+def cost_grad(x, target_y, transfer=sigmoid, transfer_prime=sigmoid_prime):
+ '''Return (∂C/∂w, ∂C/∂b) for a particular input and desired output vector'''
+
+ # forward pass, remember all z vectors and activations for every layer
+ z_s, a_s = feed_forward(x, transfer)
+
+ # backward pass
+ deltas = [None] * len(weights) # delta = dC/dz error for each layer
+ # insert the last layer error
+ deltas[-1] = transfer_prime(z_s[-1]) * 2 * (a_s[-1] - target_y)
+ for i in reversed(range(len(deltas) - 1)):
+ deltas[i] = (weights[i + 1].T @ deltas[i + 1]) * transfer_prime(z_s[i])
+
+ dw = [d @ a_s[i+1] for i, d in enumerate(deltas)]
+ db = deltas
+ return dw, db
+
+
+def label_vector(label):
+ x = np.zeros(10)
+ x[label] = 1.0
+ return x
+
+
+def backpropagate(image_batch, label_batch, eta):
+ '''Update NN with gradient descent and backpropagation to a batch of inputs
+
+ eta is the learning rate.
+ '''
+ global weights, biases
+
+ num_images = image_batch.shape[1]
+ for i in range(num_images):
+ y = label_vector(label_batch[i])
+ dws, dbs = cost_grad(image_batch[:, i], y)
+ weights = [w + eta * dw for w, dw in zip(weights, dws)]
+ biases = [b + eta * db for b, db in zip(biases, dbs)]
+
+