text: """
  network.py
  ~~~~~~~~~~
  
  A module to implement the stochastic gradient descent learning
  algorithm for a feedforward neural network.  Gradients are calculated
  using backpropagation.  Note that I have focused on making the code
  simple, easily readable, and easily modifiable.  It is not optimized,
  and omits many desirable features.
  """
  
  #### Libraries
  # Standard library
  import random
  
  # Third-party libraries
  import numpy as np
  
  xrange = range
  
  class Network(object):
  
      def __init__(self, sizes):
          """The list ``sizes`` contains the number of neurons in the
          respective layers of the network.  For example, if the list
          was [2, 3, 1] then it would be a three-layer network, with the
          first layer containing 2 neurons, the second layer 3 neurons,
          and the third layer 1 neuron.  The biases and weights for the
          network are initialized randomly, using a Gaussian
          distribution with mean 0, and variance 1.  Note that the first
          layer is assumed to be an input layer, and by convention we
          won't set any biases for those neurons, since biases are only
          ever used in computing the outputs from later layers."""
          np.random.seed(10)
          self.num_layers = len(sizes)
          self.sizes = sizes
          self.biases = [np.random.randn(y, 1) for y in sizes[1:]]
          self.weights = [np.random.randn(y, x)
                          for x, y in zip(sizes[:-1], sizes[1:])]
          
          # nik debug
          self.nabla_b = []
          self.nabla_w = []
          self.debug_zs_by_batch = []
          self.debug_acts_by_batch = []
  
      def feedforward(self, a):
          """Return the output of the network if ``a`` is input."""
          for b, w in zip(self.biases, self.weights):
              a = sigmoid(np.dot(w, a)+b)
          return a
  
      def SGD(self, training_data, epochs, mini_batch_size, eta,
              test_data=None):
          """Train the neural network using mini-batch stochastic
          gradient descent.  The ``training_data`` is a list of tuples
          ``(x, y)`` representing the training inputs and the desired
          outputs.  The other non-optional parameters are
          self-explanatory.  If ``test_data`` is provided then the
          network will be evaluated against the test data after each
          epoch, and partial progress printed out.  This is useful for
          tracking progress, but slows things down substantially."""
          np.random.seed(1) 
          if test_data: n_test = len(test_data)
          n = len(training_data)
          for j in xrange(epochs):
  #             random.shuffle(training_data)
              mini_batches = [
                  training_data[k:k+mini_batch_size]
                  for k in xrange(0, n, mini_batch_size)]
              for mini_batch in mini_batches:
                  self.update_mini_batch(mini_batch, eta)
              if test_data:
                  print (f'Epoch {j}: {self.evaluate(test_data)} over {n_test}')
              else:
                  pass
  #                 print (f'Epoch {j} complete')
  
      def update_mini_batch(self, mini_batch, eta):
          """Update the network's weights and biases by applying
          gradient descent using backpropagation to a single mini batch.
          The ``mini_batch`` is a list of tuples ``(x, y)``, and ``eta``
          is the learning rate."""
          nabla_b = [np.zeros(b.shape) for b in self.biases]
          nabla_w = [np.zeros(w.shape) for w in self.weights]
          for x, y in mini_batch:
              delta_nabla_b, delta_nabla_w = self.backprop(x, y)
              nabla_b = [nb+dnb for nb, dnb in zip(nabla_b, delta_nabla_b)]
              nabla_w = [nw+dnw for nw, dnw in zip(nabla_w, delta_nabla_w)]
              
          self.nabla_b.append(nabla_b)
          self.nabla_w.append(nabla_w)
          
          self.weights = [w-(eta/len(mini_batch))*nw
                          for w, nw in zip(self.weights, nabla_w)]
          self.biases = [b-(eta/len(mini_batch))*nb
                         for b, nb in zip(self.biases, nabla_b)]
  
      def backprop(self, x, y):
          """Return a tuple ``(nabla_b, nabla_w)`` representing the
          gradient for the cost function C_x.  ``nabla_b`` and
          ``nabla_w`` are layer-by-layer lists of numpy arrays, similar
          to ``self.biases`` and ``self.weights``."""
          nabla_b = [np.zeros(b.shape) for b in self.biases]
          nabla_w = [np.zeros(w.shape) for w in self.weights]
          
          # feedforward
          activation = x
          activations = [x] # list to store all the activations, layer by layer
          zs = [] # list to store all the z vectors, layer by layer
          for b, w in zip(self.biases, self.weights):
              z = np.dot(w, activation)+b
              zs.append(z)
              activation = sigmoid(z)
              activations.append(activation)
              
          self.debug_zs_by_batch.append(zs)
          self.debug_acts_by_batch.append(activations)
              
          # backward pass
          delta = self.cost_derivative(activations[-1], y) * \
              sigmoid_prime(zs[-1])
          nabla_b[-1] = delta
          nabla_w[-1] = np.dot(delta, activations[-2].transpose())
          # Note that the variable l in the loop below is used a little
          # differently to the notation in Chapter 2 of the book.  Here,
          # l = 1 means the last layer of neurons, l = 2 is the
          # second-last layer, and so on.  It's a renumbering of the
          # scheme in the book, used here to take advantage of the fact
          # that Python can use negative indices in lists.
          for l in xrange(2, self.num_layers):
              z = zs[-l]
              sp = sigmoid_prime(z)
              delta = np.dot(self.weights[-l+1].transpose(), delta) * sp
              nabla_b[-l] = delta
              nabla_w[-l] = np.dot(delta, activations[-l-1].transpose())
          return (nabla_b, nabla_w)
  
      def evaluate(self, test_data):
          """Return the number of test inputs for which the neural
          network outputs the correct result. Note that the neural
          network's output is assumed to be the index of whichever
          neuron in the final layer has the highest activation."""
          test_results = [(np.argmax(self.feedforward(x)), y)
                          for (x, y) in test_data]
          return sum(int(x == y) for (x, y) in test_results)
  
      def cost_derivative(self, output_activations, y):
          """Return the vector of partial derivatives \partial C_x /
          \partial a for the output activations."""
          return (output_activations-y)
  
  #### Miscellaneous functions
  def sigmoid(z):
      """The sigmoid function."""
      return 1.0/(1.0+np.exp(-z))
  
  def sigmoid_prime(z):
      """Derivative of the sigmoid function."""
      return sigmoid(z)*(1-sigmoid(z))