statistical learning theory and classification based on support vector machines

STATISTICAL LEARNING THEORY AND CLASSIFICATION BASED ON SUPPORT VECTOR MACHINES

Based on:

The Nature of Statistical Learning Theory by V. Vapnick

2009 Presentation by John DiMona

and some slides based on lectures given by Professor Andrew Moore of Carnegie Mellon University

Presentation by Michael Sullivan

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EMPIRICAL DATA MODELING Observations of a system are collected Based on these observations a process

of induction is used to build up a model of the system

This model is used to deduce responses of the system not yet observed

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EMPIRICAL DATA MODELING Data obtained through observation is

finite and sampled by nature Typically this sampling is non-uniform Due to the high dimensional nature of

some problems the data will form only a sparse distribution in the input space

Creating a model from this type of data is an ill posed problem

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EMPIRICAL DATA MODELING

Selected model

Globally Optimal Model

Best Reachable Model

The goal in modeling is to choose a model from the hypothesis space, which is closest (with respect to some error measure) to the underlying function in the target space.

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MODELING ERROR Approximation Error is a consequence of the

hypothesis space not exactly fitting target space, The underlying function may lie outside the

hypothesis space A poor choice of the model space will result in a large

approximation error (model mismatch) Estimation Error is the error due to the learning

procedure converging to a non-optimal model in the hypothesis space

Together these form the Generalization Error

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EMPIRICAL DATA MODELING

Selected model

Globally Optimal Model

Best Reachable Model

The goal in modeling is to choose a model from the hypothesis space, which is closest (with respect to some error measure) to the underlying function in the target space.

Generalization Error

Approximation Error

Estimation Error

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WHAT IS STATISTICAL LEARNING? Definition: “Consider the learning problem as a

problem of finding a desired dependence using a limited number of observations.” (Vapnick 17)

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MODEL OF SUPERVISED LEARNING

Training: The supervisor takes each generated x value and returns an output value y.

Each (x,y) pair is part of the training set: F(x,y) = F(x),F(y|x) = (x1, y1), ….., (xl, yl)

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MODEL OF SUPERVISED LEARNING

Goal: For each (x,y) pair, we want to choose the LM’s estimation function : that closest estimates according the supervisor’s response, y.

Once we have the estimation function, we can classify new and unseen data.

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RISK MINIMIZATION To find the best function, we need to

measure loss. is the discrepancy function based on the y’s generated by the supervisor and the ‘s generated by the estimation functions.

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RISK MINIMIZATIONTo do this, we calculate the risk functional:

We choose the function, f(x, α) that minimizes the risk functional R(α) over the class functions f(x, α), α ϵ Λ

Remember, F(x,y) is unknown except for the information contained in the training set.

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RISK MINIMIZATION WITH PATTERN RECOGNITION

With pattern recognition, the supervisor’s output y can only take on 2 values, y = {0, 1} and the loss takes the following values.

So the risk functional determines the probability of different answers being given by the supervisor and the estimation function.

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RISK MINIMIZATION

The expected value of loss with regards to some estimation function :

where

Problem: We still don’t don’t know

f (x,)

R() L(y, f (x,))d P(x,y)

P(x,y) P(x)P(y | x)

P(x,y)

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TO SIMPLIFY THESE TERMS…From this point on, we’ll refer to the training set,

{(x1, y1), (x2, y2),…,(xl, yl) }, as

{z1, z2, …, zl}

And we’ll refer to the loss functional, , as

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EMPIRICAL RISK MINIMIZATION (ERM) Instead of measuring risk over the set of all

just measure it over just the training set giving the empirical risk functional of

The empirical risk must converge uniformly to the actual risk over the set of loss functions

in both directions15

SO WHAT DOES LEARNING THEORY NEED TO ADDRESS? i. What are the (necessary and sufficient) conditions

for consistency of a learning process based on the ERM principle?

ii. How fast is the rate of convergence of the learning process?

iii. How can one control the rate of convergence (the generalization ability) of the learning process?

iv. How can one construct algorithms that can control the generalization ability?

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VC DIMENSION (VAPNIK–CHERVONENKIS) The VC dimension is a scalar value that measures

the capacity of a set of functions.

The VC dimension of a set of functions is responsible for the generalization ability of learning machines.

The VC dimension of a set of indicator functions α ϵ Λ is the maximum number h of vectors z1, …zh that can be separated into two classes in all possible ways using functions of the set.

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VC DIMENSION

3 vectors can be shattered, but not 4 since vectors z2, z4 cannot be separated by a line from vectors z1, z3Rule: The set of linear indicator functions in n dimensional space has a VC dimension h = n + 1

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UPPER BOUND FOR RISK It can be shown that ,

where is the confidence interval and h is the VC dimension

ERM only minimizes and , the confidence interval, is fixed based on the VC dimension of the set of functions determined a priori

When implementing ERM one must tune the confidence interval based on the problem to avoid underfitting/overfitting the data 19

STRUCTURAL RISK MINIMIZIATION (SRM) SRM attempts to minimize the right hand side of

the inequality over both terms simultaneously

The first term is dependent upon a specific function’s error and the second depends on the VC dimension of the space that function is in

Therefore VC dimension must be a controlling variable

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STRUCTURAL RISK MINIMIZATION (SRM) We define our hypothesis space S to be the set

of functions

We say that is the hypothesis space of VC dimension, k, such that:

For a set of observations SRM chooses the function minimizing the empirical risk in subset for which the guaranteed risk is minimal

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STRUCTURAL RISK MINIMIZATION (SRM) SRM defines a trade-off between the quality of

the approximation of the given data and the complexity of the approximating function

As VC dimension increases the minima of the empirical risks decrease but the confidence interval increases

SRM is more general than ERM because it uses the subset for which minimizing yields the best bound on

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SUPPORT VECTOR CLASSIFICATION Uses the SRM principal to separate two

classes by a linear indicator function which is induced from available examples in the training set.

The goal is to produce a classifier that will work well on unseen test examples. We want to the classifier with the maximum generalizing capacity i.e. the lowest risk.

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SIMPLEST CASE: LINEAR CLASSIFIERS

How would you classify this data?

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All of these lines work as linear classifiersWhich one is the best?

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Define the margin of a linear classifier as the width the boundary can be increased by before hitting a datapoint.

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We want the maximum margin linear classifier.

This is the simplest SVM called a linear SVM

Support vectors are the datapoints the margin pushes up against

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-1 zone

+1 zone

Plus Plane

Minus Plane

We can define these two planes by x, the y-intercept, b, and w, a vector perpendicular to the lines they lie on so that the dot product gives the perpendicular planes 28

THE OPTIMAL SEPARATING HYPERPLANE But how can we find M in terms of w

and b when the planes are defined as:

Positive plane = (w * x) + b = 1 Negative plane = (w * x) +b = -1

Note: Linear classifier plane: (w * x) + b = 0

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THE OPTIMAL SEPARATING HYPERPLANE

(w * x) + b ≥ 1 (w * x) + b ≤ 1

The margin is defined the distance from any point on the minus plane to the closest point on the plus plane

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Why?

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=

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=

=So

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So we want to maximize

Or minimize

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GENERALIZED OPTIMAL HYPERPLANE Possible to extend to non-separable

training sets by adding a error parameter and minimizing:

Data can be split into more than two classifications by using successive runs on the resulting classes

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QUADRATIC PROGRAMMING Optimization algorithms used to maximize a quadratic function of some real-valued variables subject to linear constraints. If we were working in the linear world, we’d want to minimize

Now, we want to maximize:

In the nonnegative quadrant Under the constraint

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SUPPORT VECTOR MACHINES (SVM)Maps the input vectors x into a high-dimensional feature space using a kernel function

In this feature space the optimal separating hyperplane is constructed

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HOW DO SV MACHINES HANDLE DATA IN DIFFERENT CIRCUMSTANCES?

Basic one dimensional example?

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Easy!43


Harder one dimensional example?

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Project the lower dimensional training points into higher dimensional space

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SV MACHINES How are SV Machines implemented?

Polynomial Learning Machines Radial Basis Functions Machines Two Layer Neural Networks

Each of these methods and all SV Machine implementation techniques use a different kernel function.

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TWO-LAYER NEURAL NETWORK APPROACHThe kernel is a sigmoid function:

Implementing the rules:

Using this technique the following are found automatically:

i. Architecture of the two layer machine, determining the number N of units in the first layer (the number of support vectors)

ii. The vectors of the weights in the first layer

iii. The vector of weights for the second layer (values of )

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TWO-LAYER NEURAL NETWORK APPROACH

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HANDWRITTEN DIGIT RECOGNITIONData used from the U.S. Postal Service Database (1990)

Purpose was to experiment on learning the recognition of handwritten digits using different SV machines

-7300 training patterns-2000 test patterns collected from real-life zip codes

16X16 pixel resolution of database 256 dimensional input space

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HANDWRITTEN DIGIT RECOGNITION

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HANDWRITTEN DIGIT RECOGNITION

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CONCLUDING REMARKS ON SV MACHINES When implementing, the quality of a

learning machine is characterized by three main components:

1. How rich and universal is the set of functions that the LM can approximate?

2. How well can the machine generalize? 3. How fast does the learning process for

this machine converge?

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EXAM QUESITON 1 What are the two components of

Generalization Error?

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EXAM QUESITON 1 What are the two components of

Generalization Error?

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Approximation Error and Estimation Error

EXAM QUESTION 2 What is the main difference between

Empirical Risk Minimization and Structural Risk Minimization?

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EXAM QUESTION 2 What is the main difference between

Empirical Risk Minimization and Structural Risk Minimization?

ERM: Keep the confidence interval fixed (chosen a priori) while minimizing empirical risk

SRM: Minimize both the confidence interval and the empirical risk simultaneously

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EXAM QUESTION 3 What differs between SVM

implementations? i.e. Polynomial, radial basis learning

machines, neural network LM’s?

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EXAM QUESTION 3 What differs between SVM

implementations? i.e. Polynomial, radial basis learning

machines, neural network LM’s?

The kernel function

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ANY QUESTIONS?

statistical learning theory and classification based on support vector machines

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