上一章完成了一个KNN Classifier，这一章就来到了熟悉又陌生的SVM...感觉自己虽然以前用过SVM，但是从来没有真正搞懂过，就着这门好课巩固一下吧！
1. Preprocessing
和上一章不同的是，先visualize mean image:
Paste_Image.png
然后从所有image中减去这个mean image，这个数据预处理过程是为了统一数据的量级。对于图像而言，像素值一定在0-255之间，所以直接减去mean就可以了，但是如果是其他数据，通常用标准化或者归一化来做。下面代码为预处理过程：
# 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
增加bias
# 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
2. implement a fully-vectorized loss function for the SVM
注意SVM的loss function, 这里的delta设置为１（即SVM所要求的间隔）：
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naive implementation
首先看一下naive的for loop解法，只要正常计算 dL/dW就可以了，这个没什么难度。需要稍微解释一下的是，代码中的dW其实是实际意义的dL/dW。
按照求导表达式为：
dW[:,j] += X[i].transpose()
dW[:,y[i]] -= X[i]
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 xrange(num_train):
scores = X[i].dot(W)
correct_class_score = scores[y[i]]
for j in xrange(num_classes):
if j == y[i]:
continue
margin = scores[j] - correct_class_score + 1 # note delta = 1
# dW[:,j] += X[i].transpose()
if margin > 0:
loss += margin
dW[:,j] += X[i].transpose()
dW[:,y[i]] -= X[i]
# 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 += 0.5 * reg * np.sum(W * W)
dW += reg * np.sum(W)
return loss, dW
vectorized implementation
接下来看vectorized version：
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
num_train = X.shape[0]
num_classes = W.shape[1]
#############################################################################
# TODO: #
# Implement a vectorized version of the structured SVM loss, storing the #
# result in loss. #
#############################################################################
scores = X.dot(W)
# true labels
s_yi = scores[np.arange(num_train), y]
mat = scores - np.tile(s_yi, (num_classes,1)).transpose() + 1
loss_mat = np.maximum(np.zeros((num_train, num_classes)), mat)
# loss_mat[loss_mat<0] = 0 # this worked out as well
loss_mat[np.arange(num_train), y] = 0
loss = np.sum(loss_mat)/num_train
loss += 0.5 * reg * np.sum(W * W)
#############################################################################
# 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. #
#############################################################################
# I don't know what's wrong about the following commented code
#############################################################################
# loss_pos = np.array(np.nonzero(loss_mat))
# print loss_pos, loss_pos.shape
# dW[ :, y[loss_pos[0,:]] ] -= X[ loss_pos[0,:],: ].transpose()
# dW[ :, loss_pos[1,:] ] += X[ loss_pos[0,:],: ].transpose()
# dW /= num_train
# dW += reg * W
# Binarize into integers
binary = loss_mat
binary[loss_mat > 0] = 1
# Perform the two operations simultaneously
# (1) for all j: dW[j,:] = sum_{i, j produces positive margin with i} X[:,i].T
# (2) for all i: dW[y[i],:] = sum_{j != y_i, j produces positive margin with i} -X[:,i].T
col_sum = np.sum(binary, axis=1)
binary[range(num_train), y] = -col_sum[range(num_train)]
dW = np.dot(X.T, binary)
# Divide
dW /= num_train
# Regularize
dW += reg*W
return loss, dW
一开始我自己实现的代码是代码中我说我不知道哪里错了的部分..到现在我也觉得是对的，还没有看出到底哪里错了，如果有人刚好看到这篇文章愿意指正出来，我会非常感谢您。第二种方法是在github上看到的一种方法，也很巧妙，搬过来用了发现代码跑的结果是对的，但是还是不明白为什么自己那个方法是错的QAQ...
3. Stochastic Gradient Descent (SGD)
每一次迭代训练的时候随机选取一个batch_size的数据个数。在给定的样本集合M中，随机取出副本N代替原始样本M来作为全集，对模型进行训练，这种训练由于是抽取部分数据，所以有较大的几率得到的是，一个局部最优解，但是一个明显的好处是，如果在样本抽取合适范围内，既会求出结果，而且速度还快。这个理解摘自：http://www.cnblogs.com/gongxijun/p/5890548.html
顺便发现了一个这个课程的bug，就是前面单纯实现svm的optimization的时候和后面做SGD的时候要求输入的数据维度是反着的……所以做这里的时候还把前面给改了……anyway……
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. #
#########################################################################
num_random = np.random.choice(num_train, batch_size, replace=True)
X_batch = X[num_random, :].transpose()
# print X_batch.shape
y_batch = y[num_random]
#########################################################################
# 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 += -grad * learning_rate
#########################################################################
# END OF YOUR CODE #
#########################################################################
if verbose and it % 100 == 0:
print 'iteration %d / %d: loss %f' % (it, num_iters, loss)
return loss_history
注意按照grad的反方向调整W!你想啊，这个grad代表的意义是，每正向改变w的值，会造成最终的目标函数有多大的改变。比如这个grad是一个正数，那么自变量就是正向作用，它越大则目标函数值越大，那么我们为了得到极小值，是不是就应该按照反方向来对权值（自变量）进行调整呢？恩～
Paste_Image.png
4. Play with hyperparameters
至于怎么决定那些learning_rate, regularization_parameter的大小，就是用cross-validation集做验证的事了，将不同的参数用train来训练好后用X_val, y_val来做验证，然后选出正确率最高的一组参数。这里就不细说了，一些dirty work...最后的正确率也就是0.35-0.4罢了，这个svm的单层训练器的有效性可想而知嘛。想说的是最后的一步可视化：
Paste_Image.png
的确是，很难看呢！！！
5 Hinge Loss
The Hinge Loss 定义为 E(z) = max(0,1-z)。Hinge loss是凸的，但是因为导数不连续，还有一些变种，比如 Squared Hing Loss (L2 SVM)。Hinge loss是下图的绿线。黑线是0-1函数，红线是log loss（基于最大似然的负log）。
4DFDU.png
一般来讲，Hinge于soft-margin svm算法；log于LR算法（Logistric Regression）；squared loss，也就是最小二乘法，于线性回归（Liner Regression）；基于指数函数的loss于Boosting。
