Homework 6 Solution

You may complete this homework assignment either individually or in teams up to 2 people.




3-layer neural network [70 points]: In this problem you will implement and train a 3-layer neural network to classify images of hand-written digits from the MNIST dataset. Similarly to Homework 3, the input to the network will be a 28 28-pixel image (converted into a 784-dimensional vector); the output will be a vector of 10 probabilities (one for each digit). Speci cally, the network you create should implement a function g : R784 ! R10, where:
z(1)

h(1)

z(2)




y^ = g(x)

W(1)x + b(1)



relu(z(1)))
W(2)h(1) + b(2)



softmax(z(2))






Computing each of the intermediate outputs z(1), h(1), z(2), and y^ is known as forwards propagation since it follows the direction of the edges in the directed graph shown below:













x
z1
h1
z2
y


W(1)


W(2)


...
...
...
...
...


b(1)


b(2)










Layer 1


Layer 2


Layer 3






Loss function: For the MNIST dataset you should use the cross-entropy loss function:




fCE(W(1); b(1); W(2); b(2)) =
1
n 10
y(i) log y^(i)




X X
k
k






n

i=1 k=1




where n is the number of examples.




Gradient descent: To train the neural network, you should use stochastic gradient descent (SGD). The hard part is computing the individual gradient terms. This can be done e ciently using backwards propagation (\backprop"), which is called as such because it proceeds opposite the direction of the edges in the network graph above. You should start by initializing the weights randomly, and initializing the bias terms to small positive numbers. The reason is that { due to the relu activation function which has a gradient of 0 whenever its argument is less than 0 { we want to give enough bias to \encourage" the argument of relu to be positive. This is already performed for you in the starter code. Then, update the weights according to SGD using the gradient expressions shown below. (Note that these expressions are obtained by deriving and multiplying the Jacobian matrices as described in class, and









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then simplifying the result analytically.)




rW(2) fCE = (y^ y)h(1)




rb(2) fCE = (y^ y)

rW(1) fCE = gx




rb(1) fCE = g




where column-vector g is de ned so that




g = (y^ y)W(2) relu0(z(1))




In the equation above, relu0 is the derivative of relu. Also, make sure that you follow the transposes exactly!




Hyperparameter tuning: In this problem, there are several di erent hyperparameters that will impact the network’s performance:




Number of units in the hidden layer (suggestions: f30; 40; 50g)




Learning rate (suggestions: f0:001; 0:005; 0:01; 0:05; 0:1; 0:5g)




Minibatch size (suggestions: 16; 32; 64; 128; 256) Number of epochs




Regularization strength




In order not to \cheat" { and thus overestimate the performance of the network { it is crucial to optimize the hyperparameters only on the validation set; do not use the test set. (The training set would be ok but typically leads to worse performance.)




Your task: Use stochastic gradient descent to minimize the cross-entropy with respect to W(1); W(2); b(1), and b(2). Speci cally:




Implement stochastic gradient descent for the network shown above. [40 points]



Verify that your implemented cost and gradient functions are correct (the discrepancy should be less than 0.01) using a numerical derivative approximation { see the call to check grad in the starter code. [10 points]



Optimize the hyperparameters by training on the training set and selecting the parameter settings that optimize performance on the validation set. You should systematically (i.e., in code) try at least 10 (in total, not for each hyperparameter) di erent hyperparameter settings; accordingly, make sure there is a method called findBestHyperparameters (and please name it as such to help us during grading) [15 points]. Include a screenshot showing the progress and nal output (selected hyperparameter values) of your hyperparameter optimization.



After you have optimized your hyperparameters, then run your trained network on the test set and report (1) the cross-entropy and (2) the accuracy (percent correctly classi ed images). Include a screenshot showing both these values during the last 20 epochs of SGD. The (unreg-ularized) cross-entropy cost on the test set should be less than 0:16, and the accuracy (percentage correctly classi ed test images) should be at least 95%. [5 points]






Datasets: You should use the following datasets, which are a superset of what I gave you for previous assignments:




https://s3.amazonaws.com/jrwprojects/mnist_train_images.npy







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https://s3.amazonaws.com/jrwprojects/mnist_train_labels.npy https://s3.amazonaws.com/jrwprojects/mnist_test_images.npy https://s3.amazonaws.com/jrwprojects/mnist_test_labels.npy




https://s3.amazonaws.com/jrwprojects/mnist_validation_images.npy https://s3.amazonaws.com/jrwprojects/mnist_validation_labels.npy




Mountains and valleys [25 points]: Visualize the SGD trajectory of your network when trained on



MNIST:




Plot in 3-D the cross-entropy loss fCE as a function of the neural network parameters (weights and bias terms). Rather than showing the loss as a function of any particular parameters (e.g., the third component of b(2)), use the x- and y-axes to span the two directions along which your parameters vary the most during SGD training { i.e., you will need to use PCA. (In this assignment, you are free to use sklearn.decomposition.PCA.) The z-axis should represent the (unregularized) fCE on training data. For each point (x; y) on a grid, compute fCE and then interpolate between the points. (The interpolation and rendering are handled for you completely by the plot surface function; see the starter code.)



Superimpose a scatter plot of the di erent points (in the neural network’s parameter space) that were reached during SGD (just sample a few times per epoch). To accelerate the rendering, train the network and plot the surface using just a small subset of the training data (e.g., 2500 examples) to estimate fCe. See the starter code, in particular the plotPath function, for an example of 3-D plotting. Submit your graph in the PDF le and the code in the Python le. Here is what I get when I superimpose two scatter plots corresponding to two di erent initializations and SGD runs (note that you only need to include a scatter plot for one run); as you can see, the two SGD trajectories descended into distinct local minima (though with similar cost):





































































In addition to your Python code (homework6 WPIUSERNAME1.py







or homework6 WPIUSERNAME1 WPIUSERNAME2.py for teams), create a PDF le (homework6 WPIUSERNAME1.pdf or homework6 WPIUSERNAME1 WPIUSERNAME2.pdf for teams) containing the screenshots described above. Please submit both the PDF and Python les in a single Zip le.



















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