1. Chapter 5 NEURAL NETWORKS
by S. Betul Ceran

2. Outline Introduction Feed-forward Network Functions Network Training
Error Backpropagation
Regularization

3. Introduction

4. Multi-Layer Perceptron (1)
Layered perceptron networks can realize any logical function, however there is no simple way to estimate the parameters/generalize the (single layer) Perceptron convergence procedure
Multi-layer perceptron (MLP) networks are a class of models that are formed from layered sigmoidal nodes, which can be used for regression or classification purposes.
They are commonly trained using gradient descent on a mean squared error performance function, using a technique known as error back propagation in order to calculate the gradients.
Widely applied to many prediction and classification problems over the past 15 years.

5. Multi-Layer Perceptron (2)
XOR (exclusive OR) problem
0+0=0
1+1=2=0 mod 2
1+0=1
0+1=1
Perceptron does not work here!
Single layer generates a linear decision boundary

6. Universal Approximation
1st layer
2nd layer
3rd layer
Universal Approximation: Three-layer network can in principle
approximate any function with any accuracy!

7. Feed-forward Network Functions
(1)
f: nonlinear activation function
Extensions to previous linear models by hidden units:
Make basis function Φ depend on the parameters
Adjust these parameters during training
Construct linear combinations of the input variables x1, …, xD.
(2)
Transform each of them using a nonlinear activation function
(3)

8. Weight-space symmetry
Cont’d
Linearly combine them to give output unit activations
(4)
Key difference with perceptron is the continuous sigmoidal nonlinearities in the hidden units i.e. neural network function is differentiable w.r.t network parameters
Whereas perceptron uses step-functions
Weight-space symmetry
Network function is unchanged by certain permutations and the sign flips in the weight space.
E.g. tanh(– a) = –tanh(a) ………flip the sign of all weights out of that hidden unit

9. Two-layer neural network
zj: hidden unit

10. A multi-layer perceptron fitting into different functions
f(x)=x2
f(x)=sin(x)
f(x)=H(x)
f(x)=|x|

11. Network Training
Problem of assigning ‘credit’ or ‘blame’ to individual elements involved in forming overall response of a learning system (hidden units)
In neural networks, problem relates to deciding which weights should be altered, by how much and in which direction.
Analogous to deciding how much a weight in the early layer contributes to the output and thus the error
We therefore want to find out how weight wij affects the error ie we want:

12. Error Backpropagation

18. Two phases of back-propagation

19. Activation and Error back-propagation

20. Weight updates

21. Other minimization procedures

22. Two schemes of training
There are two schemes of updating weights
Batch: Update weights after all patterns have been presented (epoch).
Online: Update weights after each pattern is presented.
Although the batch update scheme implements the true gradient descent, the second scheme is often preferred since
it requires less storage,
it has more noise, hence is less likely to get stuck in a local minima (which is a problem with nonlinear activation functions). In the online update scheme, order of presentation matters!

23. Problems of back-propagation
It is extremely slow, if it does converge.
It may get stuck in a local minima.
It is sensitive to initial conditions.
It may start oscillating.

24. Regularization (1)
How to adjust the number of hidden units to get the best performance while avoiding over-fitting
Add a penalty term to the error function
The simplest regularizer is the weight decay:

25. Changing number of hidden units
Over-fitting
Sinusoidal data set

26. Error vs. Number of hidden units
Regularization (2)
One approach is to choose the specific solution having the smallest validation set error
Error vs. Number of hidden units

27. Consistent Gaussian Priors
One disadvantage of weight decay is its inconsistency with certain scaling properties of network mappings
A linear transformation in the input would be reflected to the weights such that the overall mapping unchanged

28. Cont’d
A similar transformation can be achieved in the output by changing the 2nd layer weights accordingly
Then a regularizer of the following form would be invariant under the linear transformations:
W1: set of weights in 1st layer
W2: set of weights in 2nd layer

29. Effect of consistent gaussian priors

30. Early Stopping A method to
obtain good generalization performance and
control the effective complexity of the network
Instead of iteratively reducing the error until a minimum of the training data set has been reached
Stop at the point of smallest error w.r.t. the validation data set

31. Effect of early stopping
Training Set
Error vs. Number of iterations
Validation Set
A slight increase in
the validation set error

32. Invariances
Alternative approaches for encouraging an adaptive model to exhibit the required invariances
E.g. position within the image, size

33. Various approaches
Augment the training set using transformed replicas according to the desired invariances
Add a regularization term to the error function; tangent propagation
Extract the invariant features in the pre-processing for later use.
Build the invariance properteis into the network structure; convolutional networks

34. Tangent Propagation (Simard et al., 1992)
A continuous transformation on a particular input vextor xn can be approximated by the tangent vector τn
A regularization function can be derived by differentiating the output function y w.r.t. the transformation parameter, ξ

35. Tangent vector implementation
corresponding to
a clockwise rotation
Original image x
True image rotated
Adding a small
contribution from
the tangent vector
x+ετ

36. References Neurocomputing course slides by Erol Sahin. METU, Turkey.
Backpropagation of a Multi-Layer Perceptron by Alexander Samborskiy. University of Missouri, Columbia.
Neural Networks - A Systematic Introduction by Raul Rojas. Springer.
Introduction to Machine Learning by Ethem Alpaydin. MIT Press.
Neural Networks course slides by Andrew Philippides. University of Sussex, UK.