title: ‘cs231n作业：assignment1 - svm’
id: cs231n-1h-2
tags:
- cs231n
- homework
categories: - AI
- Deep Learning
date: 2018-09-27 14:17:45
GitHub地址：https://github.com/ZJUFangzh/cs231n
个人博客：fangzh.top
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 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()
# 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
# 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。
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
第i个数据图像为 ，正确分类为 ，然后根据 来计算不同分类的值，将分类简写为 ，那么第j类的得分就是 ，针对第i个数据的多类SVM的损失函数定义为：
如：假设有3个分类， ，第一个分类是正确的，也就是 ，假设 ，那么把所有不正确的分类加起来( )，
因为SVM只关心差距至少要大于10，所以
那么把公式套入：
加入正则后：
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：
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_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)
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：
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 0.084761s
Vectorized loss: 8.577034e+00 computed in 0.001029s
difference: -0.000000
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()
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: 0.376633
validation 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
- 多看看cs231n的note文档
- 多学习学习grad的推倒