Description

Rosenbrock’s Valley Problem

Consider the Rosenbrock’s Valley function:

which has a global minimum at (x,y) = (1,1) where f(x,y) = 0.  Now suppose the starting point is randomly initialized in the open interval (0, 0.5) for x and y, find the global minimum using:

a). Steepest (Gradient) descent method

w(k1) w(k)g(k)

with learning rate η = 0.001. Record the number of iterations when f(x,y) converges to (or very close to) 0 and plot out the trajectory of (x,y) in the 2-dimensional space. Also plot out the function value as it approaches the global minimum. What would happen if a larger learning rate, say η = 0.2, is used?

b). Newton’s method (as discussed on page 13 in the slides of lecture Four)

w(n)H^1(n)g(n)

Record the number of iterations when f(x,y) converges to (or very close to) 0 and plot out the trajectory of (x,y) in the 2-dimensional space. Also plot out the function value as it approaches the global minimum.

 

Q2. Function Approximation

Consider using MLP to approximate the following function:

y1.2sin(x)cos(2.4x) ,  for x [ 1,1].

The training set is generated by dividing the domain [-1, 1] using a uniform step length 0.05, while the test set is constructed by dividing the domain [-1, 1] using a uniform step length 0.01. You may use the MATLAB neural network toolbox to implement a MLP (see the Appendix for guidance) and do the following experiments:

 

a). Use the sequential mode with BP algorithm and experiment with the following different structures of the MLP: 1-n-1 (where n = 1,2,…,10, 20, 50). For each architecture plot out the outputs of the MLP for the test samples after training and compare them to the desired outputs. Try to determine whether it is under-fitting, proper fitting or over-fitting.  Identify the minimal number of hidden neurons from the experiments, and check if the result is consistent with the guideline given in the lecture slides. Compute the outputs of the MLP when x=-3 and +3, and see if the MLP can make reasonable predictions outside of the domain of the input limited by the training set.

b). Use the batch mode with trainlm algorithm to repeat the above procedure.

 

c). Use the batch mode with trainbr algorithm to repeat the above procedure.

 

Q3. Scene Classification

Multi-layer perceptron (MLP) can be used to solve real-world pattern recognition problems. In this assignment, MLP will be designed to handle a binary classification task, i.e. nature scenes vs. man-made scenes. Specifically, students are divided into 4 groups based on matric numbers and each group is assigned with different dataset as illustrated in the following Table.

 

Group ID	Nature Scenes [1]	Man-Made Scenes [0]
1	Open Country	Highway
2	Mountain	Street
3	Coast	Inside City
4	Forest	Tall Building

 

You may download the zipped dataset (e.g. group_1.zip) from LumiNUS. After unzipping, you will find two folders: train and val. The training set consists of around 500 images and validation set consists of around 165 images. Filename of each image follows the format of “imageID_label_category.jpg” (e.g. 0001_0_highway.jpg), where ‘label’ is either 1 or 0 indicating the image captures a nature scene or man-made scene; ‘category’ represents the human-readable class name of this image.

 

In order to find your group, you need to calculate “mod(LWD, 4) + 1” where LWD is the last two digits of your matric number, e.g. A1234567X is assigned to group mod(67, 4) + 1 = 4 (Forest vs. Tall Building).

 

Please specify the group ID that has been assigned to you! Take note that if you have selected wrongly, there will be some mark deduction!

 

All the images are provided in grayscale format with size 256*256. You can use I = imread(filename) to read these image files, where filename specifies the path to an image (you may use function dir() to get the filenames of images inside a folder for code efficiency). The returned value I is an array (256-by-256 in this assignment) containing the image data. For example,

I = imread(‘group_1/train/0001_0_highway.jpg’);

will read image ‘0001_0_highway.jpg’ from the training set into MATLAB workspace. Then, you could display this image using: imshow(I, []);

In order to efficiently process all the image data, you may need to convert the matrix form data I into a vector by:

V = I(:);

and the resulting V is a column vector whose elements are taken column-wisely from I. You could group all the training images together using train_images = [V1, V2, …] and all the test images together following the same way. In the next, these matrixes (of size (256*256-by-image_number)) are used as input to the networks.

 

The label information is stored in the filename of each image and can be extracted by: tmp = strsplit(‘0001_0_highway.jpg’, {‘_’, ‘.’});

L(i)= str2num(tmp{2});

where L is an array (of size (1-by-image_number)) with each element holding the ground-truth label of corresponding image.

 

You are required to complete the following tasks:

 

1. a) Apply Rosenblatt’s perceptron (single layer perceptron) to the dataset of your assigned group. After the training procedure, calculate the classification accuracy for both the training set and validation set, and evaluate the performance of the network.

b) The original input dimension is 65536 (256*256), which may be redundant and contain space for reduction. Try to naively downsample the images into 128*128, 64*64, 32*32, or apply a more sophisticated technique like PCA to these images. Then, retrain the perceptron in a) with these dimensionally reduced images and compare their performance. (you may use imresize() and processpca() or pca() in this task)

 c) Apply MLP to the dataset of your assigned group using batch mode training. After the training procedure, calculate the classification accuracy for both the training set and validation set, and evaluate the performance of the network.

 d) Please determine whether your trained MLP in c) is overfitting. If so, please specify when (i.e. after which training epoch) it becomes overfitting. Try weights regularization and observe if it helps. (you may set the regularization strength by ‘performParam.regularization’)

 e)Apply MLP to the dataset of your assigned group using sequential mode training. After the training procedure, calculate the classification accuracy for both training set and validation set, and evaluate the performance of the network. Compare the result to part c), and make your recommendation on the two approaches.

 f) Try to propose a scheme that you believe could help to improve the performance of your MLP and please explain the reason briefly.  

Important note: There are many design and training issues to be considered when you apply neural networks to solve real world problems. We have discussed most of them in the lecture four. Some of them have clear answers, some of them may rely on empirical rules, and some of them have to be determined by trial and error. I believe that you will have more fun playing with these design parameters and making your own judgment rather than solving the problem with a prescribed set of parameters. Hence, there is no standard answer to this problem, and the marking will be based upon the whole procedure rather than the final classification accuracy. (Use “help” and “doc” commands in MATLAB to get familiar with the functions that you don’t know and Google everything that confuses you.)

 

Appendix

1. Create a feed-forward back propagation network using MATLAB toolbox using: net = patternnet(hiddenSizes, trainFcn, performFcn)

where the arguments are specified as follows:           hiddenSizes — Row vector of one or more hidden layer sizes (default = 10);       trainFcn — Training function (default = ‘trainscg’);    performFcn — Performance function (default = ‘crossentropy’).

 

trainFcn specifies the optimization algorithm based on which the network is updated during training, and there are many choices:

 

• Backpropagation training functions that use Jacobian derivatives (these algorithms can be faster but require more memory than gradient backpropation):

 

trainlm — Levenberg-Marquardt backpropagation. trainbr — Bayesian Regulation backpropagation.

 

• Backpropagation training functions that use gradient derivatives (these algorithms may not be as fast as Jacobian backpropagation):

 

trainbfg — BFGS quasi-Newton backpropagation.

traincgb — Conjugate gradient backpropagation with Powell-Beale restarts.     traincgf –Conjugate gradient backpropagation with Fletcher-Reeves updates.     traincgp — Conjugate gradient backpropagation with Polak-Ribiere updates.     traingd — Gradient descent backpropagation.

traingda — Gradient descent with adaptive lr backpropagation.     traingdm — Gradient descent with momentum.

traingdx — Gradient descent w/momentum & adaptive lr backpropagation.     trainoss — One step secant backpropagation.

trainrp — RPROP backpropagation.

trainscg — Scaled conjugate gradient backpropagation.

 

performFcn specifies the cost/objective function that measures the performance of network during training, and there are many choices:

 

mae — Mean absolute error performance function. mse — Mean squared error performance function. sae — Sum absolute error performance function. sse — Sum squared error performance function. crossentropy — Cross-entropy performance. msesparse — Mean squared error performance function with L2 weight and sparsity regularizers.

 

It is very difficult to know which training function and performance function guarantee the best performance for a given problem. It depends on many factors, including the complexity of the problem, the number of samples in training set, the number of weights and biases in the network, and whether the network is being used for pattern recognition or function approximation (regression), etc. You are encouraged to try different training functions and compare their performance.

 

The following example shows how to design a pattern recognition network to classify iris flowers.

[x,t] = iris_dataset; net = patternnet(10);

net = train(net, x, t); view(net) y = net(x); perf = perform(net, t, y);

classes = vec2ind(y);

 

More details about patternnet() can be found by typing ‘help patternnet’ and ‘doc patternnet’ in MATLAB command line.

 

2. Different training functions have different parameters which are stored in

‘net.trainParam’. For example, the function parameters for ‘traincgf’ are

 

Show Training Window Feedback —              showWindow: true

Show Command Line Feedback —        showCommandLine: false

Command Line Frequency —                                            show: 25

Maximum Epochs —                                                   epochs: 1000

Maximum Training Time —                                              time: Inf

Performance Goal —                                                            goal: 0

Minimum Gradient —                                            min_grad: 1e-06

Maximum Validation Checks —                                   max_fail: 6

Sigma —                                                                        sigma: 5e-05

Lambda —                                                                   lambda: 5e-07

 

Similarly, the parameter of performance functions are stored in ‘net.performParam’.

For example, the function parameters for ‘crossentropy’ are

 

Regularization Ratio —                                        regularization: 0

Normalization —                                           normalization: ‘none’

 

The choosing of these parameters are task-dependent. You can keep the default values since they could guarantee a moderate performance; however, in order to achieve a better performance, you are encouraged modify these parameters based on your tasks.

 

3. You can train the network using MATLAB toolbox:

[net, tr] = train(net, X, T)

where the input arguments are net –Network X — Network inputs

T — Network targets (default = zeros) and returns net — Newly trained network tr — Training record (epoch and perf)

More details about train() can be found by typing ‘help train’ and ‘doc train’ in MATLAB command line.

 

4. After training, the weights in the hidden neurons are stored in the ‘net’ object. For example, for the same problem mentioned above, after training, type ‘net’ in the command line of MATLAB, you may obtain the following message:

net =

 

Neural Network

 

name: ‘Pattern Recognition Neural Network’           userdata: (your custom info)

dimensions:

 

numInputs: 1

numLayers: 2         numOutputs: 1     numInputDelays: 0     numLayerDelays: 0  numFeedbackDelays: 0  numWeightElements: 10

sampleTime: 1

connections:

biasConnect: [1; 1]       inputConnect: [1; 0]       layerConnect: [0 0; 1 0]      outputConnect: [0 1]

subobjects:

 

input: Equivalent to inputs{1}             output: Equivalent to outputs{2}

inputs: {1×1 cell array of 1 input}             layers: {2×1 cell array of 2 layers}            outputs: {1×2 cell array of 1 output}             biases: {2×1 cell array of 2 biases}       inputWeights: {2×1 cell array of 1 weight}       layerWeights: {2×2 cell array of 1 weight}

functions:

 

adaptFcn: ‘adaptwb’         adaptParam: (none)           derivFcn: ‘defaultderiv’          divideFcn: ‘dividerand’

divideParam: .trainRatio, .valRatio, .testRatio

divideMode: ‘sample’            initFcn: ‘initlay’         performFcn: ‘crossentropy’       performParam: .regularization, .normalization           plotFcns: {‘plotperform’, plottrainstate, ploterrhist,

plotconfusion, plotroc}         plotParams: {1×5 cell array of 5 params}           trainFcn: ‘trainscg’         trainParam: .showWindow, .showCommandLine, .show, .epochs,

.time, .goal, .min_grad, .max_fail, .sigma,

.lambda

weight and bias values:

 

IW: {2×1 cell} containing 1 input weight matrix                LW: {2×2 cell} containing 1 layer weight matrix                  b: {2×1 cell} containing 2 bias vectors

methods:

 

adapt: Learn while in continuous use            configure: Configure inputs & outputs             gensim: Generate Simulink model                init: Initialize weights & biases            perform: Calculate performance                sim: Evaluate network outputs given inputs               train: Train network with examples               view: View diagram

unconfigure: Unconfigure inputs & outputs

evaluate:       outputs = net(inputs)

 

You may use ‘net.LW’ and ‘net.b’ to check the detailed values of weights and biases. Besides, all the information of the trained network is stored in the object ‘net’. You may type ‘doc’ command to open the help manual and search for ‘net’ (network properties) to find more details.

 

5. The ‘train()’ function mentioned above provides batch learning mode only. In order to enable the sequential/incremental learning mode, please refer to http://www.mathworks.com/help/nnet/ug/neural–network–training–concepts.html

The most important step is to make sure that the inputs are presented as a cell array of sequential vectors.

A sample MATLAB code for sequential training is as follows:

 

function [ net, accu_train, accu_val ] = train_seq( n, images, labels, train_num, val_num, epochs )

% Construct a 1-n-1 MLP and conduct sequential training.

%

% Args:

%   n: int, number of neurons in the hidden layer of MLP. %   images: matrix of (image_dim, image_num), containing possibly  %           preprocessed image data as input.

%   labels: vector of (1, image_num), containing corresponding label of %           each image.

%   train_num: int, number of training images.

%   val_num: int, number of validation images.

%   epochs: int, number of training epochs.

%

% Returns:

%   net: object, containing trained network.

%   accu_train: vector of (epochs, 1), containing the accuracy on training %               set of each eopch during trainig.

%   accu_val: vector of (epochs, 1), containing the accuracy on validation %               set of each eopch during trainig.

 

 

% 1. Change the input to cell array form for sequential training     images_c = num2cell(images, 1);     labels_c = num2cell(labels, 1);

 

% 2. Construct and configure the MLP     net = patternnet(n);

 

net.divideFcn = ‘dividetrain’;  % input for training only     net.performParam.regularization = 0.25;  % regularization strength     net.trainFcn = ‘traingdx’;  % ‘trainrp’ ‘traingdx’     net.trainParam.epochs = epochs;

accu_train = zeros(epochs,1);  % record accuracy on training set of each epoch     accu_val = zeros(epochs,1);  % record accuracy on validation set of each epoch

% 3. Train the network in sequential mode     for i = 1 : epochs

display([‘Epoch: ‘, num2str(i)])

 

idx = randperm(train_num);  % shuffle the input

 

net = adapt(net, images_c(:,idx), labels_c(:,idx));

 

pred_train = round(net(images(:,1:train_num)));  % predictions on training set         accu_train(i) = 1 – mean(abs(pred_train-labels(1:train_num)));

 

pred_val = round(net(images(:,train_num+1:end)));  % predictions on validation set

accu_val(i) = 1 – mean(abs(pred_val-labels(train_num+1:end)));

end

end

 

You can copy this .m file into your folder and modify it according to your task.