1. EE 690 Design of Embodied Intelligence
Least-squares-based Multilayer perceptron training with weighted adaptation Software simulation project
EE 690
Design of Embodied Intelligence

2. Outline Multilayer Perceptron Least-squares based Learning Algorithm
Weighted Adaptation in training
Signal-to-Noise Ratio Figure and Overfitting
Software simulation project

3. Multilayer perceptron (MLP)
Feedforward (no recurrent connections) network with units arranged in layers
Inputs x
Outputs z
A typical 2-layer multi-layer perceptron contains input layer, hidden layer and output layer.
The inputs go into the perceptron, the data go through the connections between input layer and the hidden layer.
The weighted summation goes through the neurons on the hidden layer.
The hidden neurons implements usually nonlinear transfer functions to bring the nonlinearity of the networks.
The data from the hidden neurons go through the connections between the hidden layer and the output layer.
The weighted summation of the data from hidden neurons will be the output of the network.
The weights are trainable to implement functions.

4. Multilayer perceptron (MLP)
Efficient mapping from inputs to outputs
Powerful universal function approximation
Number of inputs and outputs determined by the data
Number of hidden neurons
Number of hidden layers
MLP
outputs
inputs
The weights are trainable to map the relations between input and output data.
The MLP can be a powerful tool in function approximation in the sense that it finds the weights to combine nonlinear basis function.
In function approximation, the number of basis function to use affects the learning ability of the function approximator.
The number of the input and output neurons are determined by the dimensions of the data, so that only the number of hidden neurons is left to be decided by users.
And the number of hidden neurons, according to number of basis functions to use in approximation, affects the fitting accuracy. In such sense, it is a critical parameter for NNs.

5. Multilayer Perceptron Learning
Back-propagation (BP) training algorithm: how much each weight is responsible for the error signal
BP has two phases:
Forward pass phase: feedforward propagation of input signals through network
Backward pass phase: propagates the error backwards through network
hidden layer
input layer
output layer

6. Multilayer Perceptron Learning
Backward Pass
We want to know how to modify weights in order to decrease E.
Use gradient descent:
Gradient-based adjustment could go to local minima
Time-consuming due to large number of learning steps and the step size needs to be configured

7. Least-squares based Learning Algorithm
Least-squared fit (LSF): to obtain the minimum sum of squared error
For underdetermined problem, LSF finds the solution with the minimum SSE
For overdetermined problem, pseudo-inverse finds the solution with minimum norm
Can be applied in the optimization for weights or signals on the layers
Optimized weights
Optimized signals

8. Least-squares based Learning Algorithm (I)
Start with desired output signal back-propagation
 signals optimization
Propagation of the desired outputs back through layers
Optimization of the weights between layers
(1). y2=f -1(z2), scale y1 to (-1, 1).
(2). Based on W2, b2: W2.z1=y2-b2.
(3). y1=f-1(z1), scale y1 to (-1, 1).
(4). Optimize W1, b1 to satisfy W1.x-b1=y1.
(5). Evaluate z1, y1 using the new W1 and bias b1.
(6). Optimize W2, b2 to satisfy W2.z1+b2=y2.
(7). Evaluate z2, y2 using the new W2 and bias b2.
(8). Evaluate the MSE z2 y2 d W1 y1 z1 b1 W2 x b2

9. Least-squares based Learning Algorithm (I)
Weights optimization with weighted LSF
The location of x on the transfer function determines its effect on output signal of this layer
dy/dx  weighting term in LSF Δy
Optimize W1, b1 to satisfy W1.x=y1-b1 Δy Δx Δx
Weighted LSF

10. Least-squares based Learning Algorithm (II)
II. Weights optimization with iterative fitting
W1 can be further adjusted based on the output error x
Each hidden neuron: basis function
Start with the 1st hidden neurons, and continue to other neurons
as long as eout exists

11. Least-squares based Learning Algorithm (III)
III. Start with input feedforward  weights optimization
Propagation of the inputs forward through layers
Optimization of the weights between layers and signals on layers
(1). Evaluate z1, y1 using the initial W1 and bias b1.
(2). y2=f -1(d).
(3). Optimize W2, b2 to satisfy W2.z1+b2=y2.
(4). Based on W2, b2, optimize z1 to satisfy
W2.z1-b2=y2.
(5). y1=f-1(z1).
(6). Optimize W1, b1 to satisfy W1.x+b1=y1.
(7). Evaluate y1, z1, y2, z2 using the new W1,W2
and bias b1,b2.
(8). Evaluate the MSE z2 y2 d W1 y1 z1 b1 W2 x b2

12. Least-squares based Learning Algorithm (III)
Signal optimization with weighted adaptation
The location of x on the transfer function determines how much the signal can be changed y x

13. Overfitting problem
Learning algorithm can adapt MLP to fit into the training data.
For the noisy training data, how well we should learn into the data?
Overfitting
Number of hidden neurons
Number of layers
 affect the training accuracy, determined by users: critical
Optimized Approximation Algorithm –SNRF criterion

14. Signal-to-noise ratio figure (SNRF)
Sampled data: function value + noise
Error signal:
approximation error component + noise component
Noise part
Should not be learned
Useful signal
Should be reduced
Without a priori knowledge of the noise characteristics of the training data, it is assumed that the training data may come with White Gaussian Noise (WGN) at an unknown level.
The error signal is the difference between the approximating function value and the training data.
In order to have a clear indication of overfitting, we need to examine this error signal. The error signal contains two components: the approximation error due to the limited learning ability or inaccuracy in approximation using the given number of hidden neurons in the network, and the WGN in the training data. The approximation error shows how much useful signal left unlearned which should be reduced. However, the WGN part shows the noise which we shouldn’t learn.
The question of whether the learning should stop or the approximation is good enough becomes the question that: whether there is still useful signal information left to be learned in the error signal or whether the noise part dominates in the error signal.
If there is, based on the assumption that the function we try to approximate is continuous and that the noise is WGN, we can estimate the level of signal and noise in the error signal. The ratio of the signal energy level over the noise energy level is defined as SNRF. The SNRF can be pre-calculated for the WGN. The comparison of SNRF of the error signal with that of WGN determines whether WGN dominates in the error signal. If it does, there is little useful information left in the error signal, and the approximation error cannot be reduced anymore.
Assumption: continuous function & WGN as noise
Signal-to-noise ratio figure (SNRF):
signal energy/noise energy
Compare SNRFe and SNRFWGN
Learning should stop – ?
If there is useful signal left unlearned
If noise dominates in the error signal

15. Signal-to-noise ratio figure (SNRF)
Training data and approximating function
Error signal
The method to estimate the signal level and noise level in the error signal so that to obtain SNRF is explained in a one-dimensional case first.
The figure on the left shows the training data and its existing fitting using quadratic polynomials. The difference between them, the error signal, is shown on the right. Obviously, the error signal doesn’t show the property of WGN.
The error signal e contains an approximation error component, which is the useful signal left unlearned, and a noise component. The level of the noise is unknown.
How can we measure the signal level and noise level.
approximation error component +
noise component

16. Optimization using SNRF
SNRFe< threshold SNRFWGN
Start with small network (small # of neurons or layers)
Train the MLP  etrain
Compare SNRFe & SNRFWGN
Add hidden neurons
Noise dominates in the error signal,
Little information left unlearned,
Learning should stop
Using SNRF, we can estimate the signal level and noise level for the error signal and then quantitatively determine the amount of useful signal information left unlearned. The noise characteristics simply serve as a reference for developing the stopping criterion. When SNRFe is smaller than the threshold about SNRFWGN, it means that the noise dominates in the error signal and there is little information left unlearned, and the learning process with certain number of hidden neurons can be stopped. Otherwise, more hidden neurons have to be used to improve the learning.
In the process of optimizing number of hidden neurons of NN, one may start with a network with a small number of hidden neurons. Examine the training error signal and obtain its SNRF. Compare the SNRF with the threshold. If it is lower than the threshold, more hidden neurons can be added until SNRF indicates overfitting. And the stopping criterion can be expressed as that the SNRF of error signal is lower than the threshold.
Stopping criterion:
SNRFe< threshold SNRFWGN

17. Optimization using SNRF
Applied in optimizing number of iterations in back-propagation training to avoid overfitting (overtraining)
Set the structure of MLP
Train the MLP with back-propagation iteration
 etrain
Compare SNRFe & SNRFWGN
Keep training with more iterations
Using SNRF, we can estimate the signal level and noise level for the error signal and then quantitatively determine the amount of useful signal information left unlearned. The noise characteristics simply serve as a reference for developing the stopping criterion. When SNRFe is smaller than the threshold about SNRFWGN, it means that the noise dominates in the error signal and there is little information left unlearned, and the learning process with certain number of hidden neurons can be stopped. Otherwise, more hidden neurons have to be used to improve the learning.
In the process of optimizing number of hidden neurons of NN, one may start with a network with a small number of hidden neurons. Examine the training error signal and obtain its SNRF. Compare the SNRF with the threshold. If it is lower than the threshold, more hidden neurons can be added until SNRF indicates overfitting. And the stopping criterion can be expressed as that the SNRF of error signal is lower than the threshold.

18. Software simulation project
Prepare the data
Data sample along the row: N samples
Features along the column: M features
Desired output in a row vector: N values
Save “features” and “values” in a training MAT file
How to recall the function
Run “main_MLP_LS.m”
Specify MAT file path and name and MLP parameters in command window.
M x N matrix: “Features”
1 x N vector: “Values”

19. Software simulation project
Input the path where data file can be found (C:*): E:\Research\MLP_LSInitial_desired\MLP_LS_package\
Input the name of data file (*.mat): mackey_glass_data.mat
There are overall 732 samples. How do you like to divide them into training and testing set?
Number of training samples: 500
Number of testing samples: 232
How many layers does MLP have? 3:2:7
How many neurons there are on each hidden layer ? 3:1:10
What kind of tranfer function you like to have on hidden neurons?
0. Linear tranfer function
1. Tangent sigmoid
2. Logrithmic sigmoid 2

20. Software simulation projectz2 y2 d W1 y1 z1 b1 W2 x b2
There are 4 types of training algorithms you can choose from. Which type you like to use?
1. Least-squared based training (I)
2. Least-squared based training with iterative neuron fitting (II)
3. Least-squared based training with weighted signal adaptation (III)
4. Back-propagation training (BP) 1
How many iterations you would like to have in the training ? 3
How many Monte-Carlo runs you would like to have for the training? 2

21. Software simulation project
Results:
J_train (num_layer, num_neuron)
J_test (num_layer, num_neuron)
SNRF (num_layer, num_neuron)
Present training and testing errors for various configurations of the MLP
Present the optimum configuration found by SNRF
Present the comparison of the results, including errors, network structure

22. Software simulation project
Typical database and literature survey
Function approximation & classification dataset
“IEEE Neural Networks Council Standards Committee Working Group on Data modeling Benchmarks”
“Neural Network Databases and Learning Data”
“UCI Machine Learning Repository”
Data are normalized
Multiple input, with signal output.
For multiple output data, use separate MLPs.
Compare results from literature which uses the same dataset (*)