CS231n的课后作业非常的好，这里记录一下自己对作业一些笔记。
一、第一个是KNN的代码，这里的trick是计算距离的三种方法，核心的话还是python和machine learning中非常实用的向量化操作，可以大大的提高计算速度。
import numpy as np
class KNearestNeighbor(object):#首先是定义一个处理KNN的类
""" a kNN classifier with L2 distance """
def __init__(self):
pass
def train(self, X, y):
"""
Train the classifier. For k-nearest neighbors this is just
memorizing the training data.
Inputs:
- X: A numpy array of shape (num_train, D) containing the training data
consisting of num_train samples each of dimension D.
- y: A numpy array of shape (N,) containing the training labels, where
y[i] is the label for X[i].
"""
self.X_train = X
self.y_train = y
def predict(self, X, k=1, num_loops=0):
"""
Predict labels for test data using this classifier.
Inputs:
- X: A numpy array of shape (num_test, D) containing test data consisting
of num_test samples each of dimension D.
- k: The number of nearest neighbors that vote for the predicted labels.
- num_loops: Determines which implementation to use to compute distances
between training points and testing points.
Returns:
- y: A numpy array of shape (num_test,) containing predicted labels for the
test data, where y[i] is the predicted label for the test point X[i].
"""
if num_loops == 0:
dists = self.compute_distances_no_loops(X)
elif num_loops == 1:
dists = self.compute_distances_one_loop(X)
elif num_loops == 2:
dists = self.compute_distances_two_loops(X)
else:
raise ValueError(‘Invalid value %d for num_loops‘ % num_loops)
return self.predict_labels(dists, k=k)
def compute_distances_two_loops(self, X):
"""
Compute the distance between each test point in X and each training point
in self.X_train using a nested loop over both the training data and the
test data.
Inputs:
- X: A numpy array of shape (num_test, D) containing test data.
Returns:
- dists: A numpy array of shape (num_test, num_train) where dists[i, j]
is the Euclidean distance between the ith test point and the jth training
point.
"""
num_test = X.shape[0]
num_train = self.X_train.shape[0]
dists = np.zeros((num_test, num_train))
for i in xrange(num_test):
for j in xrange(num_train):
dists[i][j] = np.sqrt(np.sum(np.square(self.X_train[j,:] - X[i,:])))
#####################################################################
# TODO: #
# Compute the l2 distance between the ith test point and the jth #
# training point, and store the result in dists[i, j]. You should #
# not use a loop over dimension. #
#####################################################################
#####################################################################
# END OF YOUR CODE #
#####################################################################
return dists
def compute_distances_one_loop(self, X):
"""
Compute the distance between each test point in X and each training point
in self.X_train using a single loop over the test data.
Input / Output: Same as compute_distances_two_loops
"""
num_test = X.shape[0]
num_train = self.X_train.shape[0]
dists = np.zeros((num_test, num_train))
for i in xrange(num_test):
#######################################################################
# TODO: #
# Compute the l2 distance between the ith test point and all training #
# points, and store the result in dists[i, :]. #
#######################################################################
dists[i,:] = np.sqrt(np.sum(np.square(self.X_train-X[i,:]),axis = 1))
#######################################################################
# END OF YOUR CODE #
#######################################################################
return dists
def compute_distances_no_loops(self, X):
"""
Compute the distance between each test point in X and each training point
in self.X_train using no explicit loops.
Input / Output: Same as compute_distances_two_loops
"""
num_test = X.shape[0]
num_train = self.X_train.shape[0]
dists = np.zeros((num_test, num_train))
#########################################################################
# TODO: #
# Compute the l2 distance between all test points and all training #
# points without using any explicit loops, and store the result in #
# dists. #
# #
# You should implement this function using only basic array operations; #
# in particular you should not use functions from scipy. #
# #
# HINT: Try to formulate the l2 distance using matrix multiplication #
# and two broadcast sums. #
#########################################################################
dists = np.multiply(np.dot(X,self.X_train.T),-2)
sq1 = np.sum(np.square(X),axis=1,keepdims = True)
sq2 = np.sum(np.square(self.X_train),axis=1)
dists = np.add(dists,sq1)
dists = np.add(dists,sq2)
dists = np.sqrt(dists)
#########################################################################
# END OF YOUR CODE #
#########################################################################
return dists
def predict_labels(self, dists, k=1):
"""
Given a matrix of distances between test points and training points,
predict a label for each test point.
Inputs:
- dists: A numpy array of shape (num_test, num_train) where dists[i, j]
gives the distance betwen the ith test point and the jth training point.
Returns:
- y: A numpy array of shape (num_test,) containing predicted labels for the
test data, where y[i] is the predicted label for the test point X[i].
"""
num_test = dists.shape[0]
y_pred = np.zeros(num_test)
for i in xrange(num_test):
# A list of length k storing the labels of the k nearest neighbors to
# the ith test point.
closest_y = []
#########################################################################
# TODO: #
# Use the distance matrix to find the k nearest neighbors of the ith #
# training point, and use self.y_train to find the labels of these #
# neighbors. Store these labels in closest_y. #
# Hint: Look up the function numpy.argsort. #
#########################################################################
closest_y = self.y_train[np.argsort(dists[i,:])[:k]]
#########################################################################
# TODO: #
# Now that you have found the labels of the k nearest neighbors, you #
# need to find the most common label in the list closest_y of labels. #
# Store this label in y_pred[i]. Break ties by choosing the smaller #
# label. #
#########################################################################
y_pred[i] = np.argmax(np.bincount(closest_y)) #########################################################################
# END OF YOUR CODE #
#########################################################################
return y_pred
二、softmax
同样是需要完成naive和vector的两种操作来比较速度。
import numpy as np
def softmax_loss_naive(W, X, y, reg):
# Initialize the loss and gradient to zero.
loss = 0.0
dW = np.zeros_like(W) # 得到一个和W同样shape的矩阵
dW_each = np.zeros_like(W)
num_train, dim = X.shape
num_class = W.shape[1]
f = X.dot(W) # N by C
# Considering the Numeric Stability
f_max = np.reshape(np.max(f, axis=1), (num_train, 1)) # 找到最大值然后减去，这样是为了防止后面的操作会出现数值上的一些偏差
prob = np.exp(f - f_max) / np.sum(np.exp(f - f_max), axis=1, keepdims=True) # N by C
y_trueClass = np.zeros_like(prob)
y_trueClass[np.arange(num_train), y] = 1.0
for i in xrange(num_train):
for j in xrange(num_class):
loss += -(y_trueClass[i, j] * np.log(prob[i, j])) # 损失函数的公式L = -(1/N)∑i∑j1(k=yi)log(exp(fk)/∑j exp(fj)) + λR(W)
dW_each[:, j] = -(y_trueClass[i, j] - prob[i, j]) * X[i, :]#梯度的公式 ∇Wk L = -(1/N)∑i xiT(pi,m-Pm) + 2λWk, where Pk = exp(fk)/∑j exp(fj
dW += dW_each #这是把每个类的放在了一起
loss /= num_train
loss += 0.5 * reg * np.sum(W * W) # 加上正则
dW /= num_traindW += reg * W
return loss, dW
def softmax_loss_vectorized(W, X, y, reg):
"""
Softmax loss function, vectorized version.
Inputs and outputs are the same as softmax_loss_naive.
"""
# Initialize the loss and gradient to zero.
loss = 0.0
dW = np.zeros_like(W) # D by C
num_train, dim = X.shape
f = X.dot(W) # N by C
# Considering the Numeric Stability
f_max = np.reshape(np.max(f, axis=1), (num_train, 1)) # N by 1
prob = np.exp(f - f_max) / np.sum(np.exp(f - f_max), axis=1), keepdims=True)
y_trueClass = np.zeros_like(prob)
y_trueClass[range(num_train), y] = 1.0 # N by C
loss += -np.sum(y_trueClass * np.log(prob)) / num_train + 0.5 * reg * np.sum(W * W)#向量化直接操作即可
dW += -np.dot(X.T, y_trueClass - prob) / num_train + reg * W
return loss, dW
三、SVM
import numpy as np
def svm_loss_naive(W, X, y, reg):
"""
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]: #根据公式，正确的那个不用算
continue
margin = scores[j] - correct_class_score + 1 # note delta = 1
if margin > 0:
loss += margin
dW[:, y[i]] += -X[i, :] # 根据公式：∇Wyi Li = - xiT(∑j≠yi1(xiWj - xiWyi +1>0)) + 2λWyi
dW[:, j] += X[i, :] # 根据公式： ∇Wj Li = xiT 1(xiWj - xiWyi +1>0) + 2λWj , (j≠yi)
# 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 * W
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
scores = X.dot(W) # N by C
num_train = X.shape[0]
num_classes = W.shape[1]
scores_correct = scores[np.arange(num_train), y] # 1 by N
scores_correct = np.reshape(scores_correct, (num_train, 1)) # N by 1
margins = scores - scores_correct + 1.0 # N by C
margins[np.arange(num_train), y] = 0.0
margins[margins <= 0] = 0.0
loss += np.sum(margins) / num_train
loss += 0.5 * reg * np.sum(W * W)
# compute the gradient
margins[margins > 0] = 1.0
row_sum = np.sum(margins, axis=1) # 1 by N
margins[np.arange(num_train), y] = -row_sum
dW += np.dot(X.T, margins)/num_train + reg * W # D by C
return loss, dW
四、linear_classifier
从编程思路上来看，上面三个是不同的策略，确切的说是线性分类器的集中方法，所以我们用一个LinearClassifier类来调用他们。
from linear_svm import *
from softmax import *
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=True): #注意这里传递的参数设置
"""
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
# assume y takes values 0...K-1 where K is number of classes
num_classes = np.max(y) + 1
if self.W is None:
# lazily initialize W
self.W = 0.001 * np.random.randn(dim, num_classes) # 初始化W
# Run stochastic gradient descent(Mini-Batch) to optimize W
loss_history = []
for it in xrange(num_iters): #每次随机取batch的数据来进行梯度下降
X_batch = None
y_batch = None
# Sampling with replacement is faster than sampling without replacement.
sample_index = np.random.choice(num_train, batch_size, replace=False)
X_batch = X[sample_index, :] # batch_size by D
y_batch = y[sample_index] # 1 by batch_size
# evaluate loss and gradient
loss, grad = self.loss(X_batch, y_batch, reg)
loss_history.append(loss)
# perform parameter update
self.W += -learning_rate * grad
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: D x N array of training data. Each column is a D-dimensional point.
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[1]) # 1 by N
X=X.T
y_pred = np.argmax(X.dot(self.W), axis=0) #预测直接找到最后y最大的那个值
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)
五、简单的两层神经网络
这里只是一个简单的神经网络的写法，在下次作业会有一个很好很强大的神经网络等我们去构造。
BP可以看这幅图来理解：
import numpy as np
import matplotlib.pyplot as plt
class TwoLayerNet(object):
"""
A two-layer fully-connected neural network. The net has an input dimension of
N, a hidden layer dimension of H, and performs classification over C classes.
We train the network with a softmax loss function and L2 regularization on the
weight matrices. The network uses a ReLU nonlinearity after the first fully
connected layer.
In other words, the network has the following architecture:
input - fully connected layer - ReLU - fully connected layer - softmax
The outputs of the second fully-connected layer are the scores for each class.
"""
def __init__(self, input_size, hidden_size, output_size, std=1e-4):
"""
Initialize the model. Weights are initialized to small random values and
biases are initialized to zero. Weights and biases are stored in the
variable self.params, which is a dictionary with the following keys:
W1: First layer weights; has shape (D, H)
b1: First layer biases; has shape (H,)
W2: Second layer weights; has shape (H, C)
b2: Second layer biases; has shape (C,)
Inputs:
- input_size: The dimension D of the input data.
- hidden_size: The number of neurons H in the hidden layer.
- output_size: The number of classes C.
"""
self.params = {}
self.params[‘W1‘] = std * np.random.randn(input_size, hidden_size)
self.params[‘b1‘] = np.zeros(hidden_size)
self.params[‘W2‘] = std * np.random.randn(hidden_size, output_size)
self.params[‘b2‘] = np.zeros(output_size) #初始化神经网络的参数
def loss(self, X, y=None, reg=0.0):
"""
Compute the loss and gradients for a two layer fully connected neural
network.
Inputs:
- X: Input data of shape (N, D). Each X[i] is a training sample.
- y: Vector of training labels. y[i] is the label for X[i], and each y[i] is
an integer in the range 0 <= y[i] < C. This parameter is optional; if it
is not passed then we only return scores, and if it is passed then we
instead return the loss and gradients.
- reg: Regularization strength.
Returns:
If y is None, return a matrix scores of shape (N, C) where scores[i, c] is
the score for class c on input X[i].
If y is not None, instead return a tuple of:
- loss: Loss (data loss and regularization loss) for this batch of training
samples.
- grads: Dictionary mapping parameter names to gradients of those parameters
with respect to the loss function; has the same keys as self.params.
"""
# Unpack variables from the params dictionary
W1, b1 = self.params[‘W1‘], self.params[‘b1‘]
W2, b2 = self.params[‘W2‘], self.params[‘b2‘]
N, D = X.shape
# Compute the forward pass
scores = None
#############################################################################
# TODO: Perform the forward pass, computing the class scores for the input. #
# Store the result in the scores variable, which should be an array of #
# shape (N, C). #
#############################################################################
h1=np.maximum(0,np.dot(X,W1)+b1) 这里是做了一个RELU的activition function
scores=np.dot(h1,W2)+b2
#############################################################################
# END OF YOUR CODE #
#############################################################################
# If the targets are not given then jump out, we‘re done
if y is None:
return scores
# Compute the loss
loss = None
#############################################################################
# TODO: Finish the forward pass, and compute the loss. This should include #
# both the data loss and L2 regularization for W1 and W2. Store the result #
# in the variable loss, which should be a scalar. Use the Softmax #
# classifier loss. So that your results match ours, multiply the #
# regularization loss by 0.5 #
#############################################################################
scores_max = np.max(scores, axis=1, keepdims=True) # (N,1)
# Compute the class probabilities
exp_scores = np.exp(scores - scores_max) # (N,C)
probs = exp_scores / np.sum(exp_scores, axis=1, keepdims=True) # (N,C)
# cross-entropy loss and L2-regularization
correct_logprobs = -np.log(probs[range(N), y]) # (N,1)
data_loss = np.sum(correct_logprobs) / N
reg_loss = 0.5 * reg * np.sum(W1 * W1) + 0.5 * reg * np.sum(W2 * W2)
loss = data_loss + reg_loss #计算出误差
#############################################################################
# END OF YOUR CODE #
#############################################################################
# Backward pass: compute gradients
grads = {}
#############################################################################
# TODO: Compute the backward pass, computing the derivatives of the weights #
# and biases. Store the results in the grads dictionary. For example, #
# grads[‘W1‘] should store the gradient on W1, and be a matrix of same size #
#############################################################################
dscores = probs # (N,C)
dscores[range(N), y] -= 1 # 这个是输出的误差敏感项也就是梯度的计算，具体可以看上面softmax 的计算
dscores /= N
# Backprop into W2 and b2
dW2 = np.dot(h1.T, dscores) # (H,C) BP算法的计算，下面同理
db2 = np.sum(dscores, axis=0, keepdims=True) # (1,C
# Backprop into hidden layer
dh1 = np.dot(dscores, W2.T) # (N,H)
# Backprop into ReLU non-linearity
dh1[h1 <= 0] = 0
# Backprop into W1 and b1
dW1 = np.dot(X.T, dh1) # (D,H)
db1 = np.sum(dh1, axis=0, keepdims=True) # (1,H)
# Add the regularization gradient contribution
dW2 += reg * W2
dW1 += reg * W1
grads[‘W1‘] = dW1
grads[‘b1‘] = db1
grads[‘W2‘] = dW2
grads[‘b2‘] = db2
#############################################################################
# END OF YOUR CODE #
#############################################################################
return loss, grads
def train(self, X, y, X_val, y_val,
learning_rate=1e-3, learning_rate_decay=0.95,
reg=1e-5, num_iters=100,
batch_size=200, verbose=False):
"""
Train this neural network using stochastic gradient descent.
Inputs:
- X: A numpy array of shape (N, D) giving training data.
- y: A numpy array f shape (N,) giving training labels; y[i] = c means that
X[i] has label c, where 0 <= c < C.
- X_val: A numpy array of shape (N_val, D) giving validation data.
- y_val: A numpy array of shape (N_val,) giving validation labels.
- learning_rate: Scalar giving learning rate for optimization.
- learning_rate_decay: Scalar giving factor used to decay the learning rate
after each epoch.
- reg: Scalar giving regularization strength.
- num_iters: Number of steps to take when optimizing.
- batch_size: Number of training examples to use per step.
- verbose: boolean; if true print progress during optimization.
"""
num_train = X.shape[0]
iterations_per_epoch = max(num_train / batch_size, 1)
# Use SGD to optimize the parameters in self.model
loss_history = []
train_acc_history = []
val_acc_history = []
for it in xrange(num_iters):
X_batch = None
y_batch = None
#########################################################################
# TODO: Create a random minibatch of training data and labels, storing #
# them in X_batch and y_batch respectively. #
#########################################################################
sample_index = np.random.choice(num_train, batch_size, replace=True)
X_batch = X[sample_index, :] # (batch_size,D)
y_batch = y[sample_index] # (1,batch_size)
#########################################################################
# END OF YOUR CODE #
#########################################################################
# Compute loss and gradients using the current minibatch
loss, grads = self.loss(X_batch, y=y_batch, reg=reg)
loss_history.append(loss)
#########################################################################
# TODO: Use the gradients in the grads dictionary to update the #
# parameters of the network (stored in the dictionary self.params) #
# using stochastic gradient descent. You‘ll need to use the gradients #
# stored in the grads dictionary defined above. #
#########################################################################
print grads[‘b2‘]
a=grads[‘b2‘].reshape(-1)
grads[‘b2‘]=a
a = grads[‘b1‘].reshape(-1)
grads[‘b1‘] = a
grads[‘b1‘].reshape(-1)
v_W2 = - learning_rate * grads[‘W2‘]
self.params[‘W2‘] += v_W2
self.params[‘b2‘]-=learning_rate * grads[‘b2‘]
v_W1 = - learning_rate * grads[‘W1‘]
self.params[‘W1‘] += v_W1
v_b1 = - learning_rate * grads[‘b1‘]
self.params[‘b1‘] += v_b1 #对参数进行更新
#########################################################################
# END OF YOUR CODE #
#########################################################################
if verbose and it % 100 == 0:
print ‘iteration %d / %d: loss %f‘ % (it, num_iters, loss)
# Every epoch, check train and val accuracy and decay learning rate.
if it % iterations_per_epoch == 0:
# Check accuracy
train_acc = (self.predict(X_batch) == y_batch)
val_acc = (self.predict(X_val) == y_val)
train_acc_history.append(train_acc)
val_acc_history.append(val_acc)
# Decay learning rate
learning_rate *= learning_rate_decay
return {
‘loss_history‘: loss_history,
‘train_acc_history‘: train_acc_history,
‘val_acc_history‘: val_acc_history,
}
def predict(self, X):
"""
Use the trained weights of this two-layer network to predict labels for
data points. For each data point we predict scores for each of the C
classes, and assign each data point to the class with the highest score.
Inputs:
- X: A numpy array of shape (N, D) giving N D-dimensional data points to
classify.
Returns:
- y_pred: A numpy array of shape (N,) giving predicted labels for each of
the elements of X. For all i, y_pred[i] = c means that X[i] is predicted
to have class c, where 0 <= c < C.
"""
y_pred = None
###########################################################################
# TODO: Implement this function; it should be VERY simple! #
###########################################################################
y_pred = None
h1 = np.maximum(0,(np.dot(X, self.params[‘W1‘]) + self.params[‘b1‘]))
scores = np.dot(h1, self.params[‘W2‘]) + self.params[‘b2‘]
y_pred = np.argmax(scores, axis=1)
###########################################################################
# END OF YOUR CODE #
###########################################################################
return y_pred
总结
这个作业是很久之前做的了，代码基本上也是借鉴的网上的，我主要就是把它搞懂和跑通，主要还是不太熟悉python这一套，numpy这个东西还是有一定的学习时间的...
整个作业一来说，难度不大，关键是把之前学到的公式转换成代码，我觉得做完这个工作很有收获。
时间： 2024-10-10 19:47:30