Welcome friends,
For the first assignment we'll be diving right in to neural networks. What are they? How do they work? Do they really require blood sacrifices on the third Saturday of odd-numbered months? We'll find out the answers to these questions and more!
During this class you'll be building out your own neural network framework. I'll take care of some of the more boring parts like loading data, stringing things together, etc. so you can focus on the important parts.
Those of you who took vision with me will recognize a lot of the image codebase as stuff that you did! Early on we'll be focussing on applications of deep learning to problems in computer vision so there's a lot of code in image.c for loading images and performing some basic operations. No need to go into the details, just glance through image.h for a quick summary of the kinds of operations you can perform on images. We won't need any of that for this homework though!
More pertinent to this homework is the matrix library described in matrix.h and fleshed out in matrix.c. As we've learned in class, the heavy lifting in deep learning is done through matrix operations. The important ones we'll be using today are matrix multiplication and an elementwise scaled addition called axpy (for ax + y). But to warm up we'll do some easy stuff. Let's get crackin!
Remember, the codebase is in C. There is a Makefile but every time you make a change to the code you have to run make for your executable to be updated.
If you check out matrix.h you see that our matrices are pretty simple, just a size in rows and cols and a field for the data. Now check out matrix.c and read through the functions for making new (zeroed out) matrices and randomly filled matrices.
Fill in the code for making a copy of a matrix. We've already created the new matrix for you, you just have to fill it in!
Fill in the code to transpose a matrix. First make sure the matrix is the right size. Then fill it in. This might be handy: https://en.wikipedia.org/wiki/Transpose
Fill in the code to perform the weighted sum. The operation you are performing is y = ax + y, element-wise over the matrices. This means the matrices should be the same size!
Implement matrix multiplication. No need to do anything crazy, 3 for loops oughta do it. However, good cache utilization will make your operation much faster! Once you've written something you can run: ./uwnet test. This doesn't actually check if your code is correct but it will time a bunch of matrix multiplications for you. Try optimizing your loop ordering to get the fastest time. You don't need to change anything except the loop order, there's just one ordering that's MUCH faster. On this test for my code it takes about 4 seconds but your processing speed may vary!
An important part of machine learning, be it linear classifiers or neural networks, is the activation function you use. We'll define our activation functions in activations.c. This is also a good time to check out uwnet.h, which gives you an overview of the structures will be using for building our models and, important for this, what activation functions we have available to us!
Fill in void activate_matrix(matrix m, ACTIVATION a) to modify m to be f(m) applied elementwise where the function f is given by what the activation a is.
Remember that for our softmax activation we will take e^x for every element x, but then we have to normalize each element by the sum as well. Each row in the matrix is a separate data point so we want to normalize over each data point separately.
As we are calculating the partial derivatives for the weights of our model we have to remember to factor in the gradient from our activation function at every layer. We will have the derivative of the loss with respect to the output of a layer (after activation) and we need the derivative of the loss with respect to the input (before activation). To do that we take each element in our delta (the partial derivative) and multiply it by the gradient of our activation function.
Normally, to calculate f'(x) we would need to remember what the input to the function, x, was. However, all of our activation functions have the nice property that we can calculate the gradient f'(x) only with the output f(x). This means we have to remember less stuff when running our model.
The gradient of a linear activation is just 1 everywhere. The gradient of our softmax will also be 1 everywhere because we will only use the softmax as our output with a cross-entropy loss function, for an explaination of why see here or here.
The gradient of our logistic function is discussed here:
f'(x) = f(x) * (1 - f(x))
I'll let you figure out on your own what f'(x) given f(x) is for RELU and Leaky RELU.
As discussed in the backpropagation slides, we just multiply our partial derivative by f'(x) to backpropagate through an activation function. Fill in void gradient_matrix(matrix m, ACTIVATION a, matrix d) to multiply the elements of d by the correct gradient information where m is the output of a layer that uses a as it's activation function.
For this homework we'll be implementing a fully connected neural network layer as our first layer type! This should be fairly straightforward. First let's check out our layer definition in src/uwnet.h:
typedef struct layer {
matrix *in; // Holds the most recent input matrix
matrix *out; // The most recent output matrix
matrix *delta; // The error made by this layer
// Weights
matrix w;
matrix dw;
// Biases
matrix b;
matrix db;
ACTIVATION activation;
LAYER_TYPE type;
matrix (*forward) (struct layer, struct matrix);
void (*backward) (struct layer, struct matrix);
void (*update) (struct layer, float rate, float momentum, float decay);
} layer;
We have pointers to matrices that will hold the input and output to our layer. We also have a matrix that will store the error made by the layer on the most recent run, delta.
Our layer outputs a matrix called out as f(in*w + b) where: in is the input, w is the weights, b is the bias, and f is the activation function.
To compute the output of the model we first will want to do a matrix multiplication involving the input and the weights for that layer. Remember, the weights are stored under l.w.
Then we'll want to add in our biases for that layer, stored under l.b. The function forward_bias may come in handy here!
Finally, we'll want to activate the output with the activation function for that layer.
This is the trickiest one probably.
To update our model we want to calculate dL/dw and dL/db.
We also want to calculate the deltas for the previous layer (aka the input to this layer) which is dL/din.
What we have to start with is dL/dout, stored in delta, which is equivalent to dL/df(in*w+b).
We need to calculate dL/d(in*w+b) which is dL/df(in*w+b) * df(in*w+b)/d(in*w+b). This is exactly what our gradient_matrix function is for!
First we need dL/db to calculate our bias updates. Actually, you can show pretty easily that dL/db = dL/d(in*w+b) so we're almost there. Except we have bias deltas for every example in our batch. So we just need to sum up the deltas for each specific bias term over all the examples in our batch to get our final dL/db, this can be done and added straight into our currently stored db term using the backward_bias function.
Then we need to get dL/d(in*w) which by the same logic as above we already have as dL/d(in*w+b). So taking our dL/d(in*w) we can calculate dL/dw and dL/din as described in lecture. Good luck!
We want to update our weights using SGD with momentum and weight decay.
Our normal SGD update with learning rate η is:
w = w + η*dL/dw
With weight decay λ we get:
w = w + η*(dL/dw - λw)
With momentum we first calculate the amount we'll change by as a scalar of our previous change to the weights plus our gradient and weight decay:
Δw = m*Δw_{prev} + dL/dw - λw
Then we apply that change:
w = w + η*Δw
In our case, we'll always try to keep track of Δw in our layer's l.dw. When we start the update method, our l.dw will store:
l.dw = m*Δw_{prev} + dL/dw
m*Δw_{prev} will come from the last updates we applied, and we added on dw during the backward computation. Our first step is to subtract off the weight decay (using axpy) so we have:
l.dw = m*Δw_{prev} + dL/dw - λw
This is the same as the weight updates we want to make, Δw. Next we need to add a scaled version of this into our current weights (also using axpy).
Before we applied the updates, l.dw = Δw, but after we apply them, l.dw = Δw_{prev}. Thus our final step is to scale this vector by our momentum term so that:
l.dw = m*Δw_{prev}
Then during our next backward phase we are ready to add in some new dL/dws!
First check out net.c to see how we run the network forward, backward, and updates. It's pretty simple.
Now, little did you know, but our C library has a Python API all ready to go! The API is defined in uwnet.py, but you don't have to worry about it too much. Mainly, check out trymnist.py for an example of how to train on the MNIST data set. But wait, you don't have any data yet!
Background: MNIST is a classic computer vision dataset of hand-drawn digits - http://yann.lecun.com/exdb/mnist/
I already have it all ready to go for you, just run:
wget https://pjreddie.com/media/files/mnist.tar.gz
tar xvzf mnist.tar.gz
Make sure your executable and libraries are up to date by running:
make
Then try training a model on MNIST!
python trymnist.py
Every batch the model will print out the loss and at the end of training your model will run on the training and testing data and give you final accuracy results. Try playing around with different model structures and hyperparameter values. Can you get >97% accuracy?
The CIFAR-10 dataset is similar in size to MNIST but much more challenging - https://www.cs.toronto.edu/~kriz/cifar.html. Instead of classifying digits, you will train a classifier to recognize these objects:
First get the dataset:
wget https://pjreddie.com/media/files/cifar.tar.gz
tar xvzf cifar.tar.gz
Then try training on CIFAR:
python trycifar.py
How do your results compare to MNIST? If you changed your model for MNIST, do similar changes affect your CIFAR results in the same way?
Navigate to: https://colab.research.google.com/
and upload the iPython notebook provided: mnist_pytorch.ipynb
Complete the notebook to train a PyTorch model on the MNIST dataset.
First run the collate.sh script by running:
bash collate.sh
This will create the file submit.tar.gz in your directory with all the code you need to submit. The command will check to see that your files have changed relative to the version stored in the git repository. If it hasn't changed, figure out why, maybe you need to download your ipynb from google?
Submit submit.tar.gz in the file upload field for Homework 0 on Canvas.