任务：
- 完成一个基于SVM的全向量化损失函数
- 完成解析梯度的全向量化表示
- 用数值梯度来验证
- 使用一个验证集去调优 learning rate 和 regularization
- 使用SGD方法
- 可视化最优的Weight
SVM的 loss function 为：
- 首先，加载原始数据并做简单的数据划分
import random
import numpy as np
from cs231n.data_utils import load_CIFAR10
import matplotlib.pyplot as plt
from cs231n.classifiers.linear_svm import svm_loss_naive
from cs231n.gradient_check import grad_check_sparse
import time
cifar10_dir = 'F:\pycharmFile\KNN\cifar-10-batches-py'
X_train, y_train, X_test, y_test = load_CIFAR10(cifar10_dir)
# 将数据分割为训练集， 验证集， 测试集， 开发集
num_training = 49000
num_validation = 1000
num_test = 1000
num_dev = 500
mask = range(num_training, num_training + num_validation)
X_val = X_train[mask]
y_val = y_train[mask]
mask = range(num_training)
X_train = X_train[mask]
y_train = y_train[mask]
# 在num_training中不放回地取出num_dev个
mask = np.random.choice(num_training, num_dev, replace=False)
X_dev = X_train[mask]
y_dev = y_train[mask]
mask = range(num_test)
X_test = X_test[mask]
y_test = y_test[mask]
这里的开发集的作用：拿出一小部分数据用于做一些基础的验证
- 数据预处理
特征缩放，将 bias 加入到数据集中
# 转换成二维数据
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))
# 预处理，减去图像的平均值
# 取每一列的平均值
mean_image = np.mean(X_train, axis=0)
# print(mean_image.shape)
# print(mean_image[:10])
# plt.figure(figsize=(4, 4))
# plt.imshow(mean_image.reshape((32,32,3)).astype('uint8')) # visualize the mean image
# plt.show()
# 训练集和测试集图像分别减去均值
X_train -= mean_image
X_val -= mean_image
X_test -= mean_image
X_dev -= mean_image
# 最后，在X中添加一列1作为bias，这样我们在优化时只考虑一个权重矩阵W即可
# np.hstack() 水平(按列顺序)把数组给堆叠起来
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)
- 下面，开始做SVM
先设置极小的W，所得的score也是接近于0，delta = 1， 所以loss的值应该约等于 分类数 - 1 = 9
# # SVM
# # generate a random SVM weight matrix of small numbers
W = np.random.randn(3073, 10) * 0.0001 # 生成一个3073 * 10的矩阵
# loss, grad = svm_loss_naive(W, X_dev, y_dev, 0.000005)
# # 检验loss function是否有错
# # 预期的loss == 9 (即10-1)
# print('loss: %f' % (loss, ))svm_loss_native在linear_svm.py中
这里是使用循环实现 svm loss，效率很低
loss function 加入 regularization 项
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
# 在loss function上分别对 W[Y[i]] 和 W[j] 求导得
dW[:,y[i]] += -X[i,:].T
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- 接下来，用数值梯度来验证上面所得结果
# 比较数值梯度和解析梯度
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
print('turn on reg')
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)grad_check_sparse在 gradient_check.py 中
def grad_check_sparse(f, x, analytic_grad, num_checks=10, h=1e-5):
"""
sample a few random elements and only return numerical
in this dimensions.
"""
for i in range(num_checks):
# 随机取出 W 中的一个数
ix = tuple([randrange(m) for m in x.shape])
# 计算数值梯度
oldval = x[ix]
x[ix] = oldval + h # increment by h
fxph = f(x) # evaluate f(x + h)
x[ix] = oldval - h # increment by h
fxmh = f(x) # evaluate f(x - h)
x[ix] = oldval # reset
# 计算数值梯度和解析梯度的差值
grad_numerical = (fxph - fxmh) / (2 * h)
grad_analytic = analytic_grad[ix]
rel_error = abs(grad_numerical - grad_analytic) / (abs(grad_numerical) + abs(grad_analytic))
print('numerical: %f analytic: %f, relative error: %e' % (grad_numerical, grad_analytic, rel_error))
- 完成基于SVM的全向量化损失函数，完成解析梯度的全向量化表示
# 完成一个基于SVM的全向量化损失函数 loss
# 完成解析梯度的全向量化表示 grad
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))
# The losses should match but your vectorized implementation should be much faster.
print('difference: %f' % (loss_naive - loss_vectorized))
# Complete the implementation of svm_loss_vectorized, and compute the gradient
# of the loss function in a vectorized way.
# The naive implementation and the vectorized implementation should match, but
# the vectorized version should still be much faster.
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))
# The loss is a single number, so it is easy to compare the values computed
# by the two implementations. The gradient on the other hand is a matrix, so
# we use the Frobenius norm to compare them.
difference = np.linalg.norm(grad_naive - grad_vectorized, ord='fro')
print('difference: %f' % difference)svm_loss_vectorized 在 linear_svm.py中实现
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. #
#############################################################################
# 实现结构化SVM损失函数的向量版本
scores = X.dot(W) # 得到一个 num_train * 10的矩阵
num_classes = W.shape[1]
num_train = X.shape[0]
scores_correct = scores[np.arange(num_train), y] # 用一个list小技巧直接取出所有的 scores[y[i]]
scores_correct = np.reshape(scores_correct, (num_train, -1))
margins = scores - scores_correct + 1
margins = np.maximum(0, margins)
margins[np.arange(num_train), y] = 0
loss += np.sum(margins) / num_train
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. #
#############################################################################
margins[margins > 0] = 1
row_sum = np.sum(margins, axis=1)
margins[np.arange(num_train), y] = -row_sum
dW += np.dot(X.T, margins) / num_train + reg * W
#############################################################################
# END OF YOUR CODE #
#############################################################################
return loss, dW
- 使用SGD，进行训练
# SGD
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loss曲线
plt.plot(loss_hist)
plt.xlabel('Iteration number')
plt.ylabel('Loss value')
plt.show()
# 查看正确率
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), ))画出损失曲线
训练过程在 linear_classifier.py 中的 train函数中， 在predict函数中预测
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 (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
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. #
###########################################################################
pred = np.dot(X, self.W)
y_pred = np.argmax(pred,axis=1)
###########################################################################
# END OF YOUR CODE #
###########################################################################
return y_pred- 接下来，调优超参 ( learning rate 和 regularization )
找出最好的 learning rate 和 regularization，构建最好的svm（W）
# 使用验证集去调超参数（lr, reg）
learning_rates = [2e-7, 1.75e-7, 1.5e-7, 1.25e-7, 1e-7, 0.75e-7]
regularization_strengths = [2e4, 2.5e4, 2.75e4, 3e4, 3.25e4, 3.5e4, 3.75e4, 4e4, 4.25e4]
results = {}
best_val = -1
best_svm = None
# 验证过程
for rate in learning_rates:
for regular in regularization_strengths:
svm = LinearSVM()
svm.train(X_train, y_train, learning_rate=rate, reg=regular, num_iters=1000) # num_iters可以设小一点
y_train_pred = svm.predict(X_train)
acc_train = np.mean(y_train_pred == y_train)
y_val_pred = svm.predict(X_val)
acc_val = np.mean(y_val_pred == y_val)
results[(rate, regular)] = (acc_train, acc_val)
if best_val < acc_val:
best_val = acc_val
best_svm = svm
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)可视化调参结果
import math
x_scatter = [math.log10(x[0]) for x in results]
y_scatter = [math.log10(x[1]) for x in results]
# plot training accuracy
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)
plt.colorbar()
plt.xlabel('log learning rate\n')
plt.ylabel('log regularization strength')
plt.title('CIFAR-10 training accuracy')
# plot validation accuracy
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)
plt.colorbar()
plt.xlabel('log learning rate')
plt.ylabel('log regularization strength')
plt.title('\nCIFAR-10 validation accuracy')
plt.show()
- 使用最优的 svm 测试
# Evaluate the best svm on test set
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)
- 可视化最优的Weight
# 最后，取出 W 中的“标准模板”查看一下
w = best_svm.W[:-1, :] # 丢掉 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)
# Rescale the weights to be between 0 and 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])
plt.show()
可以在参数W模板中看出识别目标大致的样子