神经网络实现及优化

环境配置

# A bit of setup

from __future__ import print_function
import numpy as np
import matplotlib.pyplot as plt

from cs231n.classifiers.neural_net import TwoLayerNet



%matplotlib inline
plt.rcParams['figure.figsize'] = (10.0, 8.0) # set default size of plots
plt.rcParams['image.interpolation'] = 'nearest'
plt.rcParams['image.cmap'] = 'gray'

# for auto-reloading external modules
# see http://stackoverflow.com/questions/1907993/autoreload-of-modules-in-ipython
%load_ext autoreload
%autoreload 2

def rel_error(x, y):
    """ returns relative error """
    return np.max(np.abs(x - y) / (np.maximum(1e-8, np.abs(x) + np.abs(y))))
# Create a small net and some toy data to check your implementations.
# Note that we set the random seed for repeatable experiments.

input_size = 4
hidden_size = 10
num_classes = 3
num_inputs = 5

def init_toy_model():
    np.random.seed(0)
    return TwoLayerNet(input_size, hidden_size, num_classes, std=1e-1)

def init_toy_data():
    np.random.seed(1)
    X = 10 * np.random.randn(num_inputs, input_size)
    y = np.array([0, 1, 2, 2, 1])
    return X, y

net = init_toy_model()
X, y = init_toy_data()

前向传播

计算得分

scores = net.loss(X)
print('Your scores:')
print(scores)
print()
print('correct scores:')
correct_scores = np.asarray([
  [-0.81233741, -1.27654624, -0.70335995],
  [-0.17129677, -1.18803311, -0.47310444],
  [-0.51590475, -1.01354314, -0.8504215 ],
  [-0.15419291, -0.48629638, -0.52901952],
  [-0.00618733, -0.12435261, -0.15226949]])
print(correct_scores)
print()

# The difference should be very small. We get < 1e-7
print('Difference between your scores and correct scores:')
print(np.sum(np.abs(scores - correct_scores)))

计算损失

loss, _ = net.loss(X, y, reg=0.05)
correct_loss = 1.30378789133

# should be very small, we get < 1e-12
print('Difference between your loss and correct loss:')
print(np.sum(np.abs(loss - correct_loss)))

反向传播

from cs231n.gradient_check import eval_numerical_gradient

# Use numeric gradient checking to check your implementation of the backward pass.
# If your implementation is correct, the difference between the numeric and
# analytic gradients should be less than 1e-8 for each of W1, W2, b1, and b2.

loss, grads = net.loss(X, y, reg=0.05)

# these should all be less than 1e-8 or so
for param_name in grads:
    f = lambda W: net.loss(X, y, reg=0.05)[0]
    param_grad_num = eval_numerical_gradient(f, net.params[param_name], verbose=False)
    print('%s max relative error: %e' % (param_name, rel_error(param_grad_num, grads[param_name])))

训练网络

net = init_toy_model()
stats = net.train(X, y, X, y,
            learning_rate=1e-1, reg=5e-6,
            num_iters=100, verbose=False)

print('Final training loss: ', stats['loss_history'][-1])

# plot the loss history
plt.plot(stats['loss_history'])
plt.xlabel('iteration')
plt.ylabel('training loss')
plt.title('Training Loss history')
plt.show()

载入数据

from cs231n.data_utils import load_CIFAR10

def get_CIFAR10_data(num_training=49000, num_validation=1000, num_test=1000):
    """
    Load the CIFAR-10 dataset from disk and perform preprocessing to prepare
    it for the two-layer neural net classifier. These are the same steps as
    we used for the SVM, but condensed to a single function.  
    """
    # Load the raw CIFAR-10 data
    cifar10_dir = 'cs231n/datasets/cifar-10-batches-py'
    
    X_train, y_train, X_test, y_test = load_CIFAR10(cifar10_dir)
        
    # Subsample the data
    mask = list(range(num_training, num_training + num_validation))
    X_val = X_train[mask]
    y_val = y_train[mask]
    mask = list(range(num_training))
    X_train = X_train[mask]
    y_train = y_train[mask]
    mask = list(range(num_test))
    X_test = X_test[mask]
    y_test = y_test[mask]

    # Normalize the data: subtract the mean image
    mean_image = np.mean(X_train, axis=0)
    X_train -= mean_image
    X_val -= mean_image
    X_test -= mean_image

    # Reshape data to rows
    X_train = X_train.reshape(num_training, -1)
    X_val = X_val.reshape(num_validation, -1)
    X_test = X_test.reshape(num_test, -1)

    return X_train, y_train, X_val, y_val, X_test, y_test


# Cleaning up variables to prevent loading data multiple times (which may cause memory issue)
try:
    del X_train, y_train
    del X_test, y_test
    print('Clear previously loaded data.')
except:
    pass

# Invoke the above function to get our data.
X_train, y_train, X_val, y_val, X_test, y_test = get_CIFAR10_data()
print('Train data shape: ', X_train.shape)
print('Train labels shape: ', y_train.shape)
print('Validation data shape: ', X_val.shape)
print('Validation labels shape: ', y_val.shape)
print('Test data shape: ', X_test.shape)
print('Test labels shape: ', y_test.shape)

训练网络

input_size = 32 * 32 * 3
hidden_size = 50
num_classes = 10
net = TwoLayerNet(input_size, hidden_size, num_classes)

# Train the network
stats = net.train(X_train, y_train, X_val, y_val,
            num_iters=1000, batch_size=200,
            learning_rate=1e-4, learning_rate_decay=0.95,
            reg=0.25, verbose=True)

# Predict on the validation set
val_acc = (net.predict(X_val) == y_val).mean()
print('Validation accuracy: ', val_acc)

调试网络

# Plot the loss function and train / validation accuracies
plt.subplot(2, 1, 1)
plt.plot(stats['loss_history'])
plt.title('Loss history')
plt.xlabel('Iteration')
plt.ylabel('Loss')

plt.subplot(2, 1, 2)
plt.plot(stats['train_acc_history'], label='train')
plt.plot(stats['val_acc_history'], label='val')
plt.title('Classification accuracy history')
plt.xlabel('Epoch')
plt.ylabel('Clasification accuracy')
plt.legend()
plt.show()

显示权重

from cs231n.vis_utils import visualize_grid

# Visualize the weights of the network

def show_net_weights(net):
    W1 = net.params['W1']
    W1 = W1.reshape(32, 32, 3, -1).transpose(3, 0, 1, 2)
    plt.imshow(visualize_grid(W1, padding=3).astype('uint8'))
    plt.gca().axis('off')
    plt.show()

show_net_weights(net)

优化超参数

best_net = None # store the best model into this 
best_val = 0
#################################################################################
# TODO: Tune hyperparameters using the validation set. Store your best trained  #
# model in best_net.                                                            #
#                                                                               #
# To help debug your network, it may help to use visualizations similar to the  #
# ones we used above; these visualizations will have significant qualitative    #
# differences from the ones we saw above for the poorly tuned network.          #
#                                                                               #
# Tweaking hyperparameters by hand can be fun, but you might find it useful to  #
# write code to sweep through possible combinations of hyperparameters          #
# automatically like we did on the previous exercises.                          #
#################################################################################
input_size = 32 * 32 * 3
Hidden_size = [50 ,64,128]
REG = [0.01,0.1,0.5]
num_classes = 10
for i in Hidden_size:
    for j in REG :
        net = TwoLayerNet(input_size,  i, num_classes)
# Train the network
        stats = net.train(X_train, y_train, X_val, y_val,
                    num_iters=2000, batch_size=200,
                    learning_rate=1e-3, learning_rate_decay=0.95,
                    reg = j, verbose=False)
        val_acc = (net.predict(X_val) == y_val).mean()
        print('Validation accuracy: ', val_acc)
        if best_val < val_acc:
            best_val = val_acc
            best_net = net
#################################################################################
#                               END OF YOUR CODE                                #
#################################################################################
# visualize the weights of the best network
show_net_weights(best_net)

运行测试集

test_acc = (best_net.predict(X_test) == y_test).mean()
print('Test accuracy: ', test_acc)

用到的子程序

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
        num_train = N
        # 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).                                                             #
        #############################################################################
        buf_H = np.dot(X, W1) + b1
        buf_H[buf_H<0] = 0
        buf_O = np.dot(buf_H, W2) + b2
        scores = buf_O
        #############################################################################
        #                              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.                                                          #
        #############################################################################
        scores = np.subtract( scores.T , np.max(scores , axis = 1) ).T
        scores = np.exp(scores)
        scores = np.divide( scores.T , np.sum(scores , axis = 1) ).T
        loss = - np.sum(np.log ( scores[np.arange(num_train),y] ) ) / num_train
        loss +=  0.5 * reg * (np.sum(W1*W1) + 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 #
        #############################################################################
        #get grad 
        scores[np.arange(num_train),y]  -= 1
        # 10 *5  * 5*3 >>>10*3
        dW2 = np.dot(buf_H.T,scores)/num_train + reg*W2
        # 
        db2 = np.sum(scores,axis =0)/num_train 
        #10 * 3 * 3 *5 >>10 * 5
        buf_hide = np.dot(W2,scores.T)
        #element > 0
        buf_H[buf_H>0] = 1
        #relu buf_hide
        buf_relu = buf_hide.T * buf_H
        #4*5  * 5*10 >>4*10 
        dW1 =   np.dot(X.T,buf_relu) /num_train + reg*W1
        #
        db1 = np.sum(buf_relu,axis =0) /num_train
        grads['W1'] = dW1
        grads['W2'] = dW2
        grads['b1'] = db1
        grads['b2'] = db2
        #############################################################################
        #                              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.                             #
            #########################################################################
            if batch_size > num_train:
                mask = np.random.choice(num_train, batch_size, replace=True)
            else :
                mask = np.random.choice(num_train, batch_size, replace=False)        
            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'] = self.params['W1'] - learning_rate * grads['W1']
            self.params['b1'] = self.params['b1'] - learning_rate * grads['b1']  
            self.params['W2'] = self.params['W2'] - learning_rate * grads['W2']
            self.params['b2'] = 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.
        """
        W1, b1 = self.params['W1'], self.params['b1']
        W2, b2 = self.params['W2'], self.params['b2']
        y_pred = None

        ###########################################################################
        # TODO: Implement this function; it should be VERY simple!                #
        ###########################################################################
        layer_hide = np.dot(X,W1) + b1
        layer_hide[layer_hide<0] = 0
        layer_soft = np.dot(layer_hide,W2) + b2
        scores = np.subtract( layer_soft.T , np.max(layer_soft , axis = 1) ).T
        scores = np.exp(scores)
        scores = np.divide( scores.T , np.sum(scores , axis = 1) ).T
        y_pred = np.argmax(scores , axis = 1)
        ###########################################################################
        #                              END OF YOUR CODE                           #
        ###########################################################################

        return y_pred

from __future__ import print_function

import numpy as np
from random import randrange

def eval_numerical_gradient(f, x, verbose=True, h=0.00001):
  """ 
  a naive implementation of numerical gradient of f at x 
  - f should be a function that takes a single argument
  - x is the point (numpy array) to evaluate the gradient at
  """ 

  fx = f(x) # evaluate function value at original point
  grad = np.zeros_like(x)
  # iterate over all indexes in x
  it = np.nditer(x, flags=['multi_index'], op_flags=['readwrite'])
  while not it.finished:

    # evaluate function at x+h
    ix = it.multi_index
    oldval = x[ix]
    x[ix] = oldval + h # increment by h
    fxph = f(x) # evalute f(x + h)
    x[ix] = oldval - h
    fxmh = f(x) # evaluate f(x - h)
    x[ix] = oldval # restore

    # compute the partial derivative with centered formula
    grad[ix] = (fxph - fxmh) / (2 * h) # the slope
    if verbose:
      print(ix, grad[ix])
    it.iternext() # step to next dimension

  return grad


def eval_numerical_gradient_array(f, x, df, h=1e-5):
  """
  Evaluate a numeric gradient for a function that accepts a numpy
  array and returns a numpy array.
  """
  grad = np.zeros_like(x)
  it = np.nditer(x, flags=['multi_index'], op_flags=['readwrite'])
  while not it.finished:
    ix = it.multi_index
    
    oldval = x[ix]
    x[ix] = oldval + h
    pos = f(x).copy()
    x[ix] = oldval - h
    neg = f(x).copy()
    x[ix] = oldval
    
    grad[ix] = np.sum((pos - neg) * df) / (2 * h)
    it.iternext()
  return grad


def eval_numerical_gradient_blobs(f, inputs, output, h=1e-5):
  """
  Compute numeric gradients for a function that operates on input
  and output blobs.
  
  We assume that f accepts several input blobs as arguments, followed by a blob
  into which outputs will be written. For example, f might be called like this:

  f(x, w, out)
  
  where x and w are input Blobs, and the result of f will be written to out.

  Inputs: 
  - f: function
  - inputs: tuple of input blobs
  - output: output blob
  - h: step size
  """
  numeric_diffs = []
  for input_blob in inputs:
    diff = np.zeros_like(input_blob.diffs)
    it = np.nditer(input_blob.vals, flags=['multi_index'],
                   op_flags=['readwrite'])
    while not it.finished:
      idx = it.multi_index
      orig = input_blob.vals[idx]

      input_blob.vals[idx] = orig + h
      f(*(inputs + (output,)))
      pos = np.copy(output.vals)
      input_blob.vals[idx] = orig - h
      f(*(inputs + (output,)))
      neg = np.copy(output.vals)
      input_blob.vals[idx] = orig
      
      diff[idx] = np.sum((pos - neg) * output.diffs) / (2.0 * h)

      it.iternext()
    numeric_diffs.append(diff)
  return numeric_diffs


def eval_numerical_gradient_net(net, inputs, output, h=1e-5):
  return eval_numerical_gradient_blobs(lambda *args: net.forward(),
              inputs, output, h=h)


def grad_check_sparse(f, x, analytic_grad, num_checks=10, h=1e-5):
  """
  sample a few random elements and only return numerical
  in this dimensions.
  """

  for i in range(num_checks):
    ix = tuple([randrange(m) for m in x.shape])

    oldval = x[ix]
    x[ix] = oldval + h # increment by h
    fxph = f(x) # evaluate f(x + h)
    x[ix] = oldval - h # increment by h
    fxmh = f(x) # evaluate f(x - h)
    x[ix] = oldval # reset

    grad_numerical = (fxph - fxmh) / (2 * h)
    grad_analytic = analytic_grad[ix]
    rel_error = abs(grad_numerical - grad_analytic) / (abs(grad_numerical) + abs(grad_analytic))
    print('numerical: %f analytic: %f, relative error: %e' % (grad_numerical, grad_analytic, rel_error))