Q1. Rosenbrocks Valley Problem Consider the Rosenbrocks 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(k1) w(k)g(k)

with learning rate = 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 = 0.2, is used?

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

w(n)H¹(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:

y1.2sin(x)cos(2.4x) , 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 Rosenblatts 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 dont 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: {11 cell array of 1 input} layers: {21 cell array of 2 layers} outputs: {12 cell array of 1 output} biases: {21 cell array of 2 biases} inputWeights: {21 cell array of 1 weight} layerWeights: {22 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: {15 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: {21 cell} containing 1 input weight matrix LW: {22 cell} containing 1 layer weight matrix b: {21 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/neuralnetworktrainingconcepts.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.