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# Programming Assignment 2: Neural Networks

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Programming Assignment 2
CSE 253: Neural Networks

Instructions

1. Please submit your assignment on Vocareum. There are two components to this assignment: written homework
(Problems 1 & 2a-c), and a programming part. You will be writing a report in a conference paper format
for this assignment, reporting your findings. While we won’t enforce this, We prefer the report to be written
using LATEX or Word in NIPS format (NIPS is the top machine learning conference, and it is now dominated
by deep nets – it will be good practice for you to write in that format!). You are free to choose an alternate
format, but NIPS format is strongly recommended. The templates, both in Word and LATEX are available
from the 2015 NIPS format site.
2. You may use a language of your choice (Python and MATLAB are recommended). You also need to submit
all of the source codes files and a readme.txt file that includes detailed instructions on how to run your code.
You should write clean code with consistent format, as well as explanatory comments, as this code may be
reused in the future.
3. Using the MATLAB neural network toolbox or any off-the-shelf code is strictly prohibited.
Part I
Homework problems to be solved individually, and
turned in individually
Multi-layer Neural Networks
In this problem, we will continue classifying handwritten digits from Yann LeCun’s MNIST Database. In Assignment
1, we classified the digits using a single-layer neural network with different output activation functions. (Logistic
and Softmax regression). In this assignment, we are going to classify the MNIST dataset using multi-layer neural
networks with softmax outputs.
1
Problem
1. In class we discussed two different error functions: sum-of-squared error (SSE) and cross-entropy error. We
learned that SSE is appropriate for linear regression problems where we try to fit data generated from:
t = h(x) + ✏ (1)
Here x is a K-dimensional vector, h(x) is a deterministic function of x, where x includes the bias x0, and ✏ is
random noise that has a Gaussian probability distribution with zero mean and variance 2, i.e. ✏ ⇠ N (0, 2).
Suppose we want to model this data with a linear function approximation with parameter vector w:
y = X
K
i=0
wixi (2)
Prove that finding the optimal parameter w for the above linear regression problem on the dataset D =
{(x1, t1), …,(xN , tN )} is equal to finding the w⇤ that minimizes the SSE:
w⇤ = argminw
X
N
n=1
(t
n yn)
2 (3)
Hint: This problem is solved in Bishop, Chapter 6.
2. For multiclass classification on the MNIST dataset, we previously used softmax regression with cross-entropy
error as the objective function, and learned the weights of a single-layer network to classify the digits. In
this assignment, we will add a hidden layer between the input and output, that consists of J units with the
sigmoid activation function. So this network has three layers: an input layer, a hidden layer and a softmax
output layer.
Notation: We use index k to represent a node in output layer and index j to represent a node in hidden layer
and index i to represent a node in the input layer. Additionally, the weight from node i in the input layer
to node j in the hidden layer is wij . Similarly, the weight from node j in the hidden layer to node k in the
output layer is wjk.
(a) Derivation Derive the expression for for both the units of output layer (k) and the hidden layer (j ).
For this problem, use the definition of as i = @E
@ai
, where ai is the weighted sum of the inputs to unit i.
(b) Update rule. Derive the update rule for wij and wjk using learning rate ⌘, starting with the gradient
descent rule. The rule for the output units should be the same as you found in Programming Assignment
1. For the hidden units:
wij = wij ⌘
@E
@wij
(4)
@E
@wij
= @E
@aj
@aj
@wij
(5)
The derivative should take into account all of the outputs, so:
j = @En
@an
j
=
X
k
@En
@yk
@yk
@aj
(6)
(c) Vectorize computation. The computation is much faster when you update all wij s and wjks at the
same time, using matrix multiplications rather than for loops. Please show the update rule for the
weight matrix from the hidden layer to output layer and the matrix from input layer to hidden layer,
using matrix/vector notation.
2
Part II
Team Programming Assignment
3. Classification. Classification on MNIST datatbase. Refer to your derivations from Problem 2.
functions for matlab can be found in http://ufldl.stanford.edu/wiki/resources/mnistHelper.zip. Python
helper functions can be found in https://gist.github.com/akesling/5358964.
(b) The pixel values in the digit images are in the range [0..255]. Divide by 255 so that they are in the range
[0..1], and then subtract the mean over all of the pixels in each image. I.e., if the mean pixel value in an
image of a “4” is 0.3, you would subtract 0.3 from each pixel value.
(c) While you should use the softmax activation function at the output level, for the hidden layer, you can
use the logistic, i.e., yj = 1/(1 + exp(aj )). This has the nice feature that dy/da = y(1 y).
(d) Check your code for computing the gradient using a small subset of data – so small, you can use one inputoutput pattern! You can compute the slope with respect to one weight using the numerical approximation:
d
dw En(w) ⇡ En(w + ✏) En(w ✏)
2✏
where ✏ is a small constant, e.g., 102. Compare the gradient computed using numerical approximation
with the one computed as in backpropagation. The difference of the gradients should be within big-O of
✏2, so if you used 102, your gradients should agree within 104. (See section 4.8.4 in Bishop for more
details). Note that w here is one weight in the network, so this must be repeated for every weight and
(e) Using the update rule you obtained from 2(c), perform gradient descent to learn a classifier that maps
each input data to one of the labels t 2 {0, …, 9} (but using a one-hot encoding). Use a hold-out set to
decide when to stop training. (Hint: You may choose to split the training set of 60000 images to two
subsets: one training set with 50000 training images and one validation set with 10000 images.) Stop
training when the error on the validation set goes up, or better yet, save the weights as you go, keeping
the ones from when the validation set error was at a minimum. Report your training procedure and
plot your training and testing accuracy vs. number of training iterations of gradient descent. Again, by
accuracy, we mean the percent correct on the training and testing patterns.
4. Adding the “Tricks of the Trade.” Read the first paper on the resources page under readings, “lecun98efficient.pdf”, sections 4.1-4.7. Implement the following ideas from that paper (you can do these all at
once, or incrementally – no need to try all possible combinations, however).
(a) Shuffle the examples, and then use stochastic gradient descent. For the purposes of this class, use
“minibatches”, in which you compute the total gradient over, say, 128 patterns, take the average weight
change (i.e., divide by 128), and then change the weights by that amount.
(b) Stick with the normalization you have already done in the previous part, which is a little different than
what he describes in Section 4.3 – that can be expensive to compute over a whole data set.
(c) For the output layer, use the usual softmax activation function. However, for the hidden layer(s), use
the sigmoid in Section 4.4. Note that you will need to derive the slope of that sigmoid to use when
computing deltas.
(d) Initialize the input weights to each unit using a distribution with 0 mean and standard deviation
1/sqrt(fan-in), where the fan-in is the number of inputs to the unit.
(e) Use momentum.
(f) Comment on the change in performance, which has hopefully improved, at least in terms of learning
speed.
3
5. Experiment with Network Topology. Start with the network of 4. Now, we will consider how the
topology of the neural network changes the performance.
(a) Try halving and doubling the number of hidden units. What do you observe if the number of hidden
units is too small? What if the number is too large?
(b) Change the number of hidden layers. Use two hidden layers instead of one. Create a new architecture
that uses two hidden layers of equal size and has approximately the same number of parameters, as the
previous network with one hidden layer. By that, we mean it should have roughly the same total number
of weights and biases. Report training and testing accuracy vs. number of training iterations of gradient
descent.
4