 Ç y7 New

discriminator is functionally

simpler


Data are not separable in one dimension

Not separable if you insist on using a specific class of functions

x

Feature Space


Blown Up Feature Space Data are separable in <x, x2> space

x

x2Key issue: what features to use.

Computationally, can be done implicitly (kernels)


A General Framework for Learning Goal: predict an unobserved output value y 2 Y based on an observed input vector x 2 X

Estimate a functional relationship y~f(x) from a set {(x,y)i}i=1,n

Most relevant - Classification: y {0,1} (or y {1,2,…k} )

(But, within the same framework can also talk about Regression, y 2 <

What do we want f(x) to satisfy?

We want to minimize the Loss (Risk): L(f()) = E X,Y( [f(x)y] )

Where: E X,Y denotes the expectation with respect to the true distribution.

Simply: # of mistakes

[…] is a indicator function


A General Framework for Learning (II) We want to minimize the Loss: L(f()) = E X,Y( [f(X)Y] ) Where: E X,Y denotes the expectation with respect to the true

distribution.

We cannot do that. Why not? Instead, we try to minimize the empirical classification

error. For a set of training examples {(Xi,Yi)}i=1,n

Try to minimize the observed loss (Issue I: when is this good enough? Not now)

This minimization problem is typically NP hard. To alleviate this computational problem, minimize a new function –

a convex upper bound of the classification error function

I(f(x),y) =[f(x) y] = {1 when f(x)y; 0 otherwise}


Learning as an Optimization Problem A Loss Function L(f(x),y) measures the penalty

incurred by a classifier f on example (x,y). There are many different loss functions one

could define: Misclassification Error:

L(f(x),y) = 0 if f(x) = y; 1 otherwise Squared Loss:

L(f(x),y) = (f(x) –y)2

Input dependent loss:

L(f(x),y) = 0 if f(x)= y; c(x)otherwise.

A continuous convex loss function also allows a conceptually simple optimization algorithm.

f(x) –y


How to Learn? Local search:

Start with a linear threshold function. See how well you are doing. Correct Repeat until you converge.

There are other ways that do not search directly in the hypotheses space

Directly compute the hypothesis?


Learning Linear Separators (LTU) f(x) = sgn {x ¢ w - } = sgn{i=1

n wi xi - }

x= (x1 ,x2,… ,xn) 2 {0,1}n is the feature based encoding of the data point

w= (w1 ,w2,… ,wn) 2 <n is the target function.

determines the shift with respect to the origin w


Expressivity f(x) = sgn {x ¢ w - } = sgn{i=1

n wi xi - } Many functions are Linear

Conjunctions: y = x1 Æ x3 Æ x5

y = sgn{1 ¢ x1 + 1 ¢ x3 + 1 ¢ x5 - 3} At least m of n:

y = at least 2 of {x1 ,x3, x5 }

y = sgn{1 ¢ x1 + 1 ¢ x3 + 1 ¢ x5 - 2}

Many functions are not Xor: y = x1 Æ x2 Ç x1 Æ x2

Non trivial DNF: y = x1 Æ x2 Ç x3 Æ x4

But some can be made linear

Probabilistic Classifiers as well


Canonical Representation f(x) = sgn {x ¢ w - } = sgn{i=1

n wi xi - }

sgn {x ¢ w - } ´ sgn {x’ ¢ w’} Where:

x’ = (x, -) and w’ = (w,1)

Moved from an n dimensional representation to an (n+1) dimensional representation, but now can look for hyperplans that go through the origin.


LMS: An online, local search algorithm A local search learning algorithm

requires: Hypothesis Space:

Linear Threshold Units Loss function:

Squared loss LMS (Least Mean Square, L2)

Search procedure: Gradient Descent

w


LMS: An online, local search algorithm

•Let w(j) be our current weight vector

• Our prediction on the d-th example x is therefore:

• Let td be the target value for this example (real value; represents u ¢ x)• A convenient error function of the data set is:

xwxwo (j)ii

jid

2d

Ddd

(j) )o-(t21

)w Err(

(i (subscript) – vector component; j (superscript) - time; d – example #)Assumption: x 2 Rn; u 2 Rn is the target weight vector;

the target (label) is td = u ¢ x Noise has been added; so, possibly, no weight vector is consistent with the data.


Gradient Descent We use gradient descent to determine the

weight vector that minimizes Err (w) ; Fixing the set D of examples, E is a function of

wj

At each step, the weight vector is modified in the direction that produces the steepest descent along the error surface. E(w)

ww4 w3 w2 w1


• To find the best direction in the weight space we compute the gradient of E with respect to each of the components of

• This vector specifies the direction that produces the steepest increase in E;• We want to modify in the direction of

• Where:

w

]wE

,...,wE

,wE

[)wE(n21

w

)wE(R - w

w w w

Δ

Δ

)wE(

Gradient Descent


• We have:

• Therefore:

))(-xo(t iddDd

d

2d

Ddd

(j) )o-(t21

)wErr(

)xw(tw

)o2(t21

dddi

dDd

d

)o(tw2

1 2d

Ddd

i

)o(t21

wwE 2

dDd

dii

Gradient Descent: LMS


• Weight update rule:

iddDd

di )xo(tRw



iddDd

di )xo(tRw

td

didi id xwxwo


• Gradient descent algorithm for training linear units: - Start with an initial random weight vector - For every example d with target value : - Evaluate the linear unit - update by adding to each component - Continue until E below some threshold

iww



iddDd

di )xo(tRw

td

didi id xwxwo


• Gradient descent algorithm for training linear units: - Start with an initial random weight vector - For every example d with target value : - Evaluate the linear unit - update by adding to each component - Continue until E below some threshold Because the surface contains only a single global minimum the algorithm will converge to a weight vector with minimum error, regardless of whether the examples are linearly separable

iww




idddi )xoR(tw

Incremental Gradient Descent: LMS


Incremental Gradient Descent - LMS


•Gradient descent algorithm for training linear units: - Start with an initial random weight vector - For every example d with target value : - Evaluate the linear unit - update by incrementally adding to each component - Continue until E below some threshold

In general - does not converge to global minimumDecreasing R with time guarantees convergence

Incremental algorithms are sometimes advantageous…

idddi )xoR(tw

dt

iwdidi id xwxwo

w


Learning Rates and Convergence

• In the general (non-separable) case the learning rate R must decrease to zero to guarantee convergence. It cannot decrease too quickly nor too slowly.

• The learning rate is called the step size. There are more sophisticates algorithms (Conjugate Gradient) that choose the step size automatically and converge faster.

• There is only one “basin” for linear threshold units, so a local minimum is the global minimum. However, choosing a starting point can make the algorithm converge much faster.


Computational Issues

Assume the data is linearly separable.Sample complexity:Suppose we want to ensure that our LTU has an error rate (on new examples) of less than with high probability(at least (1-))How large must m (the number of examples) be in order to achieve this? It can be shown that for n dimensional problems m = O(1/ [ln(1/ ) + (n+1) ln(1/ ) ].

Computational complexity:What can be said?It can be shown that there exists a polynomial time algorithm for finding consistent LTU (by reduction from linear programming). (On-line algorithms have inverse quadratic dependence on the margin)


Other methods for LTUs

• Direct Computation: Set J(w) = 0 and solve for w . Can be accomplished using SVD methods.

• Fisher Linear Discriminant: A direct computation method.

• Probabilistic methods (naive Bayes): Produces a stochastic classifier that can be viewed as a linear threshold unit.

• Winnow: A multiplicative update algorithm with the property that it can handle large numbers of irrelevant attributes.


Summary of LMS algorithms for LTUs

Local search: Begins with initial weight vector. Modifies iteratively to minimizeand error function. The error function is loosely related to the goal of minimizing the number of classification errors.

Memory: The classifier is constructed from the training examples. The examples can then be discarded.

Online or Batch:Both online and batch variants of the algorithms can be used.


Fisher Linear Discriminant This is a classical method for discriminant analysis. It is based on dimensionality reduction – finding a

better representation for the data. Notice that just finding good representations for

the data may not always be good for discrimination. [E.g., O, Q]

Intuition: Consider projecting data from d dimensions to the

line. Likely results in a mixed set of points and poor

separation. However, by moving the line around we might be

able to find an orientation for which the projected samples are well separated.


Fisher Linear Discriminant Sample S= {x1, x2, … xn } 2 <d

P, N are the positive, negative examples, resp.

Let w 2 <d. And assume ||w||=1. Then: The projection of a vector x on a line in the

direction w, is wt ¢ x.

If the data is linearly separable, there exists a good direction w.

(all vectors are column vectors)


Finding a Good Direction (2) Scaling w isn’t the solution. We want the difference to be

large relative to some measure of standard deviation for each class.

S2p = P (y-mp )2 s2

N = N (y-mN )2

1/ ( S2p + s2

N ) within class scatter: estimates the variances of the sample.

The Fischer linear discriminant employs the linear function wt ¢ x for which

J(w) = | mP – mN|2 / S2p + s2

N is maximized.

How to make this a classifier? How to find the optimal w?

Some Algebra


J as an explicit function of w (1) Compute the scatter matrices:

Sp = P (x-Mp )(x-Mp )t SN = N (x-MN )(x-MN )t

and SW = Sp + Sp

We can write: S2

p = P (y-mp )2 = P (wt x -wt Mp )2 =

= P wt (x- Mp ) (x- Mp )t w = wt Sp w

Therefore:S2

p + S2N = wt SW

w

SW is the within-class scatter matrix. It is proportional to the sample covariance matrix for the d-dimensional sample.


J as an explicit function of w (2) We can do a similar computation for the means:

SB = (MP-MN )(MP-MN )t

and we can write: (mP-mN )2 = (wt MP-wt MN )2 =

= wt (MP-MN) (MP-MN) t w = wt SB w

Therefore:SB is the between-class scatter matrix. It is the

outer product of two vectors and therefore its rank is at most 1.

SB w is always in the direction of (MP-MN )


J as an explicit function of w (3) Now we can compute explicitly: We can do a similar

computation for the means:J(w) = | mP – mN|2 / S2

p + s2N = wt SB

w / wt SW w

We are looking for a the value of w that maximizes this expression.

This is a generalized eigenvalue problem; when SW is nonsingular, it is just a eigenvalue problem. The solution can be written without solving the problem, as:

w=S-1W

(MP-MN )

This is the Fisher Linear Discriminant.1: We converted a d-dimensional problem to a 1-dimensional

problem and suggested a solution that makes some sense.2: We have a solution that makes sense; how to make it a

classifier? And, how good it is?


Fisher Linear Discriminant - Summary It turns out that both problems can be solved if we make

assumptions. E.g., if the data consists of two classes of points, generated according to a normal distribution, with the same covariance. Then:

The solution is optimal. Classification can be done by choosing a threshold, which

can be computed. Is this satisfactory?


Introduction - Summary We introduced the technical part of the class by giving

two examples for (very different) approaches to linear discrimination.

There are many other solutions. Questions 1: But this assumes that we are linear. Can we

learn a function that is more flexible in terms of what it does with the features space?

Question 2: Can we say something about the quality of what we learn (sample complexity, time complexity; quality)