code: from __future__ import print_function
  
  import numpy as np
  import matplotlib.pyplot as plt
  
  class TwoLayerNet(object):
    """
    A two-layer fully-connected neural network. The net has an input dimension of
    N, a hidden layer dimension of H, and performs classification over C classes.
    We train the network with a softmax loss function and L2 regularization on the
    weight matrices. The network uses a ReLU nonlinearity after the first fully
    connected layer.
  
    In other words, the network has the following architecture:
  
    input - fully connected layer - ReLU - fully connected layer - softmax
  
    The outputs of the second fully-connected layer are the scores for each class.
    """
  
    def __init__(self, input_size, hidden_size, output_size, std=1e-4):
      """
      Initialize the model. Weights are initialized to small random values and
      biases are initialized to zero. Weights and biases are stored in the
      variable self.params, which is a dictionary with the following keys:
  
      W1: First layer weights; has shape (D, H)
      b1: First layer biases; has shape (H,)
      W2: Second layer weights; has shape (H, C)
      b2: Second layer biases; has shape (C,)
  
      Inputs:
      - input_size: The dimension D of the input data.
      - hidden_size: The number of neurons H in the hidden layer.
      - output_size: The number of classes C.
      """
      self.params = {}
      self.params['W1'] = std * np.random.randn(input_size, hidden_size)
      self.params['b1'] = np.zeros(hidden_size)
      self.params['W2'] = std * np.random.randn(hidden_size, output_size)
      self.params['b2'] = np.zeros(output_size)
  
    def loss(self, X, y=None, reg=0.0):
      """
      Compute the loss and gradients for a two layer fully connected neural
      network.
  
      Inputs:
      - X: Input data of shape (N, D). Each X[i] is a training sample.
      - y: Vector of training labels. y[i] is the label for X[i], and each y[i] is
        an integer in the range 0 <= y[i] < C. This parameter is optional; if it
        is not passed then we only return scores, and if it is passed then we
        instead return the loss and gradients.
      - reg: Regularization strength.
  
      Returns:
      If y is None, return a matrix scores of shape (N, C) where scores[i, c] is
      the score for class c on input X[i].
  
      If y is not None, instead return a tuple of:
      - loss: Loss (data loss and regularization loss) for this batch of training
        samples.
      - grads: Dictionary mapping parameter names to gradients of those parameters
        with respect to the loss function; has the same keys as self.params.
      """
      # Unpack variables from the params dictionary
      W1, b1 = self.params['W1'], self.params['b1']
      W2, b2 = self.params['W2'], self.params['b2']
      N, D = X.shape
  
      # Compute the forward pass
      scores = None
      #############################################################################
      # TODO: Perform the forward pass, computing the class scores for the input. #
      # Store the result in the scores variable, which should be an array of      #
      # shape (N, C).                                                             #
      #############################################################################
      out1 = np.maximum(0, X.dot(W1) + b1) # relu, (N, H)
      scores = out1.dot(W2) + b2 # (N, C)
      #############################################################################
      #                              END OF YOUR CODE                             #
      #############################################################################
      
      # If the targets are not given then jump out, we're done
      if y is None:
        return scores
  
      # Compute the loss
      loss = None
      #############################################################################
      # TODO: Finish the forward pass, and compute the loss. This should include  #
      # both the data loss and L2 regularization for W1 and W2. Store the result  #
      # in the variable loss, which should be a scalar. Use the Softmax           #
      # classifier loss.                                                          #
      #############################################################################
      correct_class_score = scores[np.arange(N), y].reshape(N, 1)
      exp_sum = np.sum(np.exp(scores), axis=1).reshape(N, 1)
      loss = np.sum(np.log(exp_sum) - correct_class_score)
      loss /= N
      loss += 0.5 * reg * np.sum(W1 * W1)+  0.5 * reg * np.sum(W2 * W2)
      #############################################################################
      #                              END OF YOUR CODE                             #
      #############################################################################
  
      # Backward pass: compute gradients
      grads = {}
      #############################################################################
      # TODO: Compute the backward pass, computing the derivatives of the weights #
      # and biases. Store the results in the grads dictionary. For example,       #
      # grads['W1'] should store the gradient on W1, and be a matrix of same size #
      #############################################################################
      margin = np.exp(scores) / exp_sum
      margin[np.arange(N), y] += -1
      margin /= N #(N, C)
      dW2 = out1.T.dot(margin) #(H ,C)
      dW2 += reg * W2 
      grads['W2'] = dW2
      grads['b2'] = np.sum(margin, axis = 0)
      
      margin1 = margin.dot(W2.T) #(N, H)
      margin1[out1 <= 0] = 0
      dW1 = X.T.dot(margin1) #(D, H)
      dW1 += reg * W1 
      grads['W1'] = dW1
      grads['b1'] = np.sum(margin1, axis = 0)
      #############################################################################
      #                              END OF YOUR CODE                             #
      #############################################################################
  
      return loss, grads
  
    def train(self, X, y, X_val, y_val,
              learning_rate=1e-3, learning_rate_decay=0.95,
              reg=5e-6, num_iters=100,
              batch_size=200, verbose=False):
      """
      Train this neural network using stochastic gradient descent.
  
      Inputs:
      - X: A numpy array of shape (N, D) giving training data.
      - y: A numpy array f shape (N,) giving training labels; y[i] = c means that
        X[i] has label c, where 0 <= c < C.
      - X_val: A numpy array of shape (N_val, D) giving validation data.
      - y_val: A numpy array of shape (N_val,) giving validation labels.
      - learning_rate: Scalar giving learning rate for optimization.
      - learning_rate_decay: Scalar giving factor used to decay the learning rate
        after each epoch.
      - reg: Scalar giving regularization strength.
      - num_iters: Number of steps to take when optimizing.
      - batch_size: Number of training examples to use per step.
      - verbose: boolean; if true print progress during optimization.
      """
      num_train = X.shape[0]
      iterations_per_epoch = max(num_train / batch_size, 1)
  
      # Use SGD to optimize the parameters in self.model
      loss_history = []
      train_acc_history = []
      val_acc_history = []
  
      for it in range(num_iters):
        X_batch = None
        y_batch = None
  
        #########################################################################
        # TODO: Create a random minibatch of training data and labels, storing  #
        # them in X_batch and y_batch respectively.                             #
        #########################################################################
        mask = np.random.choice(num_train, batch_size, replace=True)
        X_batch = X[mask]
        y_batch = y[mask]
        #########################################################################
        #                             END OF YOUR CODE                          #
        #########################################################################
  
        # Compute loss and gradients using the current minibatch
        loss, grads = self.loss(X_batch, y=y_batch, reg=reg)
        loss_history.append(loss)
  
        #########################################################################
        # TODO: Use the gradients in the grads dictionary to update the         #
        # parameters of the network (stored in the dictionary self.params)      #
        # using stochastic gradient descent. You'll need to use the gradients   #
        # stored in the grads dictionary defined above.                         #
        #########################################################################
        self.params['W1'] -= learning_rate * grads['W1']
        self.params['W2'] -= learning_rate * grads['W2']
        self.params['b1'] -= learning_rate * grads['b1']
        self.params['b2'] -= learning_rate * grads['b2']
        #########################################################################
        #                             END OF YOUR CODE                          #
        #########################################################################
  
        if verbose and it % 100 == 0:
          print('iteration %d / %d: loss %f' % (it, num_iters, loss))
  
        # Every epoch, check train and val accuracy and decay learning rate.
        if it % iterations_per_epoch == 0:
          # Check accuracy
          train_acc = (self.predict(X_batch) == y_batch).mean()
          val_acc = (self.predict(X_val) == y_val).mean()
          train_acc_history.append(train_acc)
          val_acc_history.append(val_acc)
  
          # Decay learning rate
          learning_rate *= learning_rate_decay
  
      return {
        'loss_history': loss_history,
        'train_acc_history': train_acc_history,
        'val_acc_history': val_acc_history,
      }
  
    def predict(self, X):
      """
      Use the trained weights of this two-layer network to predict labels for
      data points. For each data point we predict scores for each of the C
      classes, and assign each data point to the class with the highest score.
  
      Inputs:
      - X: A numpy array of shape (N, D) giving N D-dimensional data points to
        classify.
  
      Returns:
      - y_pred: A numpy array of shape (N,) giving predicted labels for each of
        the elements of X. For all i, y_pred[i] = c means that X[i] is predicted
        to have class c, where 0 <= c < C.
      """
      y_pred = None
  
      ###########################################################################
      # TODO: Implement this function; it should be VERY simple!                #
      ###########################################################################
      out1 = np.maximum(0, X.dot(self.params['W1']) + self.params['b1']) # relu, (N, H)
      y_pred =  np.argmax(out1.dot(self.params['W2']) + self.params['b2'],axis = 1) # (N, C)
      ###########################################################################
      #                              END OF YOUR CODE                           #
      ###########################################################################
  
      return y_pred
  
  
  
apis: numpy.sum numpy.zeros numpy.random.randn numpy.random.choice numpy.exp numpy.arange numpy.log