defzero_pad(X, pad): """ Pad with zeros all images of the dataset X. The padding is applied to the height and width of an image, as illustrated in Figure 1. Argument: X -- python numpy array of shape (m, n_H, n_W, n_C) representing a batch of m images pad -- integer, amount of padding around each image on vertical and horizontal dimensions Returns: X_pad -- padded image of shape (m, n_H + 2*pad, n_W + 2*pad, n_C) """ ### START CODE HERE ### (≈ 1 line) X_pad = np.pad(X, ((0,0),(pad,pad),(pad,pad),(0,0)), 'constant', constant_values=(0,0)) ### END CODE HERE ### return X_pad
defconv_single_step(a_slice_prev, W, b): """ Apply one filter defined by parameters W on a single slice (a_slice_prev) of the output activation of the previous layer. Arguments: a_slice_prev -- slice of input data of shape (f, f, n_C_prev) W -- Weight parameters contained in a window - matrix of shape (f, f, n_C_prev) b -- Bias parameters contained in a window - matrix of shape (1, 1, 1) Returns: Z -- a scalar value, result of convolving the sliding window (W, b) on a slice x of the input data """
### START CODE HERE ### (≈ 2 lines of code) # Element-wise product between a_slice and W. Do not add the bias yet. s = a_slice_prev * W # Sum over all entries of the volume s. Z = np.sum(s) # Add bias b to Z. Cast b to a float() so that Z results in a scalar value. Z = Z + float(b) ### END CODE HERE ###
return Z
3.Convolutional Neural Networks - Forward pass
创建一次完整的卷积过程，也就是利用上面的一次卷积，进行for循环。进行切片的时候，注意边界vert_start, vert_end, horiz_start and horiz_end
这一步应该先弄清楚A_prev，A，W，b的维度，超参数项包括了stride和pad
$$ n_H = \lfloor \frac{n_{H_{prev}} - f + 2 \times pad}{stride} \rfloor +1 $$ $$ n_W = \lfloor \frac{n_{W_{prev}} - f + 2 \times pad}{stride} \rfloor +1 $$ $$ n_C = \text{number of filters used in the convolution}$$
defconv_forward(A_prev, W, b, hparameters): """ Implements the forward propagation for a convolution function Arguments: A_prev -- output activations of the previous layer, numpy array of shape (m, n_H_prev, n_W_prev, n_C_prev) W -- Weights, numpy array of shape (f, f, n_C_prev, n_C) b -- Biases, numpy array of shape (1, 1, 1, n_C) hparameters -- python dictionary containing "stride" and "pad" Returns: Z -- conv output, numpy array of shape (m, n_H, n_W, n_C) cache -- cache of values needed for the conv_backward() function """ ### START CODE HERE ### # Retrieve dimensions from A_prev's shape (≈1 line) (m, n_H_prev, n_W_prev, n_C_prev) = A_prev.shape # Retrieve dimensions from W's shape (≈1 line) (f, f, n_C_prev, n_C) = W.shape # Retrieve information from "hparameters" (≈2 lines) stride = hparameters['stride'] pad = hparameters['pad'] # Compute the dimensions of the CONV output volume using the formula given above. Hint: use int() to floor. (≈2 lines) n_H = int((n_H_prev + 2 * pad - f) / stride + 1) n_W = int((n_W_prev + 2 * pad - f) / stride + 1)
# Initialize the output volume Z with zeros. (≈1 line) Z = np.zeros((m, n_H, n_W, n_C)) # Create A_prev_pad by padding A_prev A_prev_pad = zero_pad(A_prev, pad) for i in range(m): # loop over the batch of training examples a_prev_pad = A_prev_pad[i] # Select ith training example's padded activation for h in range(n_H): # loop over vertical axis of the output volume for w in range(n_W): # loop over horizontal axis of the output volume for c in range(n_C): # loop over channels (= #filters) of the output volume # Find the corners of the current "slice" (≈4 lines) vert_start = h * stride vert_end = h * stride + f horiz_start = w * stride horiz_end = w * stride + f # Use the corners to define the (3D) slice of a_prev_pad (See Hint above the cell). (≈1 line) a_slice_prev = a_prev_pad[vert_start : vert_end, horiz_start : horiz_end] # Convolve the (3D) slice with the correct filter W and bias b, to get back one output neuron. (≈1 line) Z[i, h, w, c] = conv_single_step(a_slice_prev,W[:,:,:,c],b[:,:,:,c]) ### END CODE HERE ### # Making sure your output shape is correct assert(Z.shape == (m, n_H, n_W, n_C)) # Save information in "cache" for the backprop cache = (A_prev, W, b, hparameters) return Z, cache
defpool_forward(A_prev, hparameters, mode = "max"): """ Implements the forward pass of the pooling layer Arguments: A_prev -- Input data, numpy array of shape (m, n_H_prev, n_W_prev, n_C_prev) hparameters -- python dictionary containing "f" and "stride" mode -- the pooling mode you would like to use, defined as a string ("max" or "average") Returns: A -- output of the pool layer, a numpy array of shape (m, n_H, n_W, n_C) cache -- cache used in the backward pass of the pooling layer, contains the input and hparameters """ # Retrieve dimensions from the input shape (m, n_H_prev, n_W_prev, n_C_prev) = A_prev.shape # Retrieve hyperparameters from "hparameters" f = hparameters["f"] stride = hparameters["stride"] # Define the dimensions of the output n_H = int(1 + (n_H_prev - f) / stride) n_W = int(1 + (n_W_prev - f) / stride) n_C = n_C_prev # Initialize output matrix A A = np.zeros((m, n_H, n_W, n_C)) ### START CODE HERE ### for i in range(m): # loop over the training examples for h in range(n_H): # loop on the vertical axis of the output volume for w in range(n_W): # loop on the horizontal axis of the output volume for c in range (n_C): # loop over the channels of the output volume # Find the corners of the current "slice" (≈4 lines) vert_start = h * stride vert_end = vert_start + f horiz_start = w * stride horiz_end = horiz_start + f # Use the corners to define the current slice on the ith training example of A_prev, channel c. (≈1 line) a_prev_slice = A_prev[i, vert_start : vert_end, horiz_start : horiz_end, c] # Compute the pooling operation on the slice. Use an if statment to differentiate the modes. Use np.max/np.mean. if mode == "max": A[i, h, w, c] = np.max(a_prev_slice) elif mode == "average": A[i, h, w, c] = np.mean(a_prev_slice) ### END CODE HERE ### # Store the input and hparameters in "cache" for pool_backward() cache = (A_prev, hparameters) # Making sure your output shape is correct assert(A.shape == (m, n_H, n_W, n_C)) return A, cache
defconv_backward(dZ, cache): """ Implement the backward propagation for a convolution function Arguments: dZ -- gradient of the cost with respect to the output of the conv layer (Z), numpy array of shape (m, n_H, n_W, n_C) cache -- cache of values needed for the conv_backward(), output of conv_forward() Returns: dA_prev -- gradient of the cost with respect to the input of the conv layer (A_prev), numpy array of shape (m, n_H_prev, n_W_prev, n_C_prev) dW -- gradient of the cost with respect to the weights of the conv layer (W) numpy array of shape (f, f, n_C_prev, n_C) db -- gradient of the cost with respect to the biases of the conv layer (b) numpy array of shape (1, 1, 1, n_C) """ ### START CODE HERE ### # Retrieve information from "cache" (A_prev, W, b, hparameters) = cache # Retrieve dimensions from A_prev's shape (m, n_H_prev, n_W_prev, n_C_prev) = A_prev.shape # Retrieve dimensions from W's shape (f, f, n_C_prev, n_C) = W.shape # Retrieve information from "hparameters" stride = hparameters['stride'] pad = hparameters['pad'] # Retrieve dimensions from dZ's shape (m, n_H, n_W, n_C) = dZ.shape # Initialize dA_prev, dW, db with the correct shapes dA_prev = np.zeros(A_prev.shape) dW = np.zeros(W.shape) db = np.zeros(b.shape)
# Pad A_prev and dA_prev A_prev_pad = zero_pad(A_prev, pad) dA_prev_pad = zero_pad(dA_prev, pad) for i in range(m): # loop over the training examples # select ith training example from A_prev_pad and dA_prev_pad a_prev_pad = A_prev_pad[i] da_prev_pad = dA_prev_pad[i] for h in range(n_H): # loop over vertical axis of the output volume for w in range(n_W): # loop over horizontal axis of the output volume for c in range(n_C): # loop over the channels of the output volume # Find the corners of the current "slice" vert_start = h * stride vert_end = h * stride + f horiz_start = w * stride horiz_end = w * stride + f # Use the corners to define the slice from a_prev_pad a_slice = a_prev_pad[vert_start : vert_end, horiz_start : horiz_end, : ]
# Update gradients for the window and the filter's parameters using the code formulas given above da_prev_pad[vert_start:vert_end, horiz_start:horiz_end, :] += W[:,:,:,c] * dZ[ i, h, w ,c]
dW[:,:,:,c] += a_slice * dZ[ i, h, w ,c] db[:,:,:,c] += dZ[ i, h, w ,c] # Set the ith training example's dA_prev to the unpaded da_prev_pad (Hint: use X[pad:-pad, pad:-pad, :]) dA_prev[i, :, :, :] = da_prev_pad[pad:-pad, pad:-pad, :] ### END CODE HERE ### # Making sure your output shape is correct assert(dA_prev.shape == (m, n_H_prev, n_W_prev, n_C_prev)) return dA_prev, dW, db
defcreate_mask_from_window(x): """ Creates a mask from an input matrix x, to identify the max entry of x. Arguments: x -- Array of shape (f, f) Returns: mask -- Array of the same shape as window, contains a True at the position corresponding to the max entry of x. """ ### START CODE HERE ### (≈1 line) mask = (x == np.max(x)) ### END CODE HERE ### return mask
defdistribute_value(dz, shape): """ Distributes the input value in the matrix of dimension shape Arguments: dz -- input scalar shape -- the shape (n_H, n_W) of the output matrix for which we want to distribute the value of dz Returns: a -- Array of size (n_H, n_W) for which we distributed the value of dz """ ### START CODE HERE ### # Retrieve dimensions from shape (≈1 line) (n_H, n_W) = shape # Compute the value to distribute on the matrix (≈1 line) average = n_H * n_W # Create a matrix where every entry is the "average" value (≈1 line) a = dz / average * np.ones((n_H, n_W)) ### END CODE HERE ### return a
defpool_backward(dA, cache, mode = "max"): """ Implements the backward pass of the pooling layer Arguments: dA -- gradient of cost with respect to the output of the pooling layer, same shape as A cache -- cache output from the forward pass of the pooling layer, contains the layer's input and hparameters mode -- the pooling mode you would like to use, defined as a string ("max" or "average") Returns: dA_prev -- gradient of cost with respect to the input of the pooling layer, same shape as A_prev """ ### START CODE HERE ### # Retrieve information from cache (≈1 line) (A_prev, hparameters) = cache # Retrieve hyperparameters from "hparameters" (≈2 lines) stride = hparameters['stride'] f = hparameters['f'] # Retrieve dimensions from A_prev's shape and dA's shape (≈2 lines) m, n_H_prev, n_W_prev, n_C_prev = A_prev.shape m, n_H, n_W, n_C = dA.shape # Initialize dA_prev with zeros (≈1 line) dA_prev = np.zeros(A_prev.shape) for i in range(m): # loop over the training examples # select training example from A_prev (≈1 line) a_prev = A_prev[i] for h in range(n_H): # loop on the vertical axis for w in range(n_W): # loop on the horizontal axis for c in range(n_C): # loop over the channels (depth) # Find the corners of the current "slice" (≈4 lines) vert_start = h * stride vert_end = vert_start + f horiz_start = w * stride horiz_end = horiz_start + f # Compute the backward propagation in both modes. if mode == "max": # Use the corners and "c" to define the current slice from a_prev (≈1 line) a_prev_slice = a_prev[vert_start : vert_end, horiz_start : horiz_end, c] # Create the mask from a_prev_slice (≈1 line) mask = create_mask_from_window(a_prev_slice) # Set dA_prev to be dA_prev + (the mask multiplied by the correct entry of dA) (≈1 line) dA_prev[i, vert_start: vert_end, horiz_start: horiz_end, c] += mask * dA[i, h, w, c] elif mode == "average": # Get the value a from dA (≈1 line) da = dA[i, h, w, c] # Define the shape of the filter as fxf (≈1 line) shape = (f, f) # Distribute it to get the correct slice of dA_prev. i.e. Add the distributed value of da. (≈1 line) dA_prev[i, vert_start: vert_end, horiz_start: horiz_end, c] += distribute_value(da, shape) ### END CODE ### # Making sure your output shape is correct assert(dA_prev.shape == A_prev.shape) return dA_prev
defcreate_placeholders(n_H0, n_W0, n_C0, n_y): """ Creates the placeholders for the tensorflow session. Arguments: n_H0 -- scalar, height of an input image n_W0 -- scalar, width of an input image n_C0 -- scalar, number of channels of the input n_y -- scalar, number of classes Returns: X -- placeholder for the data input, of shape [None, n_H0, n_W0, n_C0] and dtype "float" Y -- placeholder for the input labels, of shape [None, n_y] and dtype "float" """
### START CODE HERE ### (≈2 lines) X = tf.placeholder(tf.float32, shape=(None,n_H0, n_W0, n_C0)) Y = tf.placeholder(tf.float32, shape=(None,n_y)) ### END CODE HERE ### return X, Y
- Conv2D: stride 1, padding is "SAME" - ReLU - Max pool: Use an 8 by 8 filter size and an 8 by 8 stride, padding is "SAME" - Conv2D: stride 1, padding is "SAME" - ReLU - Max pool: Use a 4 by 4 filter size and a 4 by 4 stride, padding is "SAME" - Flatten the previous output. - FULLYCONNECTED (FC) layer：这里全连接层不需要有激活函数，因为后面计算cost的时候会加上softmax，因此这里不需要加
用到的函数：
tf.nn.conv2d(X,W1, strides = [1,s,s,1], padding = ‘SAME’): given an input $X$ and a group of filters $W1$, this function convolves $W1$’s filters on X. The third input ([1,f,f,1]) represents the strides for each dimension of the input (m, n_H_prev, n_W_prev, n_C_prev). You can read the full documentation here
tf.nn.max_pool(A, ksize = [1,f,f,1], strides = [1,s,s,1], padding = ‘SAME’): given an input A, this function uses a window of size (f, f) and strides of size (s, s) to carry out max pooling over each window. You can read the full documentation here
tf.nn.relu(Z1): computes the elementwise ReLU of Z1 (which can be any shape). You can read the full documentation here.
tf.contrib.layers.flatten(P): given an input P, this function flattens each example into a 1D vector it while maintaining the batch-size. It returns a flattened tensor with shape [batch_size, k]. You can read the full documentation here.
tf.contrib.layers.fully_connected(F, num_outputs): given a the flattened input F, it returns the output computed using a fully connected layer. You can read the full documentation here.
In the last function above (tf.contrib.layers.fully_connected), the fully connected layer automatically initializes weights in the graph and keeps on training them as you train the model. Hence, you did not need to initialize those weights when initializing the parameters.
defforward_propagation(X, parameters): """ Implements the forward propagation for the model: CONV2D -> RELU -> MAXPOOL -> CONV2D -> RELU -> MAXPOOL -> FLATTEN -> FULLYCONNECTED Arguments: X -- input dataset placeholder, of shape (input size, number of examples) parameters -- python dictionary containing your parameters "W1", "W2" the shapes are given in initialize_parameters Returns: Z3 -- the output of the last LINEAR unit """ # Retrieve the parameters from the dictionary "parameters" W1 = parameters['W1'] W2 = parameters['W2'] ### START CODE HERE ### # CONV2D: stride of 1, padding 'SAME' Z1 = tf.nn.conv2d(X, filter=W1, strides=[1,1,1,1],padding='SAME') # RELU A1 = tf.nn.relu(Z1) # MAXPOOL: window 8x8, sride 8, padding 'SAME' P1 = tf.nn.max_pool(A1,ksize=[1, 8, 8, 1], strides=[1, 8, 8, 1],padding='SAME') # CONV2D: filters W2, stride 1, padding 'SAME' Z2 = tf.nn.conv2d(P1, filter=W2, strides=[1, 1, 1, 1],padding='SAME') # RELU A2 = tf.nn.relu(Z2) # MAXPOOL: window 4x4, stride 4, padding 'SAME' P2 = tf.nn.max_pool(A2,ksize=[1, 4, 4, 1], strides=[1, 4, 4, 1],padding='SAME') # FLATTEN P2 = tf.contrib.layers.flatten(P2) # FULLY-CONNECTED without non-linear activation function (not not call softmax). # 6 neurons in output layer. Hint: one of the arguments should be "activation_fn=None" Z3 = tf.contrib.layers.fully_connected(P2, 6,activation_fn=None) ### END CODE HERE ###
return Z3
4. Compute cost
tf.nn.softmax_cross_entropy_with_logits(logits = Z3, labels = Y): computes the softmax entropy loss. This function both computes the softmax activation function as well as the resulting loss. You can check the full documentation here.这个函数已经包含了计算softmax，还有求cross-entropy两件事了。
tf.reduce_mean: computes the mean of elements across dimensions of a tensor. Use this to sum the losses over all the examples to get the overall cost. You can check the full documentation here.
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# GRADED FUNCTION: compute_cost
defcompute_cost(Z3, Y): """ Computes the cost Arguments: Z3 -- output of forward propagation (output of the last LINEAR unit), of shape (6, number of examples) Y -- "true" labels vector placeholder, same shape as Z3 Returns: cost - Tensor of the cost function """ ### START CODE HERE ### (1 line of code) cost = tf.reduce_mean(tf.nn.softmax_cross_entropy_with_logits(logits=Z3,labels=Y)) ### END CODE HERE ### return cost
defmodel(X_train, Y_train, X_test, Y_test, learning_rate = 0.009, num_epochs = 100, minibatch_size = 64, print_cost = True): """ Implements a three-layer ConvNet in Tensorflow: CONV2D -> RELU -> MAXPOOL -> CONV2D -> RELU -> MAXPOOL -> FLATTEN -> FULLYCONNECTED Arguments: X_train -- training set, of shape (None, 64, 64, 3) Y_train -- test set, of shape (None, n_y = 6) X_test -- training set, of shape (None, 64, 64, 3) Y_test -- test set, of shape (None, n_y = 6) learning_rate -- learning rate of the optimization num_epochs -- number of epochs of the optimization loop minibatch_size -- size of a minibatch print_cost -- True to print the cost every 100 epochs Returns: train_accuracy -- real number, accuracy on the train set (X_train) test_accuracy -- real number, testing accuracy on the test set (X_test) parameters -- parameters learnt by the model. They can then be used to predict. """ ops.reset_default_graph() # to be able to rerun the model without overwriting tf variables tf.set_random_seed(1) # to keep results consistent (tensorflow seed) seed = 3# to keep results consistent (numpy seed) (m, n_H0, n_W0, n_C0) = X_train.shape n_y = Y_train.shape[1] costs = [] # To keep track of the cost # Create Placeholders of the correct shape ### START CODE HERE ### (1 line) X, Y = create_placeholders(n_H0, n_W0,n_C0,n_y) ### END CODE HERE ###
# Initialize parameters ### START CODE HERE ### (1 line) parameters = initialize_parameters() ### END CODE HERE ### # Forward propagation: Build the forward propagation in the tensorflow graph ### START CODE HERE ### (1 line) Z3 = forward_propagation(X,parameters) ### END CODE HERE ### # Cost function: Add cost function to tensorflow graph ### START CODE HERE ### (1 line) cost = compute_cost(Z3, Y) ### END CODE HERE ### # Backpropagation: Define the tensorflow optimizer. Use an AdamOptimizer that minimizes the cost. ### START CODE HERE ### (1 line) optimizer = tf.train.AdamOptimizer(learning_rate).minimize(cost) ### END CODE HERE ### # Initialize all the variables globally init = tf.global_variables_initializer() # Start the session to compute the tensorflow graph with tf.Session() as sess: # Run the initialization sess.run(init) # Do the training loop for epoch in range(num_epochs):
minibatch_cost = 0. num_minibatches = int(m / minibatch_size) # number of minibatches of size minibatch_size in the train set seed = seed + 1 minibatches = random_mini_batches(X_train, Y_train, minibatch_size, seed)
for minibatch in minibatches:
# Select a minibatch (minibatch_X, minibatch_Y) = minibatch # IMPORTANT: The line that runs the graph on a minibatch. # Run the session to execute the optimizer and the cost, the feedict should contain a minibatch for (X,Y). ### START CODE HERE ### (1 line) _ , temp_cost = sess.run([optimizer,cost],feed_dict={X:minibatch_X,Y:minibatch_Y}) ### END CODE HERE ### minibatch_cost += temp_cost / num_minibatches
# Print the cost every epoch if print_cost == Trueand epoch % 5 == 0: print ("Cost after epoch %i: %f" % (epoch, minibatch_cost)) if print_cost == Trueand epoch % 1 == 0: costs.append(minibatch_cost) # plot the cost plt.plot(np.squeeze(costs)) plt.ylabel('cost') plt.xlabel('iterations (per tens)') plt.title("Learning rate =" + str(learning_rate)) plt.show()
# Calculate the correct predictions predict_op = tf.argmax(Z3, 1) correct_prediction = tf.equal(predict_op, tf.argmax(Y, 1)) # Calculate accuracy on the test set accuracy = tf.reduce_mean(tf.cast(correct_prediction, "float")) print(accuracy) train_accuracy = accuracy.eval({X: X_train, Y: Y_train}) test_accuracy = accuracy.eval({X: X_test, Y: Y_test}) print("Train Accuracy:", train_accuracy) print("Test Accuracy:", test_accuracy) return train_accuracy, test_accuracy, parameters
