cs231n作业：assignment1 - svm

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title: ‘cs231n作业：assignment1 - svm’

id: cs231n-1h-2

tags:

cs231n

homework
categories:

Deep Learning
date: 2018-09-27 14:17:45

GitHub地址：

个人博客：

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:

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：

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

# Preprocessing: subtract the mean image# first: compute the image mean based on the training datamean_image = np.mean(X_train, axis=0)print(mean_image[:10]) # print a few of the elementsplt.figure(figsize=(4,4))plt.imshow(mean_image.reshape((32,32,3)).astype('uint8')) # visualize the mean imageplt.show()

# second: subtract the mean image from train and test dataX_train -= mean_imageX_val -= mean_imageX_test -= mean_imageX_dev -= mean_image

# 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)

(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

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( $\neq y_i$ )，

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

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：

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

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

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：

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:

# 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_sparsef = 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)

numerical: 34.663598 analytic: 34.663598, relative error: 6.995024e-13numerical: 21.043334 analytic: 21.043334, relative error: 5.147242e-12numerical: 1.334055 analytic: 1.334055, relative error: 5.315420e-11numerical: 16.611704 analytic: 16.611704, relative error: 6.908581e-12numerical: 25.327188 analytic: 25.327188, relative error: 1.552987e-11numerical: -12.867717 analytic: -12.867717, relative error: 1.966004e-11numerical: 15.066285 analytic: 15.066285, relative error: 7.012975e-12numerical: -3.752014 analytic: -3.752014, relative error: 7.502607e-11numerical: 9.927043 analytic: 9.927043, relative error: 9.010584e-13numerical: 33.071345 analytic: 33.071345, relative error: 1.305438e-12numerical: -19.227144 analytic: -19.227851, relative error: 1.836495e-05numerical: 31.392728 analytic: 31.391611, relative error: 1.778034e-05numerical: -10.450509 analytic: -10.456860, relative error: 3.037629e-04numerical: -1.346690 analytic: -1.345625, relative error: 3.953276e-04numerical: 7.843501 analytic: 7.846486, relative error: 1.902216e-04numerical: 20.635011 analytic: 20.628368, relative error: 1.609761e-04numerical: 23.654254 analytic: 23.652696, relative error: 3.294745e-05numerical: 37.706709 analytic: 37.703260, relative error: 4.573495e-05numerical: 9.558804 analytic: 9.566079, relative error: 3.804143e-04numerical: 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：

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：

Naive loss: 8.577034e+00 computed in 0.084761sVectorized loss: 8.577034e+00 computed in 0.001029sdifference: -0.000000

Naive loss and gradient: computed in 0.082744sVectorized loss and gradient: computed in 0.002027sdifference: 0.000000

Stochastic Gradient Descent

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

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

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

training accuracy: 0.376633validation accuracy: 0.384000

然后调一调learning_rate和regularization:

# 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