Welcome friends,
It's time for neural networks! This time, the testing is done in Python, but the code you need to write is mostly in C. Download this repo, enter it and simply type make in the terminal. After editing the code, type make clean and then make. The repo is tested to work with Linux and MacOS; feel free to contact the course staff for compilation issues.
We've added a bunch of new data structures and types to your image library as we begin to implement some machine learning algorithms. Check them out in src/image.h.
An important part of machine learning, be it linear classifiers or neural networks, is the activation function you use. 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.
Now we can fill in how to forward-propagate information through a layer. First check out our layer struct:
typedef struct {
matrix in; // Saved input to a layer
matrix w; // Current weights for a layer
matrix dw; // Current weight updates
matrix v; // Past weight updates (for use with momentum)
matrix out; // Saved output from the layer
ACTIVATION activation; // Activation the layer uses
} layer;
During forward propagation we will do a few things. We'll multiply the input matrix by our weight matrix w. We'll apply the activation function activation. We'll also save the input and output matrices in the layer so that we can use them during backpropagation. But this is already done for you.
Fill in the TODO in matrix forward_layer(layer *l, matrix in). You will need the matrix multiplication function matrix matrix_mult_matrix(matrix a, matrix b) provided by our matrix library.
Using matrix operations we can batch-process data. Each row in the input in is a separate data point and each row in the returned matrix is the result of passing that data point through the layer.
We have to backpropagate error through our layer. We will be given dL/dy, the derivative of the loss with respect to the output of the layer, y. The output y is given by y = f(xw) where x is the input, w is our weights for that layer, and f is the activation function. What we want to calculate is dL/dw so we can update our weights and dL/dx which is the backpropagated loss to the previous layer. You'll be filling in matrix backward_layer(layer *l, matrix delta)
First we need to calculate dL/d(xw) using the gradient of our activation function. Recall:
dL/d(xw) = dL/dy * dy/d(xw)
= dL/dy * df(xw)/d(xw)
= dL/dy * f'(xw)
Use your void gradient_matrix(matrix m, ACTIVATION a, matrix d) function to change delta from dL/dy to dL/d(xw)
Next we want to calculate the derivative with respect to our weights, dL/dw. Recall:
dL/dw = dL/d(xw) * d(xw)/dw
= dL/d(xw) * x
but remember from the slides, to make the matrix dimensions work out right we acutally do the matrix operiation of xt * dL/d(xw) where xt is the transpose of the input matrix x.
In our layer we saved the input as l->in. Calculate xt using that and the matrix transpose function in our library, matrix transpose_matrix(matrix m). Then calculate dL/dw and save it into l->dw (free the old one first to not leak memory!). We'll use this later when updating our weights.
Next we want to calculate the derivative with respect to the input as well, dL/dx. Recall:
dL/dx = dL/d(xw) * d(xw)/dx
= dL/d(xw) * w
again, we have to make the matrix dimensions line up so it actually ends up being dL/d(xw) * wt where wt is the transpose of our weights, w. Calculate wt and then calculate dL/dx. This is the matrix we will return.
After we've performed a round of forward and backward propagation we want to update our weights. We'll fill in void update_layer(layer *l, double rate, double momentum, double decay) to do just that.
With learning rate (η), momentum (m) and decay (λ) our weight update rule is:
Δw_t = dL/dw_t - λw_t + mΔw_{t-1}
w_{t+1} = w_t + ηΔw_t
We'll be doing gradient ascent (the partial derivative component is positive) instead of descent because we'll flip the sign of our partial derivative when we calculate it at the output. We saved dL/dw_t as l->dw and by convention we'll use l->v to store the previous weight change Δw_{t-1}.
Calculate the current weight change as a weighted sum of l->dw, l->w, and l->v. The function matrix axpy_matrix(double a, matrix x, matrix y) will be useful here, it calculates the result of the operation ax+y where a is a scalar and x and y are matrices. Save the current weight change in l->v for next round (remember to free the current l->v first).
Finally, apply the weight change to your weights by adding a scaled amount of the change based on the learning rate.
Check out the remainder of the functions which string layers together to make a model, run the model forward and backward, update it, train it, etc. Notable highlights:
Makes a new layer that takes input number of inputs and produces output number of outputs. Note that we randomly initialize the weight vector, in this case with a uniform random distribution between [-sqrt(2/input), sqrt(2/input)]. The weights could be initialize with other random distributions but this one works pretty well. This is one of those secret tricks you just have to know from experience!
Will be useful to test out our model once we are done training it.
Calculates the cross-entropy loss for the ground-truth labels y and our predictions p. For classification of an image into one of say N object classes, y is an N-length vector with all zero values except for the object class present in the image, for which it is 1. Cross-entropy loss is negative log-likelihood (log-likelihood is for logistic regression) for multi-class classification. Since we added the negative sign it becomes a loss function that we want to minimize. Cross-entropy loss is given by:
L = Σ(-y_i log(p_i))
for a single data point, summing across the different classes. Cross-entropy loss is nice to use with our softmax activation because the partial derivatives are nice. If the output of our final layer uses a softmax: y = σ(wx) then:
dL/d(wx) = (y - truth)
During training, we'll just calculate dL/dy as (y - truth) and set the gradient of the softmax to be 1 everywhere to have the same effect, as discussed earlier. This avoids numerical instability with calculating the "true" dL/dy and dσ(x)/dx for cross-entropy loss and the softmax function independently and multiplying them together.
Our training code to implement SGD. First we get a random subset of the data using data random_batch(data d, int n). Then we run our model forward and calculate the error we make dL/dy. Finally we backpropagate the error through the model and update the weights at every layer.
We'll be testing out your implementation on the MNIST dataset!
When there are questions during this section please answer them in the tryml.py file, you will submit this file in Canvas as well as classifier.c. Answers can be short, 1-2 sentences.
To run your model you'll need the dataset. The training images can be found here and the test images are here, they are preprocessed for you into PNG format. To get the data you can run:
wget https://pjreddie.com/media/files/mnist_train.tar.gz
wget https://pjreddie.com/media/files/mnist_test.tar.gz
tar xzf mnist_train.tar.gz
tar xzf mnist_test.tar.gz
We'll also need a list of the images in our training and test set. To do this you can run:
find train -name \*.png > mnist.train
find test -name \*.png > mnist.test
Or something similar.
Check out tryml.py to see how we're actually going to run the machine learning code you wrote. There are a couple example models in here for softmax regression and neural networks. Run the file with: python tryml.py to train a softmax regression model on MNIST. You will see a bunch of numbers scrolling by, this is the loss calculated by the model for the current batch. Hopefully over time this loss goes down as the model improves.
After training our python script will calculate the accuracy the model gets on both the training dataset and a separate testing dataset.
Why might we be interested in both training accuracy and testing accuracy? What do these two numbers tell us about our current model?
Try varying the model parameter for learning rate to different powers of 10 (i.e. 10^1, 10^0, 10^-1, 10^-2, 10^-3) and training the model. What patterns do you see and how does the choice of learning rate affect both the loss during training and the final model accuracy?
Try varying the parameter for weight decay to different powers of 10: (10^0, 10^-1, 10^-2, 10^-3, 10^-4, 10^-5). How does weight decay affect the final model training and test accuracy?
Now change the training code to use the neural network model instead of the softmax model.
Currently the model uses a logistic activation for the first layer. Try using all the other activation functions we programmed. How well do they perform? What's best?
Using the same activation, find the best (power of 10) learning rate for your model. What is the training accuracy and testing accuracy?
Right now the regularization parameter decay is set to 0. Try adding some decay to your model. What happens, does it help? Why or why not may this be?
Modify your model so it has 3 layers instead of 2. The layers should be inputs -> 64, 64 -> 32, and 32 -> outputs. Also modify your model to train for 3000 iterations instead of 1000. Look at the training and testing accuracy for different values of decay (powers of 10, 10^-4 -> 10^0). Which is best? Why?
Modify your model so it has 4 layers instead of 2. The layers should be inputs -> 128, 128 -> 64, 64 -> 32, and 32 -> outputs. Do the same analysis as in 2.3.4.
Use the 2 layer model with the best activation for layer 1 but linear activation for layer 2. Now implement the functions l1_loss and l2_loss and change the necessary code in classifier.c to use these loss functions. Observe the output values and accuracy of the model and write down your observations for both the loss functions compared to cross-entropy loss. P.S. L2 and L1 losses are generally used for regression, but this is a classification problem.
#UPDATE#: Note that you need to calculate the gradient of the new loss with respect to model output and change dL in train_model() accordingly. While derivative of L2 loss is straightforward, the gradient of L1 loss is constant and will affect the training (either the accuracy will be low or the model will converge to a large loss within a few iterations.) To avoid this, compute the Huber loss instead of L1 and write Huber loss equation in l1_loss(). In simple words, Huber loss is equal to L1 loss when y - p is > 1 and equal to L2 loss when <= 1. You may have to be careful about the sign (i.e. y-p or p-y).
The CIFAR-10 dataset is meant to be similar to MNIST but much more challenging. Using the same model we just created, let's try training on CIFAR.
We have to do a similar process as last time, getting the data and creating files to hold the paths. Run:
wget http://pjreddie.com/media/files/cifar.tgz
tar xzf cifar.tgz
find cifar/train -name \*.png > cifar.train
find cifar/test -name \*.png > cifar.test
Notice that the file of possible labels can be found in cifar/labels.txt
Modify tryml.py to use CIFAR. This should mostly involve swapping out references to mnist with cifar during dataset loading. Then try training the network. You may have to fiddle with the learning rate to get it to train well.
How well does your network perform on the CIFAR dataset?
Turn in your classifier.c and tryml.py on canvas under Assignment 4.