Initialize the parameters for a two-layer network and for an $L$-layer neural network.
Implement the forward propagation module (shown in purple in the figure below).
Complete the LINEAR part of a layer’s forward propagation step (resulting in $Z^{[l]}$).
We give you the ACTIVATION function (relu/sigmoid).
Combine the previous two steps into a new [LINEAR->ACTIVATION] forward function.
Stack the [LINEAR->RELU] forward function L-1 time (for layers 1 through L-1) and add a [LINEAR->SIGMOID] at the end (for the final layer $L$). This gives you a new L_model_forward function.
Compute the loss.
Implement the backward propagation module (denoted in red in the figure below).
Complete the LINEAR part of a layer’s backward propagation step.
We give you the gradient of the ACTIVATE function (relu_backward/sigmoid_backward)
Combine the previous two steps into a new [LINEAR->ACTIVATION] backward function.
Stack [LINEAR->RELU] backward L-1 times and add [LINEAR->SIGMOID] backward in a new L_model_backward function
def linear_forward(A, W, b): """ Implement the linear part of a layer's forward propagation.
Arguments: A -- activations from previous layer (or input data): (size of previous layer, number of examples) W -- weights matrix: numpy array of shape (size of current layer, size of previous layer) b -- bias vector, numpy array of shape (size of the current layer, 1)
Returns: Z -- the input of the activation function, also called pre-activation parameter cache -- a python dictionary containing "A", "W" and "b" ; stored for computing the backward pass efficiently """ ### START CODE HERE ### (≈ 1 line of code) Z = np.dot(W,A) + b ### END CODE HERE ### assert(Z.shape == (W.shape[0], A.shape[1])) cache = (A, W, b) return Z, cache
Linear-Activation Forward
Implement the forward propagation of the LINEAR->ACTIVATION layer. Mathematical relation is: $A^{[l]} = g(Z^{[l]}) = g(W^{[l]}A^{[l-1]} +b^{[l]})$ where the activation “g” can be sigmoid() or relu(). Use linear_forward() and the correct activation function.
def linear_activation_forward(A_prev, W, b, activation): """ Implement the forward propagation for the LINEAR->ACTIVATION layer
Arguments: A_prev -- activations from previous layer (or input data): (size of previous layer, number of examples) W -- weights matrix: numpy array of shape (size of current layer, size of previous layer) b -- bias vector, numpy array of shape (size of the current layer, 1) activation -- the activation to be used in this layer, stored as a text string: "sigmoid" or "relu"
Returns: A -- the output of the activation function, also called the post-activation value cache -- a python dictionary containing "linear_cache" and "activation_cache"; stored for computing the backward pass efficiently """ if activation == "sigmoid": # Inputs: "A_prev, W, b". Outputs: "A, activation_cache". ### START CODE HERE ### (≈ 2 lines of code) Z, linear_cache = linear_forward(A_prev,W,b) A, activation_cache = sigmoid(Z) ### END CODE HERE ### elif activation == "relu": # Inputs: "A_prev, W, b". Outputs: "A, activation_cache". ### START CODE HERE ### (≈ 2 lines of code) Z, linear_cache = linear_forward(A_prev,W,b) A, activation_cache = relu(Z) ### END CODE HERE ### assert (A.shape == (W.shape[0], A_prev.shape[1])) cache = (linear_cache, activation_cache)
return A, cache
L-Layer Model
Implement the forward propagation of the above model.
def L_model_forward(X, parameters): """ Implement forward propagation for the [LINEAR->RELU]*(L-1)->LINEAR->SIGMOID computation Arguments: X -- data, numpy array of shape (input size, number of examples) parameters -- output of initialize_parameters_deep() Returns: AL -- last post-activation value caches -- list of caches containing: every cache of linear_relu_forward() (there are L-1 of them, indexed from 0 to L-2) the cache of linear_sigmoid_forward() (there is one, indexed L-1) """
caches = [] A = X L = len(parameters) // 2 # number of layers in the neural network # Implement [LINEAR -> RELU]*(L-1). Add "cache" to the "caches" list. for l in range(1, L): A_prev = A ### START CODE HERE ### (≈ 2 lines of code) A, cache = linear_activation_forward(A_prev,parameters['W'+str(l)],parameters['b'+str(l)],activation='relu') caches.append(cache) ### END CODE HERE ### # Implement LINEAR -> SIGMOID. Add "cache" to the "caches" list. ### START CODE HERE ### (≈ 2 lines of code) AL, cache = linear_activation_forward(A,parameters['W'+str(L)],parameters['b'+str(L)],activation='sigmoid') caches.append(cache) ### END CODE HERE ### assert(AL.shape == (1,X.shape[1])) return AL, caches
Cost Function
Compute the cross-entropy cost $J$, using the following formula: $$-\frac{1}{m} \sum\limits_{i = 1}^{m} (y^{(i)}\log\left(a^{[L] (i)}\right) + (1-y^{(i)})\log\left(1- a^{L}\right)) \tag{7}$$
def compute_cost(AL, Y): """ Implement the cost function defined by equation (7).
Arguments: AL -- probability vector corresponding to your label predictions, shape (1, number of examples) Y -- true "label" vector (for example: containing 0 if non-cat, 1 if cat), shape (1, number of examples)
Returns: cost -- cross-entropy cost """ m = Y.shape[1]
# Compute loss from aL and y. ### START CODE HERE ### (≈ 1 lines of code) #cost = -1./m * (np.dot(Y,np.log(AL).T) + (1-Y) * np.log(1 - AL).T) cost = -1 / m * (np.sum(Y*np.log(AL) + (1 - Y) * np.log(1-AL),axis=1,keepdims=True)) ### END CODE HERE ### cost = np.squeeze(cost) # To make sure your cost's shape is what we expect (e.g. this turns [[17]] into 17). assert(cost.shape == ()) return cost
Backward propagation module
Linear backward
The three outputs $(dW^{[l]}, db^{[l]}, dA^{[l]})$ are computed using the input $dZ^{[l]}$.Here are the formulas you need: $$ dW^{[l]} = \frac{\partial \mathcal{L} }{\partial W^{[l]}} = \frac{1}{m} dZ^{[l]} A^{[l-1] T} \tag{8}$$ $$ db^{[l]} = \frac{\partial \mathcal{L} }{\partial b^{[l]}} = \frac{1}{m} \sum_{i = 1}^{m} dZ^{l}\tag{9}$$ $$ dA^{[l-1]} = \frac{\partial \mathcal{L} }{\partial A^{[l-1]}} = W^{[l] T} dZ^{[l]} \tag{10}$$
Use the 3 formulas above to implement linear_backward().
def linear_backward(dZ, cache): """ Implement the linear portion of backward propagation for a single layer (layer l)
Arguments: dZ -- Gradient of the cost with respect to the linear output (of current layer l) cache -- tuple of values (A_prev, W, b) coming from the forward propagation in the current layer
Returns: dA_prev -- Gradient of the cost with respect to the activation (of the previous layer l-1), same shape as A_prev dW -- Gradient of the cost with respect to W (current layer l), same shape as W db -- Gradient of the cost with respect to b (current layer l), same shape as b """ A_prev, W, b = cache m = A_prev.shape[1] ### START CODE HERE ### (≈ 3 lines of code) dW = 1/m * np.dot(dZ,A_prev.T) db = 1/m * np.sum(dZ,axis=1,keepdims=True) dA_prev = np.dot(W.T,dZ) ### END CODE HERE ### assert (dA_prev.shape == A_prev.shape) assert (dW.shape == W.shape) assert (db.shape == b.shape) return dA_prev, dW, db
Linear-Activation backward
If $g(.)$ is the activation function, sigmoid_backward and relu_backward compute $$dZ^{[l]} = dA^{[l]} * g’(Z^{[l]}) \tag{11}$$.
Implement the backpropagation for the LINEAR->ACTIVATION layer.
def linear_activation_backward(dA, cache, activation): """ Implement the backward propagation for the LINEAR->ACTIVATION layer. Arguments: dA -- post-activation gradient for current layer l cache -- tuple of values (linear_cache, activation_cache) we store for computing backward propagation efficiently activation -- the activation to be used in this layer, stored as a text string: "sigmoid" or "relu" Returns: dA_prev -- Gradient of the cost with respect to the activation (of the previous layer l-1), same shape as A_prev dW -- Gradient of the cost with respect to W (current layer l), same shape as W db -- Gradient of the cost with respect to b (current layer l), same shape as b """ linear_cache, activation_cache = cache if activation == "relu": ### START CODE HERE ### (≈ 2 lines of code) dZ = relu_backward(dA,activation_cache) dA_prev, dW, db = linear_backward(dZ,linear_cache) ### END CODE HERE ### elif activation == "sigmoid": ### START CODE HERE ### (≈ 2 lines of code) dZ = sigmoid_backward(dA,activation_cache) dA_prev, dW, db = linear_backward(dZ,linear_cache) ### END CODE HERE ### return dA_prev, dW, db
L-Model Backward
Initializing backpropagation: To backpropagate through this network, we know that the output is, $A^{[L]} = \sigma(Z^{[L]})$. Your code thus needs to compute dAL $= \frac{\partial \mathcal{L}}{\partial A^{[L]}}$. To do so, use this formula (derived using calculus which you don’t need in-depth knowledge of):
1
dAL = - (np.divide(Y, AL) - np.divide(1 - Y, 1 - AL)) # derivative of cost with respect to AL
Implement backpropagation for the [LINEAR->RELU] $\times$ (L-1) -> LINEAR -> SIGMOID model.
def L_model_backward(AL, Y, caches): """ Implement the backward propagation for the [LINEAR->RELU] * (L-1) -> LINEAR -> SIGMOID group Arguments: AL -- probability vector, output of the forward propagation (L_model_forward()) Y -- true "label" vector (containing 0 if non-cat, 1 if cat) caches -- list of caches containing: every cache of linear_activation_forward() with "relu" (it's caches[l], for l in range(L-1) i.e l = 0...L-2) the cache of linear_activation_forward() with "sigmoid" (it's caches[L-1]) Returns: grads -- A dictionary with the gradients grads["dA" + str(l)] = ... grads["dW" + str(l)] = ... grads["db" + str(l)] = ... """ grads = {} L = len(caches) # the number of layers m = AL.shape[1] Y = Y.reshape(AL.shape) # after this line, Y is the same shape as AL
# Initializing the backpropagation ### START CODE HERE ### (1 line of code) dAL = - (np.divide(Y,AL) - np.divide(1-Y,1-AL)) ### END CODE HERE ### # Lth layer (SIGMOID -> LINEAR) gradients. Inputs: "AL, Y, caches". Outputs: "grads["dAL"], grads["dWL"], grads["dbL"] ### START CODE HERE ### (approx. 2 lines) current_cache = caches[L-1] grads["dA" + str(L)], grads["dW" + str(L)], grads["db" + str(L)] = linear_activation_backward(dAL,current_cache,activation='sigmoid') ### END CODE HERE ### for l in reversed(range(L - 1)): # lth layer: (RELU -> LINEAR) gradients. # Inputs: "grads["dA" + str(l + 2)], caches". Outputs: "grads["dA" + str(l + 1)] , grads["dW" + str(l + 1)] , grads["db" + str(l + 1)] ### START CODE HERE ### (approx. 5 lines) current_cache = caches[L-l-2] dA_prev_temp, dW_temp, db_temp = linear_activation_backward(grads['dA'+str(l+2)],current_cache,activation='relu') grads["dA" + str(l + 1)] = dA_prev_temp grads["dW" + str(l + 1)] = dW_temp grads["db" + str(l + 1)] = db_temp ### END CODE HERE ###
return grads
Update Parameters
Implement update_parameters() to update your parameters using gradient descent.
def update_parameters(parameters, grads, learning_rate): """ Update parameters using gradient descent Arguments: parameters -- python dictionary containing your parameters grads -- python dictionary containing your gradients, output of L_model_backward Returns: parameters -- python dictionary containing your updated parameters parameters["W" + str(l)] = ... parameters["b" + str(l)] = ... """ L = len(parameters) // 2 # number of layers in the neural network
# Update rule for each parameter. Use a for loop. ### START CODE HERE ### (≈ 3 lines of code) for l in range(L): parameters["W" + str(l+1)] = parameters['W'+str(l+1)] - learning_rate*grads['dW'+str(l+1)] parameters["b" + str(l+1)] = parameters['b'+str(l+1)] - learning_rate*grads['db'+str(l+1)] ### END CODE HERE ### return parameters
