cs231n svm作业笔记
1.线性分类基本概念
现在假设有一个包含很多图像的训练集
，每个图像都有一个对应的分类标签
。对到
构建一个简单的映射 ：
假设图像只有4个像素（都是黑白像素，这里不考虑RGB通道），有3个分类（红色代表猫，绿色代表狗，蓝色代表船，注意，这里的红、绿和蓝3种颜色仅代表分类，和RGB通道没有关系），列如
2.svm损失函数
代表错误标签通过
预测所得值，
代表正确标签所得值，
是超参数项，以猫猫图为例：
正则化项 ：
完整公式为：
3.svm损失函数梯度代码
位于cs231n/classifiers/linear_svm.py文件中，分为标准版与向量版：
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] #种类数
num_train = X.shape[0] #训练集图片数
loss = 0.0 #损失值
for i in range(num_train):
scores = X[i].dot(W)
correct_class_score = scores[y[i]] #正确标签得分
for j in range(num_classes):
if j == y[i]:
continue #跳过正确标签
margin = scores[j] - correct_class_score + 1 # note delta = 1
# 损失值计算
if margin > 0: # max计算
loss += margin
dW[:, j] += X[i]*1.0 # 错误标签梯度计算
dW[:, y[i]] -= X[i]*1.0 # 正确标签梯度计算
# 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) 正则化项计算
#############################################################################
# TODO: #
# Compute the gradient of the loss function and store it dW. #
# Rather than 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)*****
'''
# verbose version:
# try exchanging the loop of k and j !
for i in range(num_train):
scores = X[i].dot(W)
for k in range(num_classes):
if k == y[i]:
for j in range(num_classes):
if j == y[i]:
continue
margin = scores[j] - scores[y[i]] + 1
if margin > 0:
dW[:, k] -= X[i]*1.0
else:
margin = scores[k] - scores[y[i]] + 1
if margin > 0:
dW[:, k] += X[i]*1.0
dW = dW / num_train + 2*reg*W
'''
dW /= num_train
dW += 2*reg*W #正则化梯度项
pass
# *****END OF YOUR CODE (DO NOT DELETE/MODIFY THIS LINE)*****
return loss, dW向量形式为：
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)
margin = np.maximum(0, scores.T - scores[range(num_train), y] + 1).T # note delta = 1
margin[range(num_train), y] = 0 #相同标签置0
data_loss = np.sum(margin) * 1.0 / num_train
reg_loss = reg*np.sum(np.square(W)) #正则化值
loss = data_loss + reg_loss
pass
# *****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)*****
X_effect = (margin > 0).astype('float')
# 每个样本i在非y[i]的类上产生X[i]的梯度 ,si的梯度为1
X_effect[range(num_train), y] -= np.sum(X_effect, axis=1)
# 每个样本i在y[i]的类上产生sigma(margin gt 0)*X[i]（除y[i]的margin）的梯度
dW = X.T.dot(X_effect)
dW /= num_train
dW += 2*reg*W
''' verbose version:
margin_chara = (margin > 0).astype('float')
margin_chara_sum = np.sum(margin_chara, axis=1).astype('float')
for i in range(num_train):
dW += (margin_chara[i][:, np.newaxis]*X[i]).T # broadcast
# dW[:, y[i]] -= margin_chara[i, y[i]]*X[i] # margin_chara[i, y[i]] == 0 is always
dW[:, y[i]] -= margin_chara_sum[i]*X[i]
dW /= num_train
dW += 2*reg*W
'''
pass
# *****END OF YOUR CODE (DO NOT DELETE/MODIFY THIS LINE)*****
return loss, dW4.LinearClassifier代码：
该部分主要用于实现SGD（随机梯度下降）
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 # 假设 y 取值 0...K-1 其中 K 是类数
if self.W is None:
# lazily 初始化W
self.W = 0.001 * np.random.randn(dim, num_classes)
# 运行随机梯度下降以优化 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)*****
batch_idx = np.random.choice(num_train, size=batch_size, replace=False)
# 在num_train数目中随机采样batch_size个元素，分别存储在X_batch和y_batch中
X_batch = X[batch_idx]
y_batch = y[batch_idx]
pass
# *****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)*****
self.W -= learning_rate*grad # 进行梯度下降更新
pass
# *****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 (either training, validating or testing 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)*****
# be careful about self.W.shape = (dim, num_classes) !!
y_pred = np.argmax(X.dot(self.W), axis=1)
# argmax查找X.dot(self.W)最大元素在第一维中的索引值
pass
# *****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)5.验证最佳超参数：
超参数性能由val_acc衡量，越大越好
# 使用验证集来调整超参数（正则化强度和学习率）。 您应该尝试不同的学习率和正则化强度范围； 如果你小心，你应该能够在验证集上获得大约 0.39 的分类准确度。
# 注意：您可能会在超参数搜索期间看到运行时/溢出警告。 这可能是由极端值引起的，而不是错误。
#结果是将表单（learning_rate，regularization_strength）的字典映射到表单（training_accuracy，validation_accuracy）的元组。
#准确度只是被正确分类的数据点的分数。
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. #
################################################################################
# Provided as a reference. You may or may not want to change these hyperparameters
learning_rates = [1e-7, 1e-6] #选取学习率校验范围
regularization_strengths = [1e4, 4e4] #正则化强度校验范围
# *****START OF YOUR CODE (DO NOT DELETE/MODIFY THIS LINE)*****
# some notes I took for myself:
# 1. first tested with small reg (=2.5e-6), I change lr interval from [1e-7, 1e-5] to [5e-8, 1e-6] cause 5e-6 dose not converge
# 2. then observe the tendency relationship of `val_acc` and `reg` for any given `lr` to shrink interval of `reg`
# 3. np.mean() can calculate accurancy in one short line !!!! not even using the list comprehension
# 4. finally, remember to train model with a larger value for num_iters
# 5. but why can he/she do so well ... orz: https://tomaxent.com/2017/03/03/cs231n-Assignment-1-svm/
for lr in np.linspace(learning_rates[0], learning_rates[1], 5):
for reg in np.linspace(regularization_strengths[0], regularization_strengths[1], 8):
svm = LinearSVM()
svm.train(X_train, y_train, learning_rate=lr, reg=reg, num_iters=1500)
y_train_pred = svm.predict(X_train)
train_acc = np.mean(y_train_pred == y_train)
y_val_pred = svm.predict(X_val)
val_acc = np.mean(y_val_pred == y_val)
results[(lr, reg)] = (train_acc, val_acc)
print('lr %e reg %e train accuracy: %f val accuracy: %f' % (
lr, reg, train_acc, val_acc))
if best_val < val_acc:
best_val = val_acc
best_svm = svm
pass
# *****END OF YOUR CODE (DO NOT DELETE/MODIFY THIS LINE)*****
print('best validation accuracy achieved during cross-validation: %f' % best_val)