上一章完成了一個(gè)KNN Classifier兴猩，這一章就來(lái)到了熟悉又陌生的SVM...感覺自己雖然以前用過(guò)SVM鞭衩，但是從來(lái)沒有真正搞懂過(guò)敲茄，就著這門好課鞏固一下吧充包！
1. Preprocessing
和上一章不同的是秕岛，先visualize mean image:
然后從所有image中減去這個(gè)mean image，這個(gè)數(shù)據(jù)預(yù)處理過(guò)程是為了統(tǒng)一數(shù)據(jù)的量級(jí)误证。對(duì)于圖像而言，像素值一定在0-255之間修壕，所以直接減去mean就可以了愈捅，但是如果是其他數(shù)據(jù)，通常用標(biāo)準(zhǔn)化或者歸一化來(lái)做慈鸠。下面代碼為預(yù)處理過(guò)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, 這里的delta設(shè)置為１（即SVM所要求的間隔）：
naive implementation
首先看一下naive的for loop解法蓝谨，只要正常計(jì)算 dL/dW就可以了，這個(gè)沒什么難度。需要稍微解釋一下的是譬巫，代碼中的dW其實(shí)是實(shí)際意義的dL/dW咖楣。
按照求導(dǎo)表達(dá)式為：
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
接下來(lái)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
一開始我自己實(shí)現(xiàn)的代碼是代碼中我說(shuō)我不知道哪里錯(cuò)了的部分..到現(xiàn)在我也覺得是對(duì)的，還沒有看出到底哪里錯(cuò)了芦昔，如果有人剛好看到這篇文章愿意指正出來(lái)诱贿，我會(huì)非常感謝您。第二種方法是在github上看到的一種方法咕缎，也很巧妙珠十，搬過(guò)來(lái)用了發(fā)現(xiàn)代碼跑的結(jié)果是對(duì)的，但是還是不明白為什么自己那個(gè)方法是錯(cuò)的QAQ...
3. Stochastic Gradient Descent (SGD)
每一次迭代訓(xùn)練的時(shí)候隨機(jī)選取一個(gè)batch_size的數(shù)據(jù)個(gè)數(shù)凭豪。在給定的樣本集合M中焙蹭，隨機(jī)取出副本N代替原始樣本M來(lái)作為全集，對(duì)模型進(jìn)行訓(xùn)練嫂伞，這種訓(xùn)練由于是抽取部分?jǐn)?shù)據(jù)孔厉，所以有較大的幾率得到的是，一個(gè)局部最優(yōu)解帖努，但是一個(gè)明顯的好處是撰豺，如果在樣本抽取合適范圍內(nèi)，既會(huì)求出結(jié)果然磷，而且速度還快郑趁。這個(gè)理解摘自：http://www.cnblogs.com/gongxijun/p/5890548.html
順便發(fā)現(xiàn)了一個(gè)這個(gè)課程的bug，就是前面單純實(shí)現(xiàn)svm的optimization的時(shí)候和后面做SGD的時(shí)候要求輸入的數(shù)據(jù)維度是反著的……所以做這里的時(shí)候還把前面給改了……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的反方向調(diào)整W!你想啊姿搜，這個(gè)grad代表的意義是寡润，每正向改變w的值，會(huì)造成最終的目標(biāo)函數(shù)有多大的改變舅柜。比如這個(gè)grad是一個(gè)正數(shù)梭纹，那么自變量就是正向作用，它越大則目標(biāo)函數(shù)值越大致份，那么我們?yōu)榱说玫綐O小值变抽，是不是就應(yīng)該按照反方向來(lái)對(duì)權(quán)值（自變量）進(jìn)行調(diào)整呢？恩～
4. Play with hyperparameters
至于怎么決定那些learning_rate, regularization_parameter的大小氮块，就是用cross-validation集做驗(yàn)證的事了绍载，將不同的參數(shù)用train來(lái)訓(xùn)練好后用X_val, y_val來(lái)做驗(yàn)證，然后選出正確率最高的一組參數(shù)滔蝉。這里就不細(xì)說(shuō)了击儡，一些dirty work...最后的正確率也就是0.35-0.4罷了，這個(gè)svm的單層訓(xùn)練器的有效性可想而知嘛蝠引。想說(shuō)的是最后的一步可視化：
的確是阳谍，很難看呢Ｖ瘛！矫夯！
5 Hinge Loss
The Hinge Loss 定義為 E(z) = max(0,1-z)鸽疾。Hinge loss是凸的，但是因?yàn)閷?dǎo)數(shù)不連續(xù)训貌，還有一些變種制肮，比如 Squared Hing Loss (L2 SVM)。Hinge loss是下圖的綠線旺订。黑線是0-1函數(shù)弄企，紅線是log loss（基于最大似然的負(fù)log）。
一般來(lái)講区拳，Hinge于soft-margin svm算法拘领；log于LR算法（Logistric Regression）；squared loss樱调，也就是最小二乘法约素，于線性回歸（Liner Regression）；基于指數(shù)函數(shù)的loss于Boosting笆凌。
