Introduction to Radial Basis Function Networks

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Introduction to Radial Basis Function Networks

• ??? ???

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Content

• Overview
• The Models of Function Approximator
• The Radial Basis Function Networks
• RBFNs for Function Approximation
• The Projection Matrix
• Learning the Kernels
• Bias-Variance Dilemma
• The Effective Number of Parameters
• Model Selection
• Incremental Operations

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Introduction to Radial Basis Function Networks

• Overview

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Typical Applications of NN

• Pattern Classification
• Function Approximation
• Time-Series Forecasting

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Function Approximation
Unknown
Approximator
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Supervised Learning
Unknown Function
Neural Network
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Neural Networks as Universal Approximators

• Feedforward neural networks with a single hidden
  layer of sigmoidal units are capable of
  approximating uniformly any continuous
  multivariate function, to any desired degree of
  accuracy.
• Hornik, K., Stinchcombe, M., and White, H.
  (1989). "Multilayer Feedforward Networks are
  Universal Approximators," Neural Networks, 2(5),
  359-366.
• Like feedforward neural networks with a single
  hidden layer of sigmoidal units, it can be shown
  that RBF networks are universal approximators.
• Park, J. and Sandberg, I. W. (1991). "Universal
  Approximation Using Radial-Basis-Function
  Networks," Neural Computation, 3(2), 246-257.
• Park, J. and Sandberg, I. W. (1993).
  "Approximation and Radial-Basis-Function
  Networks," Neural Computation, 5(2), 305-316.

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Statistics vs. Neural Networks
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Introduction to Radial Basis Function Networks

• The Model of
• Function Approximator

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Linear Models
Weights
Fixed Basis Functions
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Linear Models
Linearly weighted output
Output Units

• Decomposition
• Feature Extraction
• Transformation

Hidden Units
Inputs
Feature Vectors
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Linear Models
Can you say some bases?
y
Linearly weighted output
Output Units
w2
w1
wm

• Decomposition
• Feature Extraction
• Transformation

Hidden Units
?1
?2
?m
Inputs
Feature Vectors
x1
x2
xn
x
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Example Linear Models
Are they orthogonal bases?

• Polynomial
• Fourier Series

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Single-Layer Perceptrons as Universal
Aproximators
With sufficient number of sigmoidal units, it can
be a universal approximator.
Hidden Units
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Radial Basis Function Networks as Universal
Aproximators
With sufficient number of radial-basis-function
units, it can also be a universal approximator.
Hidden Units
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Non-Linear Models
Weights
Adjusted by the Learning process
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Introduction to Radial Basis Function Networks

• The Radial Basis Function Networks

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Radial Basis Functions
Three parameters for a radial function
?i(x)? (x ? xi)
xi

• Center
• Distance Measure
• Shape

r x ? xi
?
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Typical Radial Functions

• Gaussian
• Hardy Multiquadratic
• Inverse Multiquadratic

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Gaussian Basis Function (?0.5,1.0,1.5)
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Inverse Multiquadratic
c5
c4
c3
c2
c1
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Most General RBF
Basis ?i i 1,2, is near orthogonal.
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Properties of RBFs

• On-Center, Off Surround
• Analogies with localized receptive fields found
  in several biological structures, e.g.,
• visual cortex
• ganglion cells

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The Topology of RBF
As a function approximator
Output Units
Interpolation
Hidden Units
Projection
Feature Vectors
Inputs
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The Topology of RBF
As a pattern classifier.
Output Units
Classes
Hidden Units
Subclasses
Feature Vectors
Inputs
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Introduction to Radial Basis Function Networks

• RBFNs for
• Function Approximation

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The idea
y
x
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The idea
y
x
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The idea
y
x
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The idea
y
x
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The idea
y
x
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Radial Basis Function Networks as Universal
Aproximators
Training set
Goal
for all k
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Learn the Optimal Weight Vector
Training set
Goal
for all k
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Regularization
Training set
If regularization is unneeded, set
Goal
for all k
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Learn the Optimal Weight Vector
Minimize
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Learn the Optimal Weight Vector
Define
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Learn the Optimal Weight Vector
Define
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Learn the Optimal Weight Vector
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Learn the Optimal Weight Vector
Design Matrix
Variance Matrix
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Summary
Training set
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Introduction to Radial Basis Function Networks

• The Projection Matrix

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The Empirical-Error Vector
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The Empirical-Error Vector
Error Vector
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Sum-Squared-Error
If ?0, the RBFNs learning algorithm is to
minimize SSE (MSE).
Error Vector
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The Projection Matrix
Error Vector
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Introduction to Radial Basis Function Networks

• Learning the Kernels

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RBFNs as Universal Approximators
Training set
Kernels
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What to Learn?

• Weights wijs
• Centers ?js of ?js
• Widths ?js of ?js
• Number of ?js ? Model Selection

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One-Stage Learning
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One-Stage Learning
The simultaneous updates of all three sets of
parameters may be suitable for non-stationary
environments or on-line setting.
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Two-Stage Training
Step 2
Determines wijs.
E.g., using batch-learning.
Step 1

• Determines
• Centers ?js of ?js.
• Widths ?js of ?js.
• Number of ?js.

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Train the Kernels
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Unsupervised Training
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Methods

• Subset Selection
• Random Subset Selection
• Forward Selection
• Backward Elimination
• Clustering Algorithms
• KMEANS
• LVQ
• Mixture Models
• GMM

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Subset Selection
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Random Subset Selection

• Randomly choosing a subset of points from
  training set
• Sensitive to the initially chosen points.
• Using some adaptive techniques to tune
• Centers
• Widths
• points

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Clustering Algorithms
Partition the data points into K clusters.
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Clustering Algorithms
Is such a partition satisfactory?
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Clustering Algorithms
How about this?
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Clustering Algorithms
?1


?2

?4

?3
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Introduction to Radial Basis Function Networks

• Bias-Variance Dilemma

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Goal Revisit

• Ultimate Goal ? Generalization

Minimize Prediction Error

• Goal of Our Learning Procedure

Minimize Empirical Error
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Badness of Fit

• Underfitting
• A model (e.g., network) that is not sufficiently
  complex can fail to detect fully the signal in a
  complicated data set, leading to underfitting.
• Produces excessive bias in the outputs.
• Overfitting
• A model (e.g., network) that is too complex may
  fit the noise, not just the signal, leading to
  overfitting.
• Produces excessive variance in the outputs.

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Underfitting/Overfitting Avoidance

• Model selection
• Jittering
• Early stopping
• Weight decay
• Regularization
• Ridge Regression
• Bayesian learning
• Combining networks

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Best Way to Avoid Overfitting

• Use lots of training data, e.g.,
• 30 times as many training cases as there are
  weights in the network.
• for noise-free data, 5 times as many training
  cases as weights may be sufficient.
• Dont arbitrarily reduce the number of weights
  for fear of underfitting.

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Badness of Fit
Underfit
Overfit
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Badness of Fit
Underfit
Overfit
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Bias-Variance Dilemma
However, it's not really a dilemma.
Underfit
Overfit
Large bias
Small bias
Small variance
Large variance
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Bias-Variance Dilemma

• More on overfitting
• Easily lead to predictions that are far beyond
  the range of the training data.
• Produce wild predictions in multilayer
  perceptrons even with noise-free data.

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Bias-Variance Dilemma
It's not really a dilemma.
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Bias-Variance Dilemma
The mean of the bias?
The variance of the bias?
The true model
bias
bias
bias
E.g., depend on hidden nodes used.
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Bias-Variance Dilemma
The mean of the bias?
The variance of the bias?
Variance
The true model
E.g., depend on hidden nodes used.
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Model Selection
Reduce the effective number of parameters.
Reduce the number of hidden nodes.
Variance
The true model
E.g., depend on hidden nodes used.
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Bias-Variance Dilemma
Goal
The true model
E.g., depend on hidden nodes used.
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Bias-Variance Dilemma
Goal
Goal
0
constant
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Bias-Variance Dilemma
0
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Bias-Variance Dilemma
Goal
bias2
variance
Minimize both bias2 and variance
noise
Cannot be minimized
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Model Complexity vs. Bias-Variance
Goal
bias2
variance
noise
Model Complexity (Capacity)
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Bias-Variance Dilemma
Goal
bias2
variance
noise
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Example (Polynomial Fits)
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Example (Polynomial Fits)
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Example (Polynomial Fits)
Degree 1
Degree 5
Degree 10
Degree 15
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Introduction to Radial Basis Function Networks

• The Effective Number of Parameters

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Variance Estimation
Mean
Variance
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Variance Estimation
Mean
Variance
Loss 1 degree of freedom
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Simple Linear Regression
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Simple Linear Regression
Minimize
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Mean Squared Error (MSE)
Minimize
Loss 2 degrees of freedom
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Variance Estimation
Loss m degrees of freedom
m parameters of the model
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The Number of Parameters
m
degrees of freedom
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The Effective Number of Parameters (?)
The projection Matrix
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The Effective Number of Parameters (?)
Facts
Pf)
The projection Matrix
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Regularization
The effective number of parameters
Penalize models with large weights
SSE
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Regularization
The effective number of parameters
Without penalty (?i0), there are m degrees of
freedom to minimize SSE (Cost). The effective
number of parameters ? m.
Penalize models with large weights
SSE
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Regularization
The effective number of parameters
With penalty (?igt0), the liberty to minimize SSE
will be reduced. The effective number of
parameters ? ltm.
Penalize models with large weights
SSE
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Variance Estimation
The effective number of parameters
Loss ? degrees of freedom
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Variance Estimation
The effective number of parameters
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Introduction to Radial Basis Function Networks

• Model Selection

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Model Selection

• Goal
• Choose the fittest model
• Criteria
• Least prediction error
• Main Tools (Estimate Model Fitness)
• Cross validation
• Projection matrix
• Methods
• Weight decay (Ridge regression)
• Pruning and Growing RBFNs

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Empirical Error vs. Model Fitness

• Ultimate Goal ? Generalization

Minimize Prediction Error

• Goal of Our Learning Procedure

Minimize Empirical Error
(MSE)
Minimize Prediction Error
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Estimating Prediction Error

• When you have plenty of data use independent test
  sets
• E.g., use the same training set to train
  different models, and choose the best model by
  comparing on the test set.
• When data is scarce, use
• Cross-Validation
• Bootstrap

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Cross Validation

• Simplest and most widely used method for
  estimating prediction error.
• Partition the original set into several different
  ways and to compute an average score over the
  different partitions, e.g.,
• K-fold Cross-Validation
• Leave-One-Out Cross-Validation
• Generalize Cross-Validation

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K-Fold CV

• Split the set, say, D of available input-output
  patterns into k mutually exclusive subsets, say
  D1, D2, , Dk.
• Train and test the learning algorithm k times,
  each time it is trained on D\Di and tested on Di.

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K-Fold CV
Available Data
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K-Fold CV
Test Set
Training Set
Available Data
D1
D2
D3
. . .
Dk
Estimate ?2
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Leave-One-Out CV
A special case of k-fold CV.

• Split the p available input-output patterns into
  a training set of size p?1 and a test set of size
  1.
• Average the squared error on the left-out pattern
  over the p possible ways of partition.

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Error Variance Predicted by LOO
A special case of k-fold CV.
The estimate for the variance of prediction error
using LOO
Error-square for the left-out element.
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Error Variance Predicted by LOO
A special case of k-fold CV.
Given a model, the function with least empirical
error for Di.
As an index of models fitness. We want to find a
model also minimize this.
The estimate for the variance of prediction error
using LOO
Error-square for the left-out element.
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Error Variance Predicted by LOO
A special case of k-fold CV.
Are there any efficient ways?
How to estimate?
The estimate for the variance of prediction error
using LOO
Error-square for the left-out element.
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Error Variance Predicted by LOO
Error-square for the left-out element.
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Generalized Cross-Validation
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More Criteria Based on CV
GCV (Generalized CV)
Akaikes Information Criterion
UEV (Unbiased estimate of variance)
FPE (Final Prediction Error)
BIC (Bayesian Information Criterio)
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More Criteria Based on CV
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More Criteria Based on CV
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Regularization
Standard Ridge Regression,
Penalize models with large weights
SSE
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Regularization
Standard Ridge Regression,
Penalize models with large weights
SSE
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Solution Review
Used to compute model selection criteria
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Example
Width of RBF r 0.5
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Example
Width of RBF r 0.5
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Example
Width of RBF r 0.5
How the determine the optimal regularization
parameter effectively?
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Optimizing the Regularization Parameter
Re-Estimation Formula
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Local Ridge Regression
Re-Estimation Formula
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Example
Width of RBF
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Example
Width of RBF
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Example
Width of RBF
There are two local-minima.
Using the about re-estimation formula, it will be
stuck at the nearest local minimum.
That is, the solution depends on the initial
setting.
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Example
Width of RBF
There are two local-minima.
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Example
Width of RBF
There are two local-minima.
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Example
Width of RBF
RMSE Root Mean Squared Error In real case, it is
not available.
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Example
Width of RBF
RMSE Root Mean Squared Error In real case, it is
not available.
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Local Ridge Regression
Standard Ridge Regression
Local Ridge Regression
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Local Ridge Regression
Standard Ridge Regression
?j ?? implies that ?j(?) can be removed.
Local Ridge Regression
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The Solutions
Used to compute model selection criteria
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Optimizing the Regularization Parameters
Incremental Operation
P The current projection Matrix.
Pj The projection Matrix obtained by removing
?j(?).
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Optimizing the Regularization Parameters
Solve
Subject to
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Optimizing the Regularization Parameters
Solve
Subject to
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Optimizing the Regularization Parameters
Remove ?j(?)
Solve
Subject to
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The Algorithm

• Initialize ?is.
• e.g., performing standard ridge regression.
• Repeat the following until GCV converges
• Randomly select j and compute
• Perform local ridge regression
• If GCV reduce remove ?j(?)

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References
Mark J. L. Orr (April 1996), Introduction to
Radial Basis Function Networks,
http//www.anc.ed.ac.uk/mjo/intro/intro.html.
Kohavi, R. (1995), "A study of cross-validation
and bootstrap for accuracy estimation and model
selection," International Joint Conference on
Artificial Intelligence (IJCAI).
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Introduction to Radial Basis Function Networks

• Incremental Operations