11/4/2015236875 visual recognition1 content linear learning machines and svm the perceptron...

04/20/23 236875 Visual Recognition 1

Content

Linear Learning Machines and SVMThe Perceptron Algorithm revisitedFunctional and Geometric MarginNovikoff theoremDual RepresentationLearning in the Feature SpaceKernel-Induced Feature SpaceMaking KernelsThe Generalization Problem Probably Approximately Correct LearningStructural Risk Minimization


Basic Notation• Input space• Output space for classification

for regression• Hypothesis • Training Set• Test error also R()• Dot product

Linear Learning Machines and SVM

nx X R

,y R

h H

1 1{( , ),..., ( , ),...}i iS x y x y,x z

{ 1,1},y Y


• Linear separation

of the input space

The algorithm requires that the input patterns are linearly separable,which means that there exist linear discriminant function which haszero training error. We assume that this is the case.

xx

wx

x

x/ || ||b w

( ) ,f x w x b

( ) sign( ( ))h x f x

The Perceptron Algorithm revisited


initialize repeat error false for i=1..l

if then

error true end if end foruntil (error==false)

return k,(wk,bk) where k is the number of mistakes

The Perceptron Algorithm (primal form)

0 0, 0, 0, , max || ||i ib k R w 0 R x

1k k

1k k i iy w w x

( , ) 0i k i ky b w x

21k k ib b y R


The Perceptron Algorithm Comments

• The perceptron works by adding misclassified positive or subtracting misclassified negative examples to an arbitrary weight vector, which (without loss of generality) we assumed to be the zero vector. So the final weight vector is a linear combination of training points

where, since the sign of the coefficient of is given by label yi, the are positive values, proportional to the number of times, misclassification of has caused the weight to be updated. It is called the embedding strength of the pattern .

1

,l

i i ii

y

w x

ixi

ix

ix


Functional and Geometric Margin

• The notion of margin of a data point w.r.t. a linear discriminant will turn out to be an important concept.

• The functional margin of a linear discriminant (w,b) w.r.t. a labeled pattern is defined as

• If the functional margin is negative, then the pattern is incorrectly classified, if it is positive then the classifier predicts the correct label.

• The larger the further away xi is from the discriminant.• This is made more precise in the notion of the geometric

margin

( , ) { 1,1}di ix y R

( , )i i iy b w x

| |i

|| ||i

i

w


Functional and Geometric Margin cont.

x

x

ix ix

x

xx

x

x

xj

The geometric margin of The margin of a training set two points

o j

ooo

o

o

o

oo o

o


Functional and Geometric Margin cont.

which measures the Euclidean distance of a point from the decision boundary.

• Finally, is called the (functional) margin of (w,b)

w.r.t. the data set S={(xi,yi)}. • The margin of a training set S is the maximum geometric

margin over all hyperplanes. A hyperplane realizing this maximum is a maximal margin hyperplane.

Maximal Margin Hyperplane

min i i


Novikoff theorem

Theorem:• Suppose that there exists a vector and a

bias term such that the margin on a (non-trivial) data set S is at least , i.e.

then the number of update steps in the perceptron algorithm is at most

where

* *, || || 1 w w*b

* *( , ) , 1,..., ,i iy b i l w x

2

* 2,

Rt

1max || || .i

i lR x


Novikoff theorem cont.

Comments:

• Novikoff theorem says that no matter how small the margin, if a data set is linearly separable, then the perceptron will find a solution that separates the two classes in a finite number of steps.

• More precisely, the number of update steps (and the runtime) will depend on the margin and is inverse proportional to the squared margin.

• The bound is invariant under rescaling of the patterns.

• The learning rate does not matter.


Dual Representation

• The decision function can be rewritten as follows:

• And also the update rule can be rewritten as follows:

• The learning rate only influences the overall scaling of the hyperplanes, it does no affect an algorithm with zero starting vector, so we can put

1

1

( ) sgn( ) sgn( )

sgn( )

l

j j ji

l

j j ji

h x b y b

y b

w x x x

x x

1

if ( ) 0 then ,l

i j j j i ii

y y b

x x

1.


Duality: First Property of SVMs

• DUALITY is the first feature of Support Vector Machines• SVM are Linear Learning Machines represented in a dual

fashion

• Data appear only inside dot products (in the decision function and in the training algorithm)• The matrix is called Gram matrix

1

( )l

j j ji

f x b y b

w x x x

, 1

l

i ji j

G

x x


Limitations of Linear Classifiers

• Linear Learning Machines (LLM) cannot deal withNon-linearly separable dataNoisy dataThis formulation only deals with vectorial data


Limitations of Linear Classifiers

• Neural networks solution: multiple layers of thresholded linear functions – multi-layer neural networks. Learning algorithms – back-propagation.

• SVM solution: kernel representation. Approximation-theoretic issues are independent of

the learning-theoretic ones. Learning algorithms are decoupled from the specifics of the application area, which is encoded into design of kernel.


Learning in the Feature Space

• Map data into a feature space where they are linearly separable (i.e. attributes features)

( )x x

x

x

x

x

( ) x( ) x

( ) x

( ) x

o

o

oo

FX

( ) o

( ) o( ) o

( ) o

f


Learning in the Feature Space cont.

Example• Consider the target function

giving gravitational force between two bodies.• Observable quantities are masses m1, m2 and

distance r. A linear machine could not represent it, but a change of coordinates

gives the representation

1 21 2 2

( , , ) ,m m

f m m r Gr

1 2 1 2( , , ) ( , , ) (ln , ln , ln )m m r x y z m m r

1 2 1 2( , , ) ln ( , , ) ln ln ln 2ln 2g x y z f m m r G m m r c x y z


Learning in the Feature Space cont.

• The task of choosing the most suitable representation is known as feature selection.

• The space X is referred to as the input space, while is called the feature space.

• Frequently one seeks to find smallest possible set of features that still conveys essential information (dimensionality reduction

{ ( ) : }F X x x

1 1( ,..., ) ( ) ( ( ),..., ( )),n dx x d n x x x x


Problems with Feature Space

• Working in high dimensional feature spaces solves the problem of expressing complex functions

BUT:• There is a computational problem (working with very large

vectors)• And a generalization theory problem (curse of dimensionality)


Implicit Mapping to Feature Space

We will introduce Kernels:• Solve the computational problem of working with

many dimensions• Can make it possible to use infinite dimensions

Efficiently in time/space• Other advantages, both practical and conceptual


Kernel-Induced Feature Space

• In order to learn non-linear relations we select non-linear features. Hence, the set of hypotheses we consider will be functions of type

where is a non-linear map from input space to feature space

• In the dual representation, the data points only appear inside dot products

1

( ) ( ), ( )l

i i ii

f x y x x b

1

( ) ( )N

i ii

f x w x b

: X F


Kernels

• Kernel is a function that returns the value of the dot product between the images of the two arguments

• When using kernels, the dimensionality of space F not necessarily important. We may not even know the map

• Given a function K, it is possible to verify that it is a kernel

1 2 1 2( , ) ( ), ( )K x x x x


Kernels cont.

• One can use LLMs in a feature space by simply rewriting it in dual representation and replacing dot products with kernels:

1 2 1 2 1 2, ( , ) ( ), ( )x x K x x x x


The Kernel Matrix (Gram Matrix)

K=

K(1,1) K(1,2) K(1,3) … K(1,m)

K(2,1) K(2,2) K(2,3) … K(2,m)

… … … … …

K(m,1) K(m,2) K(m,3) … K(m,m)


The Kernel Matrix

• The central structure in kernel machines• Information ‘bottleneck’: contains all necessary

information for the learning algorithm• Fuses information about the data AND the kernel• Many interesting properties:


Mercer’s Theorem

• The kernel matrix is Symmetric Positive Definite• Any symmetric positive definite matrix can be

regarded as a kernel matrix, that is as an inner product matrix in some space

• More formally, Mercer’s Theorem: Every (semi) positive definite, symmetric function is a kernel: i.e. there exists a mapping such that it is possible to write:

Definition of Positive Definiteness:

1 2 1 2( , ) ( ), ( )K x x x x

2( , ) ( ) ( ) 0,K x z f x f z dxdz f L


Mercer’s Theorem cont.

• Eigenvalues expansion of Mercer’s Kernels:

• That is: the eigenvalues act as features!

1 2 1 2( , ) ( ), ( )i i ii

K x x x x


Examples of Kernels

• Simple examples of kernels are:

which is a polynomial of degree d

which is Gaussian RBF

two-layer sigmoidal neural network

( , ) ( , 1)dK x z x z

2 2|| || / 2( , ) x zK x z e

( , ) tanh( , )K x z x z


Example: Polynomial Kernels

•

1 2

1 2

( , );

( , );

x x x

z z z

2 21 1 2 2

2 2 2 21 1 2 2 1 1 2 2

2 2 2 21 2 1 2 1 2 1 2

, ( )

2

( , , 2 ), ( , , 2 )

( ), ( )

x z x z x z

x z x z x z x z

x x x x z z z z

x z


Example

xxx

x

( ) x

ooo

( ) o

xx x

o( ) o ( ) o

( ) o

( ) x

( ) x

( ) x

X F


Making Kernels

The set of kernels is closed under some operations. IfK,K’ are kernels, then: • K+K’ is a kernel• cK is a kernel, if c>0• aK+bK’ is a kernel, for a,b>0• Etc…• One can make complex kernels from simple ones:

modularity!


Second Property of SVMs:

SVMs are Linear Learning Machines, that :• Use a dual representation• Operate in a kernel induced feature space (that is:

is a linear function in the feature space implicitly defined by K)

( ) ( ), ( )i iif x y x x b


A bad kernel

• .. Would be a kernel whose kernel matrix is mostly diagonal: all points orthogonal to each other, no clusters, no structure….

1 0 0 … 0

0 1 0 … 0

1

… … … … …

0 0 0 … 1


No Free Kernel

• In mapping in a space with too many irrelevant features, kernel matrix becomes diagonal

• Need some prior knowledge of target so choose a good kernel


The Generalization Problem

• The curse of dimensionality: easy to overfit in high dimensional spaces

(=regularities could be found in the training set that are accidental, that is that would not be found again in a test set)

• The SVM problem is ill posed (finding one hyperplane that separates the data: many such hyperplanes exist)

• Need principled way to choose the best possible hyperplane


The Generalization Problem cont.

• “Capacity” of the machine – ability to learn any training set without error.

• “A machine with too much capacity is like a botanist with a photographic memory who, when presented with a new tree, concludes that it is not a tree because it has a different number of leaves from anything she has seen before; a machine with too little capacity is like the botanist’s lazy brother, who declares that if it’s green, it’s a tree”

C. Burges


Probably Approximately Correct Learning

Assumptions and Definitions• Suppose:

We are given l observationsTrain and test points drawn randomly (i.i.d) from

some unknown probability distribution D(x,y) • The machine learns the mapping and

outputs a hypothesis . A particular choice of

generates “trained machine”.• The expectation of the test error or “expected risk” is

( , ).i iy x

i iyx( , )h x

1( ) | ( , ) | ( , ) (1)

2R y h dD y x x


A Bound on the Generalization Performance

• The “empirical risk” is:

• Choose some such that . With probability the following bound holds (Vapnik,1995):

where is called VC dimension is a measure of “capacity” of machine.

R.h.s. of (3) is called the “risk bound” of h(x,) in distribution D.

1

1( ) | ( , ) | . (2)

2

l

emp i ii

R y hl

x

0 1 1

(log(2 / ) 1) log( / 4)( ) ( ) (3)emp

d l dR R

l

0d


A Bound on the Generalization Performance

• The second term in the right-hand side is called VC confidence.

• Three key points about the actual risk bound:It is independent of D(x,y) It is usually not possible to compute the left

hand side.If we know d, we can compute the right hand

side.• This gives a possibility to compare learning machines!


The VC Dimension

• Definition: the VC dimension of a set of functions

is d if and only if there exists a set of points such that these points can be labeled in all 2d possible configurations, and for each labeling, a member of set H can be found which correctly assigns those labels, but that no set exists where q>d satisfying this property.

1{ }i qix

1{ }i dix

{ ( , )}H h x


The VC Dimension

Saying another way:VC dimension is size of largest subset of X shattered by H (every dichotomy implemented). VC dimension measures the capacity of a set H of functions.

• If for any number N, it is possible to find N points

that can be separated in all the 2N possible ways, we will say that the VC-dimension of the set is infinite

1,.., Nx x


The VC Dimension Example

• Suppose that the data live in space, and the set consists of oriented straight lines, (linear discriminants).

While it is possible to find three points that can be shattered by this set of functions, it is not possible to find four. Thus the VC dimension of the set of linear discriminants in is three.

2R { ( , )}h x

2R


The VC Dimension cont.

• Theorem 1 Consider some set of m points in . Choose any one of the points as origin. Then the m points can be shattered by oriented hyperplanes if and only if the position vectors of the remaining points are linearly independent.

• Corollary: The VC dimension of the set of oriented hyperplanes in is n+1, since we can always choose n+1 points, and then choose one of the points as origin, such that the position vectors of the remaining points are linearly independent, but can never choose n+2 points

nR

nR


The VC Dimension cont.

• VC dimension can be infinite even when the number of parameters of the set of hypothesis functions is low.

• Example: For any integer l with any labels we can find l points and parameter such that

those points can be shattered by Those points are:

and parameter is:

{ ( , )}h x

( , ) sgn(sin( )), ,h x x x R

1,..., , { 1,1}l iy y y

1,..., lx x( , )h x

10 , 1,..., .iix i l

1

(1 )10(1 )

2

ili

i

y


Minimizing the Bound by Minimizing d

/d l


Minimizing the Bound by Minimizing d

• VC confidence (second term in (3)) dependence on d/l given 95% confidence level ( ) and assuming training sample of size 10000.

• One should choose that learning machine whose set of functions has minimal d

• For d/l>0.37 (for and l=10000) VC confidence >1. Thus for higher d/l the bound is not tight.

0.05

0.05


Example

• Question. What is VC dimension and empirical risk of the nearest neighbor classifier?

• Any number of points, labeled arbitrarily, will be successfully learned, thus and empirical risk =0 .

So the bound provide no information in this example.d


Structural Risk Minimization

• Finding a learning machine with the minimum upper bound on the actual risk leads us to a method of choosing an optimal machine for a given task. This is the essential idea of the structural risk minimization (SRM).

• Let be a sequence of nested subsets of hypotheses whose VC dimensions satisfy

d1 < d2 < d3 < … SRM then consists of finding that subset of functions which minimizes the upper bound on the actual risk. This can be done by training a series of machines, one for each subset, where for a given subset the goal of training is to minimize the empirical risk. One then takes that trained machine in the series whose sum of empirical risk and VC confidence is minimal.

1 2 3 ...H H H

11/4/2015236875 visual recognition1 content linear learning machines and svm the perceptron...

Documents