作业一
作业内容:
实现k-NN,SVM分类器,Softmax分类器和两层神经网络,实践一个简单的图像分类流程。
多类支持向量机损失 Multiclass Support Vector Machine Loss:
SVM 的损失函数:
如何理解损失函数:
yi 代表正确类别的标签中第i个数据。举个例子来理解SVM的损失函数:假设有3个分类,并且得到了分值[13,-7,11]。其中第一个类别是正确类别,即yi =0;同时假设三角形间隔是10,上面的公式是将所有不正确分类加起来,所以我们得到两个部分:
可以看到第一个部分结果是0,这是因为[-7-13+10]得到的是负数,经过max函数处理后得到0。这一对类别分数和标签的损失值是0,这是因为正确分类的得分13与错误分类的得分-7的差为20,高于边界值10。而SVM只关心差距至少要大于10,更大的差值还是算作损失值为0。第二个部分计算[11-13+10]得到8。虽然正确分类的得分比不正确分类的得分要高(13>11),但是比10的边界值还是小了,分差只有2,这就是为什么损失值等于8。简而言之,SVM的损失函数想要正确分类类别[公式]的分数比不正确类别分数高,而且至少要高10。如果不满足这点,就开始计算损失值。
正则化:
我们希望能向某些特定的权重W添加一些偏好,对其他权重则不添加,以此来消除模糊性。这一点是能够实现的,方法是向损失函数增加一个正则化惩罚最常用的正则化惩罚是L2范式,L2范式通过对所有参数进行逐元素的平方惩罚来抑制大数值的权重:
代码
import random
import numpy as np
from cs231n.data_utils import load_CIFAR10
import matplotlib.pyplot as plt
# matplotlib 设置
%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'
%load_ext autoreload
%autoreload 2
# 加载 CIFAR-10 数据.
cifar10_dir = 'cs231n/datasets/CIFAR10'
# 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
X_train, y_train, X_test, y_test = load_CIFAR10(cifar10_dir)
# As a sanity check, we print out the size of the training and test data.
print('Training data shape: ', X_train.shape)
print('Training labels shape: ', y_train.shape)
print('Test data shape: ', X_test.shape)
print('Test labels shape: ', y_test.shape)
# 可视化一些训练样本
classes = ['plane', 'car', 'bird', 'cat', 'deer', 'dog', 'frog', 'horse', 'ship', 'truck']
num_classes = len(classes)
samples_per_class = 7
for y, cls in enumerate(classes):
idxs = np.flatnonzero(y_train == y)
idxs = np.random.choice(idxs, samples_per_class, replace=False)
for i, idx in enumerate(idxs):
plt_idx = i * num_classes + y + 1
plt.subplot(samples_per_class, num_classes, plt_idx)
plt.imshow(X_train[idx].astype('uint8'))
plt.axis('off')
if i == 0:
plt.title(cls)
plt.show()
划分训练集、验证集、测试集
# 将数据拆分为训练,验证集和测试集
#创建一个小的开发集作为训练数据的子集
num_training = 49000
num_validation = 1000
num_test = 1000
num_dev = 500
# 验证集.num_training到num_training + num_validation
mask = range(num_training, num_training + num_validation)
X_val = X_train[mask]
y_val = y_train[mask]
#训练集.
mask = range(num_training)
X_train = X_train[mask]
y_train = y_train[mask]
#开发集
mask = np.random.choice(num_training, num_dev, replace=False)
X_dev = X_train[mask]
y_dev = y_train[mask]
#测试集
mask = range(num_test)
X_test = X_test[mask]
y_test = y_test[mask]
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)
# 转换shape将图像中的像素变成一行数据 样本*特征
X_train = np.reshape(X_train, (X_train.shape[0], -1))
X_val = np.reshape(X_val, (X_val.shape[0], -1))
X_test = np.reshape(X_test, (X_test.shape[0], -1))
X_dev = np.reshape(X_dev, (X_dev.shape[0], -1))
# 输出
print('Training data shape: ', X_train.shape)
print('Validation data shape: ', X_val.shape)
print('Test data shape: ', X_test.shape)
print('dev data shape: ', X_dev.shape)
预处理阶段
# 预处理:减去平均图像
# 首先:根据训练数据计算图像均值
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()
# 第二:从训练和测试数据中减去平均图像
X_train -= mean_image
X_val -= mean_image
X_test -= mean_image
X_dev -= mean_image
# 第三:附加一个偏置项,以便我们的SVM
# 仅需担心优化单个权重矩阵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)
分类器:损失函数
from cs231n.classifiers.linear_svm import svm_loss_naive
import time
# 初始化随机权重
W = np.random.randn(3073, 10) * 0.0001
loss, grad = svm_loss_naive(W, X_dev, y_dev, 0.000005)
print('loss: %f' % (loss, ))
# 计算梯度
loss, grad = svm_loss_naive(W, X_dev, y_dev, 0.0)
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)
# 梯度检测
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)
比较两种方式:1.向量化 2。矩阵化
# 向量、矩阵形式损失函数,并比较时间
tic = time.time()
loss_naive, grad_naive = svm_loss_naive(W, X_dev, y_dev, 0.000005)
toc = time.time()
print('Naive loss: %e computed in %fs' % (loss_naive, toc - tic))
from cs231n.classifiers.linear_svm import svm_loss_vectorized
tic = time.time()
loss_vectorized, _ = svm_loss_vectorized(W, X_dev, y_dev, 0.000005)
toc = time.time()
print('Vectorized loss: %e computed in %fs' % (loss_vectorized, toc - tic))
print('difference: %f' % (loss_naive - loss_vectorized))
# 向量、矩阵形式梯度,并比较时间
tic = time.time()
_, grad_naive = svm_loss_naive(W, X_dev, y_dev, 0.000005)
toc = time.time()
print('Naive loss and gradient: computed in %fs' % (toc - tic))
tic = time.time()
_, grad_vectorized = svm_loss_vectorized(W, X_dev, y_dev, 0.000005)
toc = time.time()
print('Vectorized loss and gradient: computed in %fs' % (toc - tic))
difference = np.linalg.norm(grad_naive - grad_vectorized, ord='fro')
print('difference: %f' % difference)
训练阶段
from cs231n.classifiers import LinearSVM
svm = LinearSVM()
tic = time.time()
loss_hist = svm.train(X_train, y_train, learning_rate=1e-7, reg=2.5e4,
num_iters=1500, verbose=True)
toc = time.time()
print('That took %fs' % (toc - tic))
# 画出图像
plt.plot(loss_hist)
plt.xlabel('Iteration number')
plt.ylabel('Loss value')
plt.show()
预测阶段
# 预测
y_train_pred = svm.predict(X_train)
print('training accuracy: %f' % (np.mean(y_train == y_train_pred), ))
y_val_pred = svm.predict(X_val)
print('validation accuracy: %f' % (np.mean(y_val == y_val_pred), ))
使用验证集调整超参数
# 使用验证集调整超参数(正则化参数和学习率)
learning_rates = [1e-7, 5e-5]
regularization_strengths = [2.5e4, 5e4]
results = {}
best_val = -1
best_svm = None
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=False)
y_train_pred = svm.predict(X_train)
training_accuracy = np.mean(y_train == y_train_pred)
y_val_pred = svm.predict(X_val)
validation_accuracy = np.mean(y_val == y_val_pred)
results[(learning_rate, regularization_strength)] = (training_accuracy, validation_accuracy)
if best_val < validation_accuracy:
best_val = validation_accuracy
best_svm = svm
# 打印结果
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)
# 可视化交叉验证结果
import math
x_scatter = [math.log10(x[0]) for x in results]
y_scatter = [math.log10(x[1]) for x in results]
# 训练集精度
marker_size = 100
colors = [results[x][0] for x in results]
plt.subplot(2, 1, 1)
plt.scatter(x_scatter, y_scatter, marker_size, c=colors, cmap=plt.cm.coolwarm)
plt.colorbar()
plt.xlabel('log learning rate')
plt.ylabel('log regularization strength')
plt.title('CIFAR-10 training accuracy')
# 验证集精度
colors = [results[x][1] for x in results] # default size of markers is 20
plt.subplot(2, 1, 2)
plt.scatter(x_scatter, y_scatter, marker_size, c=colors, cmap=plt.cm.coolwarm)
plt.colorbar()
plt.xlabel('log learning rate')
plt.ylabel('log regularization strength')
plt.title('CIFAR-10 validation accuracy')
plt.show()
测试集
# 测试集
y_test_pred = best_svm.predict(X_test)
test_accuracy = np.mean(y_test == y_test_pred)
print('linear SVM on raw pixels final test set accuracy: %f' % test_accuracy)
可视化每类权重
# 可视化每类权重
w = best_svm.W[:-1,:] # strip out the bias
w = w.reshape(32, 32, 3, 10)
w_min, w_max = np.min(w), np.max(w)
classes = ['plane', 'car', 'bird', 'cat', 'deer', 'dog', 'frog', 'horse', 'ship', 'truck']
for i in range(10):
plt.subplot(2, 5, i + 1)
# 把权重映射到0-255
wimg = 255.0 * (w[:, :, :, i].squeeze() - w_min) / (w_max - w_min)
plt.imshow(wimg.astype('uint8'))
plt.axis('off')
plt.title(classes[i])
linear_classifier.py
from __future__ import print_function
from builtins import range
from builtins import object
import numpy as np
from cs231n.classifiers.linear_svm import *
from cs231n.classifiers.softmax import *
from past.builtins import xrange
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 range(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 (batch_size, dim) #
# 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. #
#########################################################################
# *****START OF YOUR CODE (DO NOT DELETE/MODIFY THIS LINE)*****
indices = np.random.choice(num_train,batch_size,replace = False)
X_batch = X[indices]
y_batch = y[indices]
# *****END OF YOUR CODE (DO NOT DELETE/MODIFY THIS LINE)*****
# 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. #
#########################################################################
# *****START OF YOUR CODE (DO NOT DELETE/MODIFY THIS LINE)*****
loss, grad = self.loss(X_batch, y_batch, reg)
loss_history.append(loss)
self.W -= learning_rate*grad
# *****END OF YOUR CODE (DO NOT DELETE/MODIFY THIS LINE)*****
if verbose and it % 100 == 0:
print('iteration %d / %d: loss %f' % (it, num_iters, loss))
return loss_history
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. #
###########################################################################
# *****START OF YOUR CODE (DO NOT DELETE/MODIFY THIS LINE)*****
scores = X.dot(self.W)
y_pred = np.argsort(scores,axis = 1)[:,-1]
# *****END OF YOUR CODE (DO NOT DELETE/MODIFY THIS LINE)*****
return y_pred
def loss(self, X_batch, y_batch, reg):
"""
Compute the loss function and its derivative.
Subclasses will override this.
Inputs:
- X_batch: A numpy array of shape (N, D) containing a minibatch of N
data points; each point has dimension D.
- y_batch: A numpy array of shape (N,) containing labels for the minibatch.
- reg: (float) regularization strength.
Returns: A tuple containing:
- loss as a single float
- gradient with respect to self.W; an array of the same shape as W
"""
pass
class LinearSVM(LinearClassifier):
""" A subclass that uses the Multiclass SVM loss function """
def loss(self, X_batch, y_batch, reg):
return svm_loss_vectorized(self.W, X_batch, y_batch, reg)
class Softmax(LinearClassifier):
""" A subclass that uses the Softmax + Cross-entropy loss function """
def loss(self, X_batch, y_batch, reg):
return softmax_loss_vectorized(self.W, X_batch, y_batch, reg)
linear_svm.py
from builtins import range
import numpy as np
from random import shuffle
from past.builtins import xrange
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
for i in range(num_train):
scores = X[i].dot(W)
correct_class_score = scores[y[i]]
for j in range(num_classes):
if j == y[i]:
continue
margin = scores[j] - correct_class_score + 1 # note delta = 1
if margin > 0:
loss += margin
# 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
# Add regularization to the loss.
loss += reg * np.sum(W * 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. #
#############################################################################
# *****START OF YOUR CODE (DO NOT DELETE/MODIFY THIS LINE)*****
for i in range(num_train):
scores = X[i].dot(W)
index = y[i]
for j in range(num_classes):
if j != index and scores[j] >= scores[index]-1:
dW[:,j] += X[i]
dW[:,index] -= X[i]
dW /= num_train
dW += 2*reg*W
# *****END OF YOUR CODE (DO NOT DELETE/MODIFY THIS LINE)*****
return loss, dW
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. #
#############################################################################
# *****START OF YOUR CODE (DO NOT DELETE/MODIFY THIS LINE)*****
num_train = X.shape[0]
scores = X.dot(W)
rightscore = scores[np.arange(num_train), y]
scores = scores - rightscore.reshape(-1, 1) + 1.
scores[scores <= 0] = 0
scores[np.arange(num_train), y] = 0
loss = np.sum(scores)/num_train
# *****END OF YOUR CODE (DO NOT DELETE/MODIFY THIS LINE)*****
#############################################################################
# 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. #
#############################################################################
# *****START OF YOUR CODE (DO NOT DELETE/MODIFY THIS LINE)*****
C = W.shape[1]
mask = np.zeros((num_train, C))
scores[scores > 0] = 1
mask += scores
mask[np.arange(num_train), y] = -np.sum(scores,axis = 1)
dW = X.T.dot(mask)/num_train
dW += 2*reg*W
# *****END OF YOUR CODE (DO NOT DELETE/MODIFY THIS LINE)*****
return loss, dW
参考资料
1.https://blog.csdn.net/Zhuanzhu22nian/article/details/92018949
2.周志华机器学习
3.https://zhuanlan.zhihu.com/p/21930884