cs231n作业：assignment1 - svm


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GitHub地址：https://github.com/ZJUFangzh/cs231n

完成了一个基于SVM的损失函数。

数据集

载入的数据集依旧是:

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Train data shape:  (49000, 32, 32, 3)
Train labels shape:  (49000,)
Validation data shape:  (1000, 32, 32, 3)
Validation labels shape:  (1000,)
Test data shape:  (1000, 32, 32, 3)
Test labels shape:  (1000,)

而后进行32 32 3的图像拉伸，得到：

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Training data shape:  (49000, 3072)
Validation data shape:  (1000, 3072)
Test data shape:  (1000, 3072)
dev data shape:  (500, 3072)

进行一下简单的预处理，减去图像的平均值

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# Preprocessing: subtract the mean image
# first: compute the image mean based on the training data
mean_image = np.mean(X_train, axis=0)
print(mean_image[:10]) # print a few of the elements
plt.figure(figsize=(4,4))
plt.imshow(mean_image.reshape((32,32,3)).astype('uint8')) # visualize the mean image
plt.show()
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# second: subtract the mean image from train and test data
X_train -= mean_image
X_val -= mean_image
X_test -= mean_image
X_dev -= mean_image
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# third: append the bias dimension of ones (i.e. bias trick) so that our SVM
# only has to worry about optimizing a single weight matrix W.
X_train = np.hstack([X_train, np.ones((X_train.shape[0], 1))])
X_val = np.hstack([X_val, np.ones((X_val.shape[0], 1))])
X_test = np.hstack([X_test, np.ones((X_test.shape[0], 1))])
X_dev = np.hstack([X_dev, np.ones((X_dev.shape[0], 1))])

print(X_train.shape, X_val.shape, X_test.shape, X_dev.shape)
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(49000, 3073) (1000, 3073) (1000, 3073) (500, 3073)

SVM分类器

然后就可以开始来编写cs231n/classifiers/linear_svm.py的SVM分类器了。在这里先介绍一下SVM的基本公式和原理。

参考CS231n:线性分类

SVM损失函数想要SVM在正确分类上的比分始终比不正确的比分高出一个边界值$\triangle$

第i个数据图像为$x_i$，正确分类为$y_i$，然后根据$f(x_i,W)$来计算不同分类的值，将分类简写为$s$，那么第j类的得分就是$s_j = f(x_i,W)_j$，针对第i个数据的多类SVM的损失函数定义为：

$$L_i = \sum_{j \neq y_i} max(0, s_j - s_{y_i} + \triangle)$$

如：假设有3个分类，$s = [ 13,-7,11]$，第一个分类是正确的，也就是$y_i = 0$，假设$\triangle=10$，那么把所有不正确的分类加起来($j \neq y_i$)，

$$L_i = max(0,-7-13+10)+max(0,11-13+10)$$

因为SVM只关心差距至少要大于10，所以$L_i = 8$

那么把公式套入：

$$L_i = \sum_{j \neq y_i} max(0, w_j x_i - w_{y_i} x_i + \triangle)$$

加入正则后：

$$L = \frac{1}{N} \sum_i \sum_{j \neq y_i}max(0, f(x_i ;W)_{j} - f(x_i ; W)_{y_i} + \triangle) + \lambda \sum_k \sum_l W^{2}_{k,l}$$

到目前为止计算了loss，然后还需要计算梯度下降的grads，

官方并没有给推导过程，这才是cs231n作业难的地方所在。。。

详细可以看这一篇文章CS 231 SVM 求导

总之就是两个公式：

而后开始编写compute_loss_naive 函数，先用循环来感受一下：

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def 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
  #逐个计算每个样本的loss
  for i in xrange(num_train):
    #计算每个样本的各个分类得分
    scores = X[i].dot(W)
    correct_class_score = scores[y[i]]
    #计算每个分类的得分，计入loss中
    for j in xrange(num_classes):
      # 根据公式，j==y[i]的就是本身的分类，不用算了
      if j == y[i]:
        continue
      margin = scores[j] - correct_class_score + 1 # note delta = 1
      #如果计算的margin > 0，那么就要算入loss，
      if margin > 0:
        loss += margin
        #公式2
        dW[:,y[i]] += -X[i,:].T
        #公式1
        dW[:,j] += X[i, :].T
  # 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 += 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, dW

写完后，用梯度检验检查一下:

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# Once you've implemented the gradient, recompute it with the code below
# and gradient check it with the function we provided for you

# Compute the loss and its gradient at W.
loss, grad = svm_loss_naive(W, X_dev, y_dev, 0.0)

# Numerically compute the gradient along several randomly chosen dimensions, and
# compare them with your analytically computed gradient. The numbers should match
# almost exactly along all dimensions.
from cs231n.gradient_check import grad_check_sparse
f = lambda w: svm_loss_naive(w, X_dev, y_dev, 0.0)[0]
grad_numerical = grad_check_sparse(f, W, grad)

# do the gradient check once again with regularization turned on
# you didn't forget the regularization gradient did you?
loss, grad = svm_loss_naive(W, X_dev, y_dev, 5e1)
f = lambda w: svm_loss_naive(w, X_dev, y_dev, 5e1)[0]
grad_numerical = grad_check_sparse(f, W, grad)
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numerical: 34.663598 analytic: 34.663598, relative error: 6.995024e-13
numerical: 21.043334 analytic: 21.043334, relative error: 5.147242e-12
numerical: 1.334055 analytic: 1.334055, relative error: 5.315420e-11
numerical: 16.611704 analytic: 16.611704, relative error: 6.908581e-12
numerical: 25.327188 analytic: 25.327188, relative error: 1.552987e-11
numerical: -12.867717 analytic: -12.867717, relative error: 1.966004e-11
numerical: 15.066285 analytic: 15.066285, relative error: 7.012975e-12
numerical: -3.752014 analytic: -3.752014, relative error: 7.502607e-11
numerical: 9.927043 analytic: 9.927043, relative error: 9.010584e-13
numerical: 33.071345 analytic: 33.071345, relative error: 1.305438e-12
numerical: -19.227144 analytic: -19.227851, relative error: 1.836495e-05
numerical: 31.392728 analytic: 31.391611, relative error: 1.778034e-05
numerical: -10.450509 analytic: -10.456860, relative error: 3.037629e-04
numerical: -1.346690 analytic: -1.345625, relative error: 3.953276e-04
numerical: 7.843501 analytic: 7.846486, relative error: 1.902216e-04
numerical: 20.635011 analytic: 20.628368, relative error: 1.609761e-04
numerical: 23.654254 analytic: 23.652696, relative error: 3.294745e-05
numerical: 37.706709 analytic: 37.703260, relative error: 4.573495e-05
numerical: 9.558804 analytic: 9.566079, relative error: 3.804143e-04
numerical: 20.450011 analytic: 20.451451, relative error: 3.521650e-05

向量化SVM

套循环肯定是最菜的做法，我们在处理图像的时候肯定都要用矩阵算的：

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def 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.                                                           #
  #############################################################################
  #scores (N,C)
  scores = X.dot(W)
  #num_classes = W.shape[1]
  num_train = X.shape[0]
  #利用np.arange(),correct_class_score变成了 (num_train,y)的矩阵
  correct_class_score = scores[np.arange(num_train),y]
  correct_class_score = np.reshape(correct_class_score,(num_train,-1))
  margins = scores - correct_class_score + 1
  margins = np.maximum(0, margins)
  #然后这里计算了j=y[i]的情形，所以把他们置为0
  margins[np.arange(num_train),y] = 0
  loss += np.sum(margins) / num_train
  loss += 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.                                                                     #
  #############################################################################
  margins[margins > 0] = 1
  #因为j=y[i]的那一个元素的grad要计算 >0 的那些次数次
  row_sum = np.sum(margins,axis=1)
  margins[np.arange(num_train),y] = -row_sum.T
  #把公式1和2合到一起计算了
  dW = np.dot(X.T,margins)
  dW /= num_train
  dW += reg * W
  #############################################################################
  #                             END OF YOUR CODE                              #
  #############################################################################

  return loss, dW

计算一下两者的时间差：

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Naive loss: 8.577034e+00 computed in 0.084761s
Vectorized loss: 8.577034e+00 computed in 0.001029s
difference: -0.000000
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Naive loss and gradient: computed in 0.082744s
Vectorized loss and gradient: computed in 0.002027s
difference: 0.000000

Stochastic Gradient Descent

编辑一下classifiers/linear_classifier/LinearClassifier.train()

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class LinearClassifier(object):

  def __init__(self):
    self.W = None

  def train(self, X, y, learning_rate=1e-3, reg=1e-5, num_iters=100,
            batch_size=200, verbose=False):
    """
    Train this linear classifier using stochastic gradient descent.

    Inputs:
    - X: A numpy array of shape (N, D) containing training data; there are N
      training samples each of dimension D.
    - y: A numpy array of shape (N,) containing training labels; y[i] = c
      means that X[i] has label 0 <= c < C for C classes.
    - learning_rate: (float) learning rate for optimization.
    - reg: (float) regularization strength.
    - num_iters: (integer) number of steps to take when optimizing
    - batch_size: (integer) number of training examples to use at each step.
    - verbose: (boolean) If true, print progress during optimization.

    Outputs:
    A list containing the value of the loss function at each training iteration.
    """
    num_train, dim = X.shape
    num_classes = np.max(y) + 1 # assume y takes values 0...K-1 where K is number of classes
    if self.W is None:
      # lazily initialize W
      self.W = 0.001 * np.random.randn(dim, num_classes)

    # Run stochastic gradient descent to optimize W
    loss_history = []
    for it in xrange(num_iters):
      X_batch = None
      y_batch = None

      #########################################################################
      # TODO:                                                                 #
      # Sample batch_size elements from the training data and their           #
      # corresponding labels to use in this round of gradient descent.        #
      # Store the data in X_batch and their corresponding labels in           #
      # y_batch; after sampling X_batch should have shape (dim, batch_size)   #
      # and y_batch should have shape (batch_size,)                           #
      #                                                                       #
      # Hint: Use np.random.choice to generate indices. Sampling with         #
      # replacement is faster than sampling without replacement.              #
      #########################################################################
      batch_inx = np.random.choice(num_train,batch_size)
      X_batch = X[batch_inx,:]
      y_batch = y[batch_inx]
      #########################################################################
      #                       END OF YOUR CODE                                #
      #########################################################################

      # evaluate loss and gradient
      loss, grad = self.loss(X_batch, y_batch, reg)
      loss_history.append(loss)

      # perform parameter update
      #########################################################################
      # TODO:                                                                 #
      # Update the weights using the gradient and the learning rate.          #
      #########################################################################
      self.W = self.W - learning_rate * grad
      #########################################################################
      #                       END OF YOUR CODE                                #
      #########################################################################

      if verbose and it % 100 == 0:
        print('iteration %d / %d: loss %f' % (it, num_iters, loss))

    return loss_history

再编辑一下predict函数

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def predict(self, X):
    """
    Use the trained weights of this linear classifier to predict labels for
    data points.

    Inputs:
    - X: A numpy array of shape (N, D) containing training data; there are N
      training samples each of dimension D.

    Returns:
    - y_pred: Predicted labels for the data in X. y_pred is a 1-dimensional
      array of length N, and each element is an integer giving the predicted
      class.
    """
    y_pred = np.zeros(X.shape[0])
    ###########################################################################
    # TODO:                                                                   #
    # Implement this method. Store the predicted labels in y_pred.            #
    ###########################################################################
    score = X.dot(self.W)
    y_pred = np.argmax(score,axis=1)
    ###########################################################################
    #                           END OF YOUR CODE                              #
    ###########################################################################
    return y_pred

得到预测值

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training accuracy: 0.376633
validation accuracy: 0.384000

然后调一调learning_rate和regularization:

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# Use the validation set to tune hyperparameters (regularization strength and
# learning rate). You should experiment with different ranges for the learning
# rates and regularization strengths; if you are careful you should be able to
# get a classification accuracy of about 0.4 on the validation set.
learning_rates = [1e-7, 3e-7,5e-7,9e-7]
regularization_strengths = [2.5e4, 1e4,3e4,2e4]

# results is dictionary mapping tuples of the form
# (learning_rate, regularization_strength) to tuples of the form
# (training_accuracy, validation_accuracy). The accuracy is simply the fraction
# of data points that are correctly classified.
results = {}
best_val = -1   # The highest validation accuracy that we have seen so far.
best_svm = None # The LinearSVM object that achieved the highest validation rate.

################################################################################
# TODO:                                                                        #
# Write code that chooses the best hyperparameters by tuning on the validation #
# set. For each combination of hyperparameters, train a linear SVM on the      #
# training set, compute its accuracy on the training and validation sets, and  #
# store these numbers in the results dictionary. In addition, store the best   #
# validation accuracy in best_val and the LinearSVM object that achieves this  #
# accuracy in best_svm.                                                        #
#                                                                              #
# Hint: You should use a small value for num_iters as you develop your         #
# validation code so that the SVMs don't take much time to train; once you are #
# confident that your validation code works, you should rerun the validation   #
# code with a larger value for num_iters.                                      #
################################################################################
for learning_rate in learning_rates:
    for regularization_strength in regularization_strengths:
        svm = LinearSVM()
        loss_hist = svm.train(X_train, y_train, learning_rate=learning_rate, reg=regularization_strength,
                      num_iters=1500, verbose=True)
        y_train_pred = svm.predict(X_train)
        y_val_pred = svm.predict(X_val)
        y_train_acc = np.mean(y_train_pred==y_train)
        y_val_acc = np.mean(y_val_pred==y_val)
        results[(learning_rate,regularization_strength)] = [y_train_acc, y_val_acc]
        if y_val_acc > best_val:
            best_val = y_val_acc
            best_svm = svm

################################################################################
#                              END OF YOUR CODE                                #
################################################################################
    
# Print out results.
for lr, reg in sorted(results):
    train_accuracy, val_accuracy = results[(lr, reg)]
    print('lr %e reg %e train accuracy: %f val accuracy: %f' % (
                lr, reg, train_accuracy, val_accuracy))
    
print('best validation accuracy achieved during cross-validation: %f' % best_val)

小结

  • 多看看cs231n的note文档
  • 多学习学习grad的推倒
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