Initialization¶
The specific problem that arises, when trying to implement the feedforward neural network, is that we are trying to transform from 784 nodes all the way down to 10 nodes. When instantiating the DeepNeuralNetwork class, we pass in an array of sizes that defines the number of activations for each layer.
dnn = DeepNeuralNetwork(sizes=[784, 64, 10])
This initializes the DeepNeuralNetwork class by the init function.
def __init__(self, sizes, activation='sigmoid'):
self.sizes = sizes
# Choose activation function
if activation == 'relu':
self.activation = self.relu
elif activation == 'sigmoid':
self.activation = self.sigmoid
# Save all weights
self.params = self.initialize()
# Save all intermediate values, i.e. activations
self.cache = {}
To smarten up our initialization, we shrink the variance of the weights in each layer. Following this nice video by Andrew Ng, we’ll set the variance for each layer to 1/n, where n is the number of inputs feeding into that layer. We use np.random.randn() function to get our initial weights, which draws from the standard normal distribution. So to adjust the variance to 1/n, we just divide by √n.
The initialization of weights in the neural network is kind of hard to think about. To really understand how and why the following approach works, you need a grasp of linear algebra, specifically dimensionality when using the dot product operation, which is beyond the scope of this class.
def initialize(self):
# number of nodes in each layer
input_layer=self.sizes[0]
hidden_layer=self.sizes[1]
output_layer=self.sizes[2]
params = {
"W1": np.random.randn(hidden_layer, input_layer) * np.sqrt(1./input_layer),
"b1": np.zeros((hidden_layer, 1)),
"W2": np.random.randn(output_layer, hidden_layer) * np.sqrt(1./hidden_layer),
"b2": np.zeros((output_layer, 1))
}
return params
Feedforward¶
The forward pass consists of the dot operation in NumPy, which turns out to be just matrix multiplication. As described in the introduction to neural networks article, we have to multiply the weights by the activations of the previous layer. Then we have to apply the activation function to the outcome.
To get through each layer, we sequentially apply the dot operation, followed by the sigmoid/relu activation function. In the last layer we use the softmax activation function, since we wish to have probabilities of each class, so that we can measure how well our current forward pass performs.
def feed_forward(self, x):
self.cache["X"] = x
self.cache["Z1"] = np.matmul(self.params["W1"], self.cache["X"].T) +\
self.params["b1"]
self.cache["A1"] = self.activation(self.cache["Z1"])
self.cache["Z2"] = np.matmul(self.params["W2"], self.cache["A1"]) +\
self.params["b2"]
self.cache["A2"] = self.softmax(self.cache["Z2"])
return self.cache["A2"]
The following are the activation functions used for this article. As you can see, we provide a derivative version of the relu and sigmoid, since we will need that later on when backpropagating through the neural network.
def relu(self, x, derivative=False):
if derivative:
x = np.where(x < 0, 0, x)
x = np.where(x >= 0, 1, x)
return x
return np.maximum(0, x)
def sigmoid(self, x, derivative=False):
if derivative:
return (np.exp(-x))/((np.exp(-x)+1)**2)
return 1/(1 + np.exp(-x))
def softmax(self, x):
# Numerically stable with large exponentials
exps = np.exp(x - x.max())
return exps / np.sum(exps, axis=0)
Note: A numerical stable version of the softmax function was chosen, you can read more from Stanford CS231n course.
Backpropagation¶
The backward pass is hard to get right, because there are so many sizes and operations that have to align, for all the operations to be successful. Here is the full function for the backward pass; we will go through each weight update below.
For people who are interested in the magic of backpropagation, please refer to this nice article. For people who want to deep dive into methmetics and understand thoroughly, please refer to NTHU CS565600 Deep Learning. Here is slide and video for backpropagation.
def back_propagate(self, y, output):
current_batch_size = y.shape[0]
dZ2 = output - y.T
dW2 = (1./current_batch_size) * np.matmul(dZ2, self.cache["A1"].T)
db2 = (1./current_batch_size) * np.sum(dZ2, axis=1, keepdims=True)
dA1 = np.matmul(self.params["W2"].T, dZ2)
dZ1 = dA1 * self.activation(self.cache["Z1"], derivative=True)
dW1 = (1./current_batch_size) * np.matmul(dZ1, self.cache["X"])
db1 = (1./current_batch_size) * np.sum(dZ1, axis=1, keepdims=True)
self.grads = {"W1": dW1, "b1": db1, "W2": dW2, "b2": db2}
return self.grads
Training (Stochastic Gradient Descent)¶
We have defined a forward and backward pass, but how can we start using them? We have to make a training loop and choose to use Stochastic Gradient Descent (SGD) as the optimizer to update the parameters of the neural network.
There are two main loops in the training function. One loop for the number of epochs, which is the number of times we run through the whole dataset, and a second loop for running through each batch one by one.
For each batch, we do a forward pass by calling self.feedforward() with x, which is one batch in an array with the length 784, as explained earlier. The output of the forward pass is used along with y, which are the one-hot encoded labels (the ground truth), in the backward pass. self.back_propagate() returns the gradients of weights.
def train(self, x_train, y_train, x_test, y_test):
for i in range(self.epochs):
# Shuffle
permutation = np.random.permutation(x_train.shape[0])
x_train_shuffled = x_train[permutation]
y_train_shuffled = y_train[permutation]
for j in range(num_batches):
# Batch
begin = j * self.batch_size
end = min(begin + self.batch_size, x_train.shape[0]-1)
x = x_train_shuffled[begin:end]
y = y_train_shuffled[begin:end]
# Forward
output = self.feed_forward(x)
# Backprop
grad = self.back_propagate(y, output)
# Optimize
self.optimize(l_rate=l_rate, beta=beta)
The optimize() function has the code for both SGD and momentum update rules. SGD algorithm is relatively straightforward, updating the networks by calculated gradient directly, while momentum make the same movement in the last iteration, corrected by negative gradient.
def optimize(self, l_rate=0.1, beta=.9):
'''
Stochatic Gradient Descent (SGD):
θ^(t+1) <- θ^t - η∇L(y, ŷ)
Momentum:
v^(t+1) <- βv^t + (1-β)∇L(y, ŷ)^t
θ^(t+1) <- θ^t - ηv^(t+1)
'''
if self.optimizer == "sgd":
for key in self.params:
self.params[key] = self.params[key] -\
l_rate*self.grads[key]
elif self.optimizer == "momentum":
for key in self.params:
self.momemtum_opt[key] = (beta*self.momemtum_opt[key] +\
(1.-beta)*self.grads[key])
self.params[key] = self.params[key] -\
l_rate * self.momemtum_opt[key]
After having updated the parameters of the neural network, we can measure the accuracy on a validation set that we conveniently prepared earlier, to validate how well our network performs after each iteration over the whole dataset.
def accuracy(self, y, output):
return np.mean(np.argmax(y, axis=-1) == np.argmax(output.T, axis=-1))
Finally, we can call the training function, after knowing what will happen. We use the training and validation data as input to the training function, and then we wait.
dnn.train(x_train, y_train, x_val, y_val)
