CS231n学习笔记--4.Backpropagation and Neural Networks

1.损失函数

其中SVM损失函数计算的是不正确分类的得分惩罚，即Syi是正确分类结果的得分，Sj是错误分类结果的得分，超参数（1）度量正确分类得分的优越性。

2.简单损失函数流程图：

3.求解损失函数的梯度矩阵

其中，绿色数值代表前向网络中的实际值，红色数值代表反向神经网络得到的梯度值。

4.常见的激活函数

sigmoid

有点落伍了（fallen out of favor and it is rarely ever used）
原因 1：梯度饱和问题（sigmoid saturate and kill gradients），即如果神经元的激活值很大，返回的梯度几乎为零，因此反向传播的时候，也会阻断（or kill）从此处流动的梯度。此外初始化的时候，也要注意，如果梯度很大的话，也很容易造成梯度饱和。
原因 2：sigmoid outputs are not zero-centered，因为输出结果在 [0,1] 之间，都是整数，所以造成了某些维度一直更新正的梯度，某些则相反。就会造成 zig-zagging 形状的参数更新。不过利用批随机梯度下降法就会缓解这个问题，没有第一个问题严重。

tanh

很明显，tanh（non-linearity ）虽然也有梯度饱和问题，但是起码是 zero-centered，因此实际中比 sigmoid 效果更好。

ReLU

好处：能够加速收敛速度，据说是因为线性，非饱和（non-saturating）的形式；运算（oprations）很实现都很简单，不用指数（exponentials）操作。
坏处：很脆弱（fragile），容易死掉（die），即如果很大的梯度经过神经元，那么就会造成此神经元不会再对任何数据点有激活。不过学习率设置小一点就不会有太大的问题。

Leaky ReLU

不是单纯的把负数置零，而是加一个很小的 slope，比如 0.01
f(x)=1(x<0)(αx)+1(x≥0)(x)。这样做是为了修复 “dying ReLU的 问题。
如果 α 作为参数，即每个神经元的 slope 都不一样，如果可以自学习的话，称为 PReLU。

Maxout

每个神经元会有两个权重，激活函数为 max(wT1x+b1,wT2+b2)
当 w1=b1=0 时，就是 ReLU
虽然和 ReLU 一样没有梯度饱和问题，也没有 dying ReLU 问题，但是参数确实原来的两倍

实践中，用 ReLU 较多，学习率要调小一点，如果 dead units 很多的话，用 PReLU 或者 Maxout 试一下。

5.minbath

深度学习的优化算法，说白了就是梯度下降。每次的参数更新有两种方式。
第一种，遍历全部数据集算一次损失函数，然后算函数对各个参数的梯度，更新梯度。这种方法每更新一次参数都要把数据集里的所有样本都看一遍，计算量开销大，计算速度慢，不支持在线学习，这称为Batch gradient descent，批梯度下降。
另一种，每看一个数据就算一下损失函数，然后求梯度更新参数，这个称为随机梯度下降，stochastic gradient descent。这个方法速度比较快，但是收敛性能不太好，可能在最优点附近晃来晃去，hit不到最优点。两次参数的更新也有可能互相抵消掉，造成目标函数震荡的比较剧烈。
为了克服两种方法的缺点，现在一般采用的是一种折中手段，mini-batch gradient decent，小批的梯度下降，这种方法把数据分为若干个批，按批来更新参数，这样，一个批中的一组数据共同决定了本次梯度的方向，下降起来就不容易跑偏，减少了随机性。另一方面因为批的样本数与整个数据集相比小了很多，计算量也不是很大。

6.Assinment 1:

参考博客：cs231n 课程作业 Assignment 1

KNN分类器

import numpy as npclass KNearestNeighbor(object):  """ a kNN classifier with L2 distance """  def __init__(self):    pass  def train(self, X, y):    """    Train the classifier. For k-nearest neighbors this is just    memorizing the training data.    Inputs:    - X: A numpy array of shape (num_train, D) containing the training data      consisting of num_train samples each of dimension D.    - y: A numpy array of shape (N,) containing the training labels, where         y[i] is the label for X[i].    """    self.X_train = X    self.y_train = y  def predict(self, X, k=1, num_loops=0):    """    Predict labels for test data using this classifier.    Inputs:    - X: A numpy array of shape (num_test, D) containing test data consisting         of num_test samples each of dimension D.    - k: The number of nearest neighbors that vote for the predicted labels.    - num_loops: Determines which implementation to use to compute distances      between training points and testing points.    Returns:    - y: A numpy array of shape (num_test,) containing predicted labels for the      test data, where y[i] is the predicted label for the test point X[i].    """    if num_loops == 0:      dists = self.compute_distances_no_loops(X)    elif num_loops == 1:      dists = self.compute_distances_one_loop(X)    elif num_loops == 2:      dists = self.compute_distances_two_loops(X)    else:      raise ValueError('Invalid value %d for num_loops' % num_loops)    return self.predict_labels(dists, k=k)  def compute_distances_two_loops(self, X):    """    Compute the distance between each test point in X and each training point    in self.X_train using a nested loop over both the training data and the    test data.    Inputs:    - X: A numpy array of shape (num_test, D) containing test data.    Returns:    - dists: A numpy array of shape (num_test, num_train) where dists[i, j]      is the Euclidean distance between the ith test point and the jth training      point.    """    num_test = X.shape[0]    num_train = self.X_train.shape[0]    dists = np.zeros((num_test, num_train))    for i in xrange(num_test):      for j in xrange(num_train):        #####################################################################        # TODO:                                                             #        # Compute the l2 distance between the ith test point and the jth    #        # training point, and store the result in dists[i, j]. You should   #        # not use a loop over dimension.                                    #        #####################################################################        pass        dists[i][j] = np.sum((X[i] - self.X_train[j]) ** 2)        #####################################################################        #                       END OF YOUR CODE                            #        #####################################################################    return dists  def compute_distances_one_loop(self, X):    """    Compute the distance between each test point in X and each training point    in self.X_train using a single loop over the test data.    Input / Output: Same as compute_distances_two_loops    """    num_test = X.shape[0]    num_train = self.X_train.shape[0]    dists = np.zeros((num_test, num_train))    for i in xrange(num_test):      #######################################################################      # TODO:                                                               #      # Compute the l2 distance between the ith test point and all training #      # points, and store the result in dists[i, :].                        #      #######################################################################      pass      dists[i] = np.sum((self.X_train - X[i]) ** 2, 1)      #######################################################################      #                         END OF YOUR CODE                            #      #######################################################################    return dists  def compute_distances_no_loops(self, X):    """    Compute the distance between each test point in X and each training point    in self.X_train using no explicit loops.    Input / Output: Same as compute_distances_two_loops    """    num_test = X.shape[0]    num_train = self.X_train.shape[0]    dists = np.zeros((num_test, num_train))    #########################################################################    # TODO:                                                                 #    # Compute the l2 distance between all test points and all training      #    # points without using any explicit loops, and store the result in      #    # dists.                                                                #    #                                                                       #    # You should implement this function using only basic array operations; #    # in particular you should not use functions from scipy.                #    #                                                                       #    # HINT: Try to formulate the l2 distance using matrix multiplication    #    #       and two broadcast sums.                                         #    #########################################################################    pass    dists += np.sum(self.X_train ** 2, axis=1).reshape(1, num_train)    dists += np.sum(X ** 2, axis=1).reshape(num_test, 1) # reshape for broadcasting    dists -= 2 * np.dot(X, self.X_train.T)    #########################################################################    #                         END OF YOUR CODE                              #    #########################################################################    return dists  def predict_labels(self, dists, k=1):    """    Given a matrix of distances between test points and training points,    predict a label for each test point.    Inputs:    - dists: A numpy array of shape (num_test, num_train) where dists[i, j]      gives the distance betwen the ith test point and the jth training point.    Returns:    - y: A numpy array of shape (num_test,) containing predicted labels for the      test data, where y[i] is the predicted label for the test point X[i].    """    num_test = dists.shape[0]    y_pred = np.zeros(num_test)    for i in xrange(num_test):      # A list of length k storing the labels of the k nearest neighbors to      # the ith test point.      closest_y = []      #########################################################################      # TODO:                                                                 #      # Use the distance matrix to find the k nearest neighbors of the ith    #      # testing point, and use self.y_train to find the labels of these       #      # neighbors. Store these labels in closest_y.                           #      # Hint: Look up the function numpy.argsort.                             #      #########################################################################      pass      closest_y = self.y_train[np.argsort(dists[i])[0:k]]      #########################################################################      # TODO:                                                                 #      # Now that you have found the labels of the k nearest neighbors, you    #      # need to find the most common label in the list closest_y of labels.   #      # Store this label in y_pred[i]. Break ties by choosing the smaller     #      # label.                                                                #      #########################################################################      pass      # to find the most common element in list, you can use np.bincount      y_pred[i] = np.bincount(closest_y).argmax()      #########################################################################      #                           END OF YOUR CODE                            #      #########################################################################    return y_pred

前向/后向神经网络

a.损失函数为SVM：

Python代码：

import numpy as np#from random import shuffledef svm_loss_naive(W, X, y, reg):  """  Structured SVM loss function, naive implementation (with loops).  Inputs have dimension D, there are C classes, and we operate on minibatches  of N examples.  Inputs:  - W: A numpy array of shape (D, C) containing weights.  - X: A numpy array of shape (N, D) containing a minibatch of data.  - y: A numpy array of shape (N,) containing training labels; y[i] = c means    that X[i] has label c, where 0 <= c < C.  - reg: (float) regularization strength  Returns a tuple of:  - loss as single float  - gradient with respect to weights W; an array of same shape as W  """  dW = np.zeros(W.shape) # initialize the gradient as zero  # compute the loss and the gradient  num_classes = W.shape[1]  num_train = X.shape[0]  loss = 0.0  for i in xrange(num_train):    scores = X[i].dot(W)    correct_class_score = scores[y[i]]    for j in xrange(num_classes):      if j == y[i]:        continue      margin = scores[j] - correct_class_score + 1 # note delta = 1      if margin > 0:        loss += margin        dW[:, j] += X[i]        dW[:, y[i]] -= X[i]  # Right now the loss is a sum over all training examples, but we want it  # to be an average instead so we divide by num_train.  loss /= num_train  dW /= num_train  # Add regularization to the loss.  loss += 0.5 * reg * np.sum(W * W)  dW += reg * W  #############################################################################  # TODO:                                                                     #  # Compute the gradient of the loss function and store it dW.                #  # Rather that first computing the loss and then computing the derivative,   #  # it may be simpler to compute the derivative at the same time that the     #  # loss is being computed. As a result you may need to modify some of the    #  # code above to compute the gradient.                                       #  #############################################################################  return loss, dWdef svm_loss_vectorized(W, X, y, reg):  """  Structured SVM loss function, vectorized implementation.  Inputs and outputs are the same as svm_loss_naive.  """  loss = 0.0  dW = np.zeros(W.shape) # initialize the gradient as zero  #############################################################################  # TODO:                                                                     #  # Implement a vectorized version of the structured SVM loss, storing the    #  # result in loss.                                                           #  #############################################################################  pass  N = X.shape[0]  #scores = np.dot(X, W)  #margin = scores - scores[range(0, N), y].reshape(N, 1) + 1  #margin[range(0, N), y] = 0  #margin = margin * (margin > 0) # max(0, s_j - s_yi + delta)  #loss += np.sum(margin) / N + 0.5 * reg * np.sum(W * W)  scores = X.dot(W) # N x C  margin = scores - scores[range(0,N), y].reshape(-1, 1) + 1 # N x C  margin[range(N), y] = 0  margin = (margin > 0) * margin  loss += margin.sum() / N  loss += 0.5 * reg * np.sum(W * W)  #############################################################################  #                             END OF YOUR CODE                              #  #############################################################################  #############################################################################  # TODO:                                                                     #  # Implement a vectorized version of the gradient for the structured SVM     #  # loss, storing the result in dW.                                           #  #                                                                           #  # Hint: Instead of computing the gradient from scratch, it may be easier    #  # to reuse some of the intermediate values that you used to compute the     #  # loss.                                                                     #  #############################################################################  pass  counts = (margin > 0).astype(int)  counts[range(N), y] = - np.sum(counts, axis = 1)  dW += np.dot(X.T, counts) / N + reg * W  #############################################################################  #                             END OF YOUR CODE                              #  #############################################################################  return loss, dW

b.损失函数为Softmax：
（不同之处仅在损失函数及其梯度计算部分）：

python代码：

import numpy as np#from random import shuffledef softmax_loss_naive(W, X, y, reg):  """  Softmax loss function, naive implementation (with loops)  Inputs have dimension D, there are C classes, and we operate on minibatches  of N examples.  Inputs:  - W: A numpy array of shape (D, C) containing weights.  - X: A numpy array of shape (N, D) containing a minibatch of data.  - y: A numpy array of shape (N,) containing training labels; y[i] = c means    that X[i] has label c, where 0 <= c < C.  - reg: (float) regularization strength  Returns a tuple of:  - loss as single float  - gradient with respect to weights W; an array of same shape as W  """  # Initialize the loss and gradient to zero.  loss = 0.0  dW = np.zeros_like(W)  #############################################################################  # TODO: Compute the softmax loss and its gradient using explicit loops.     #  # Store the loss in loss and the gradient in dW. If you are not careful     #  # here, it is easy to run into numeric instability. Don't forget the        #  # regularization!                                                           #  #############################################################################  pass  N, C = X.shape[0], W.shape[1]  for i in range(N):      f = np.dot(X[i], W)      f -= np.max(f) # f.shape = C      loss = loss + np.log(np.sum(np.exp(f))) - f[y[i]]      dW[:, y[i]] -= X[i]      s = np.exp(f).sum()      for j in range(C):          dW[:, j] += np.exp(f[j]) / s * X[i]  loss = loss / N + 0.5 * reg * np.sum(W * W)  dW = dW / N + reg * W  #############################################################################  #                          END OF YOUR CODE                                 #  #############################################################################  return loss, dWdef softmax_loss_vectorized(W, X, y, reg):  """  Softmax loss function, vectorized version.  Inputs and outputs are the same as softmax_loss_naive.  """  # Initialize the loss and gradient to zero.  loss = 0.0  dW = np.zeros_like(W)  #############################################################################  # TODO: Compute the softmax loss and its gradient using no explicit loops.  #  # Store the loss in loss and the gradient in dW. If you are not careful     #  # here, it is easy to run into numeric instability. Don't forget the        #  # regularization!                                                           #  #############################################################################  pass  N = X.shape[0]  f = np.dot(X, W) # f.shape = N, C  f -= f.max(axis = 1).reshape(N, 1)  s = np.exp(f).sum(axis = 1)  loss = np.log(s).sum() - f[range(N), y].sum()  counts = np.exp(f) / s.reshape(N, 1)  counts[range(N), y] -= 1  dW = np.dot(X.T, counts)  loss = loss / N + 0.5 * reg * np.sum(W * W)  dW = dW / N + reg * W  #############################################################################  #                          END OF YOUR CODE                                 #  #############################################################################  return loss, dW

简单的两层网络：

步骤：
1。设置learning_rate, learning_rate_decay（每个epoch（数据集）训练后learning_rate下降的倍率，因为越训练到后面学习率应该越低，以防止震荡）, num_iters,batch_size。
2。根据随机梯度下降法随机从epoch中抽取batch_size个训练样本。
3。根据batch_size个训练样本计算各中间变量（w,b…）的梯度，并根据学习率更新这些中间变量。
4。如果一个epoch训练结束，更新学习率进行下一个epoch的训练。

import numpy as np#import matplotlib.pyplot as pltclass 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).                                                             #    #############################################################################    pass    hidden_layer = np.maximum(0, np.dot(X, W1) + b1)    scores = np.dot(hidden_layer, W2) + b2    #############################################################################    #                              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. So that your results match ours, multiply the            #    # regularization loss by 0.5                                                #    #############################################################################    pass    f = scores - np.max(scores, axis = 1, keepdims = True)#负值化    loss = -f[range(N), y].sum() + np.log(np.exp(f).sum(axis = 1)).sum()    loss = loss / N + 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 #    #############################################################################    pass    dscore = np.exp(f) / np.exp(f).sum(axis = 1, keepdims = True)    dscore[range(N), y] -= 1    dscore /= N    grads['W2'] = np.dot(hidden_layer.T, dscore) + reg * W2    grads['b2'] = np.sum(dscore, axis = 0)    dhidden = np.dot(dscore, W2.T)    dhidden[hidden_layer <= 0.00001] = 0    grads['W1'] = np.dot(X.T, dhidden) + reg * W1    grads['b1'] = np.sum(dhidden, 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=1e-5, 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 xrange(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.                             #      #########################################################################      pass      indices = np.random.choice(num_train, batch_size, replace=True)      X_batch = X[indices]      y_batch = y[indices]      #########################################################################      #                             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.                         #      #########################################################################      pass      self.params['W1'] -= learning_rate * grads['W1']      self.params['b1'] -= learning_rate * grads['b1']      self.params['W2'] -= learning_rate * grads['W2']      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!                #    ###########################################################################    pass    W1, b1 = self.params['W1'], self.params['b1']    W2, b2 = self.params['W2'], self.params['b2']    hidden_layer = np.maximum(0, np.dot(X, W1) + b1)    scores = np.dot(hidden_layer, W2) + b2    y_pred = np.argmax(scores, axis = 1)    ###########################################################################    #                              END OF YOUR CODE                           #    ###########################################################################    return y_pred