CS231n作业+代码实践：Assignment1 SVM

Multiclass Support Vector Machine exercise

# Run some setup code for this notebook.

from __future__ import print_function
import random
import numpy as np
from cs231n.data_utils import load_CIFAR10
import matplotlib.pyplot as plt


# This is a bit of magic to make matplotlib figures appear inline in the
# notebook rather than in a new window.
%matplotlib inline
plt.rcParams['figure.figsize'] = (10.0, 8.0) # set default size of plots
plt.rcParams['image.interpolation'] = 'nearest'
plt.rcParams['image.cmap'] = 'gray'

# Some more magic so that the notebook will reload external python modules;
# see http://stackoverflow.com/questions/1907993/autoreload-of-modules-in-ipython
%load_ext autoreload
%autoreload 2

首先，这里还是跟上一个作业一样，要注意from __future__ import print_function要放到最上面（我也不知道为什么，看提示改的）

# Load the raw CIFAR-10 data.
cifar10_dir = 'cs231n/datasets/cifar-10-batches-py'

# Cleaning up variables to prevent loading data multiple times (which may cause memory issue)
try:
   del X_train, y_train
   del X_test, y_test
   print('Clear previously loaded data.')
except:
   pass

X_train, y_train, X_test, y_test = load_CIFAR10(cifar10_dir

# As a sanity check, we print out the size of the training and test data.
print('Training data shape: ', X_train.shape)
print('Training labels shape: ', y_train.shape)
print('Test data shape: ', X_test.shape)
print('Test labels shape: ', y_test.shape)

Training data shape:  (50000, 32, 32, 3)
Training labels shape:  (50000,)
Test data shape:  (10000, 32, 32, 3)
Test labels shape:  (10000,)

# Visualize some examples from the dataset.
# We show a few examples of training images from each class.
classes = ['plane', 'car', 'bird', 'cat', 'deer', 'dog', 'frog', 'horse', 'ship', 'truck']
num_classes = len(classes)
samples_per_class = 7
for y, cls in enumerate(classes):
    idxs = np.flatnonzero(y_train == y)
    idxs = np.random.choice(idxs, samples_per_class, replace=False)
    for i, idx in enumerate(idxs):
        plt_idx = i * num_classes + y + 1
        plt.subplot(samples_per_class, num_classes, plt_idx)
        plt.imshow(X_train[idx].astype('uint8'))
        plt.axis('off')
        if i == 0:
            plt.title(cls)
plt.show()

这上面的代码都和之前一模一样，所以不多赘述。
下面开始把样本分成三类，1.训练集 2.验证集3.测试集

# Split the data into train, val, and test sets. In addition we will
# create a small development set as a subset of the training data;
# we can use this for development so our code runs faster.
num_training = 49000
num_validation = 1000 
num_test = 1000
num_dev = 500

# Our validation set will be num_validation points from the original
# training set.
mask = range(num_training, num_training + num_validation) #range生成val索引序列
X_val = X_train[mask]
y_val = y_train[mask]

# Our training set will be the first num_train points from the original
# training set.
mask = range(num_training)#range生成train的索引序列
X_train = X_train[mask]
y_train = y_train[mask]

# We will also make a development set, which is a small subset of
# the training set.
mask = np.random.choice(num_training, num_dev, replace=False) #随机从train挑选500个dev
X_dev = X_train[mask]
y_dev = y_train[mask]

# We use the first num_test points of the original test set as our
# test set.
mask = range(num_test) #生成test索引序列
X_test = X_test[mask]
y_test = y_test[mask]

print('Train data shape: ', X_train.shape)
print('Train labels shape: ', y_train.shape)
print('Validation data shape: ', X_val.shape)
print('Validation labels shape: ', y_val.shape)
print('Test data shape: ', X_test.shape)
print('Test labels shape: ', y_test.shape)

Train data shape:  (49000, 32, 32, 3)
Train labels shape:  (49000,)
Validation data shape:  (1000, 32, 32, 3)
Validation labels shape:  (1000,)
Test data shape:  (1000, 32, 32, 3)
Test labels shape:  (1000,)

同理，我们还是把他们后面几个维度flatten，常用 np.reshape(X_train, (X_train.shape[0], -1))来把后面的维度合并（记下来）

# Preprocessing: reshape the image data into rows
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))

# As a sanity check, print out the shapes of the data
print('Training data shape: ', X_train.shape)
print('Validation data shape: ', X_val.shape)
print('Test data shape: ', X_test.shape)
print('dev data shape: ', X_dev.shape)

Training data shape:  (49000, 3072)
Validation data shape:  (1000, 3072)
Test data shape:  (1000, 3072)
dev data shape:  (500, 3072)

下面是一步预处理步骤，我们计算每个像素点在49000副图像里的均值。

mean_image = np.mean(X_train, axis=0)
print(mean_image[:10]) # 打印前10个像素点的平均值
plt.figure(figsize=(4,4)) #图像大小
plt.imshow(mean_image.reshape((32,32,3)).astype('uint8')) # visualize the mean image
plt.show()

(3072,)
[130.64189796 135.98173469 132.47391837 130.05569388 135.34804082
 131.75402041 130.96055102 136.14328571 132.47636735 131.48467347]

所有图像要减去均值图像的值,并且为了不引入b偏置项，我们把样本hstack一列全1项，这样就相当于有个b了。

# 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
# 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)

SVM分类器

本节的代码将全部写在cs231n / classifiers / linear_svm.py中。
如您所见，我们已经预填充了函数compute_loss_naive，该函数使用for循环来评估多类SVM损失函数。

# Evaluate the naive implementation of the loss we provided for you:
from cs231n.classifiers.linear_svm import svm_loss_naive
import time

# generate a random SVM weight matrix of small numbers
W = np.random.randn(3073, 10) * 0.0001 

loss, grad = svm_loss_naive(W, X_dev, y_dev, 0.000005)
print('loss: %f' % (loss, ))

打开.py文件，找到简单svm算法，发现他已经写出来了loss的算法（这里用的是线性分类和svmloss去构建模型，下面这部分反向传播求梯度有些复杂，需要耐心一点计算）
这边我们要先计算一下导数dW:(为了与编程一致，我这边写成$y=xw$（按照代码里的正向传播来写）的形式：
$$L_i=\frac{1}{N}\sum _{j\ne y_i}max(0,s_j-s_{y_i}+\Delta )+\frac{1}{2}\lambda \Vert w{\Vert }^2$$

$\Delta$为选定的某个阈值。

$$L_i=\frac{1}{N}\sum _{j\ne y_i}max(0,x_i{w}_j-x_i{w}_{y_i}+\Delta )+\frac{1}{2}\lambda \Vert w{\Vert }^2$$
于是W的每一列可以表示为：
$$\left[\begin{array}{cccc}{S}_{00}& S_{01}& ...& S_{0C}\\ S_{10}& S_{11}& ...& S_{1C}\\ ...& ...& ...& ...\\ S_{N0}& S_{N1}& ...& S_{N1}\end{array}\right]=\left[\begin{array}{ccc}...& X_0& ...\\ ...& X_1& ...\\ ...& ...& ...\\ ...& X_N& ...\end{array}\right]\left[\begin{array}{cccc}⋮& ⋮& & ⋮\\ W_0& W_1& ...& W_c\\ ⋮& ⋮& & ⋮\end{array}\right]$$

在每一张训练集i的图片循环（第一层循环）
$$L_i=\sum _{j\ne y_i}max(0,x_i{w}_j-x_i{w}_{y_i}+\Delta )$$

又在j的循环（第二层循环），求导：

$$\frac{L_i}{\partial w_j}=\{\begin{array}{rlr}{x}_i^T& & x_i{w}_j-x_i{w}_{y_i}>0\\ 0& & x_i{w}_j-x_i{w}_{y_i}⩽0\end{array}$$

$$\frac{L_i}{\partial w_{yi}}=-x_i^T{x}_i{w}_j-x_i{w}_{y_i}>0$$
如果 $x_i{w}_j-x_i{w}_{y_i}+\Delta >0$，那么根据求导有：

$$d{w}_j=d{w}_j+x_i^T$$

$$d{w}_{yi}=d{w}_{yi}-x_i^T$$

最后还要加上$\lambda w$,因此：
$$dW=\frac{1}{N}dW+\lambda w$$

这里的j代表W矩阵的每一列，$x_i$代表不同的训练样本

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]#10类
  num_train = X.shape[0] #num_train表示训练样本个数
  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]:   #j是真实样本
        continue
      margin = scores[j] - correct_class_score + 1 # note delta = 1
      if margin > 0:
        loss +=margin
        dW[:,j]+=X[i,:].T
        dW[:,y[i]]+=-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 += 0.5*reg * np.sum(W * W) #Np.sum(a*a)逐项相乘
  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

修改好上面之后，我们运行一下，检查一些输出grad的维度是不是与W的维度一样：

# Evaluate the naive implementation of the loss we provided for you:
from cs231n.classifiers.linear_svm import svm_loss_naive
import time

# generate a random SVM weight matrix of small numbers
W = np.random.randn(3073, 10) * 0.0001 

loss, grad = svm_loss_naive(W, X_dev, y_dev, 0.000005)
print('loss: %f' % (loss, ))
print('gradient:\n',grad.shape)

进行梯度检查：

# Once you've implemented the gradient, recompute it with the code below
# and gradient check it with the function we provided for you

# Compute the loss and its gradient at W.
loss, grad = svm_loss_naive(W, X_dev, y_dev, 0.0)

# Numerically compute the gradient along several randomly chosen dimensions, and
# compare them with your analytically computed gradient. The numbers should match
# almost exactly along all dimensions.
from cs231n.gradient_check import grad_check_sparse
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
# you didn't forget the regularization gradient did you?
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)

numerical: 23.085385 analytic: 23.085385, relative error: 1.172930e-11
numerical: -1.236521 analytic: -1.236521, relative error: 7.884760e-11
numerical: 6.910923 analytic: 6.845186, relative error: 4.778747e-03
numerical: -19.968839 analytic: -19.968839, relative error: 7.256444e-12
numerical: 11.331944 analytic: 11.331944, relative error: 2.770798e-11
numerical: -3.385788 analytic: -3.385788, relative error: 3.976704e-11
numerical: 20.107681 analytic: 20.107681, relative error: 7.589920e-12
numerical: 1.887162 analytic: 1.887162, relative error: 7.564024e-11
numerical: 21.488876 analytic: 21.417529, relative error: 1.662846e-03
numerical: -13.144394 analytic: -13.144394, relative error: 6.459439e-12
numerical: 30.272869 analytic: 30.272869, relative error: 5.525255e-12
numerical: 20.681273 analytic: 20.681273, relative error: 1.626950e-11
numerical: -3.407003 analytic: -3.374702, relative error: 4.762895e-03
numerical: 12.234011 analytic: 12.234011, relative error: 1.130557e-11
numerical: -2.756436 analytic: -2.756436, relative error: 9.104975e-11
numerical: 19.967492 analytic: 19.967492, relative error: 7.247770e-12
numerical: 32.355688 analytic: 32.355688, relative error: 2.117155e-11
numerical: 7.605475 analytic: 7.605475, relative error: 3.218538e-11
numerical: -3.685143 analytic: -3.685143, relative error: 2.872000e-11
numerical: 5.678984 analytic: 5.747740, relative error: 6.017125e-03

在前面的练习我们知道，用向量化的方法进行运算可以让程序快很多，所以接下来我们用向量化方法来计算loss和gradient：
这里我们还是要重点推到一下向量化的梯度 ：(用矩阵来计算梯度，常用链式求导+维度分析）
$$\frac{\partial L}{\partial W}=\frac{\partial L}{\partial S}\frac{\partial S}{\partial W}=X^T\frac{\partial L}{\partial S}=X^T\left[\begin{array}{ccc}...& \frac{\partial L}{\partial S_1}& ...\\ ...& \frac{\partial L}{\partial S_2}& ...\\ ...& ...& ...\\ ...& \frac{\partial L}{\partial S_N}& ...\end{array}\right]$$

$$\frac{\partial L}{\partial S_i}=\left[\begin{array}{cccc}\frac{\partial L}{\partial S_{i1}}& \frac{\partial L}{\partial S_{i2}}& ...& \frac{\partial L}{\partial S_{iC}}\end{array}\right]$$

因此只要将L依次对$s_{ij}$依次求导：
$$\frac{L_i}{\partial S_{ij}}=\{\begin{array}{rlr}0& & j\ne y_i,s_j-s_{y_i}+\Delta ⩽0\\ 1& & j\ne y_i,,s_j-s_{y_i}+\Delta >0\\ -(num（j\ne y_i,,s_j-s_{y_i}+\Delta >0）-1)& & \end{array}$$

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.                                                           #
  #############################################################################
  # note delta = 1
  print(X)
  num = X.shape[0]
  y_c = np.dot(X,W)
  score_y = y_c[range(num),y].reshape(num,1)
  margin = np.maximum(0,y_c-score_y+1)
  margin[range(num),y] =0
  loss += np.sum(margin)/num
  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.                                                                     #
  #############################################################################
  #first step :
  margin[margin > 0] = 1.0
  row_sum = np.sum(margin,axis = 1)
  margin[np.arange(num), y] = -row_sum
  dW += 1.0 / num * np.dot(X.T, margin) + reg * W
  #############################################################################
  #                             END OF YOUR CODE                              #
  #############################################################################

  return loss, dW

Naive loss and gradient: computed in 0.267549s
Vectorized loss and gradient: computed in 0.002870s
difference: 0.000000

果然速度还是快了很多！（以后都尽量用向量去写代码）

Stochastic Gradient Descent

from __future__ import print_function

import numpy as np
from cs231n.classifiers.linear_svm import *
from cs231n.classifiers.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=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: #初始化
      # 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) #随机选5个
      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 += -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.            #
    ###########################################################################
    y = np.dot(X,self.W)
    y_pred += np.argmax(y,axis=1)
    ###########################################################################
    #                           END OF YOUR CODE                              #
    ###########################################################################
    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)

下面这步是用来寻找最优超参数的（算法还是比较简单的，就是找最高的accuracy)

# Use the validation set to tune hyperparameters (regularization strength and
# learning rate). You should experiment with different ranges for the learning
# rates and regularization strengths; if you are careful you should be able to
# get a classification accuracy of about 0.4 on the validation set.
learning_rate = [2e-7,0.75e-7,1.5e-7,1.25e-7,0.75e-7]
regularization_strengths = [3e4,3.25e4,3.5e4,3.75e4,4e4]

# results is dictionary mapping tuples of the form
# (learning_rate, regularization_strength) to tuples of the form
# (training_accuracy, validation_accuracy). The accuracy is simply the fraction
# of data points that are correctly classified.
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.                                      #
################################################################################
for rate in learning_rate:
    for regular in regularization_strengths:
        svm = LinearSVM()
        loss_hist = svm.train(X_train, y_train, learning_rate=rate, reg=regular,num_iters=2000, verbose=True)
        y_train_pred = svm.predict(X_train)
        y_val_pred = svm.predict(X_val)
        accuracy_train = np.mean(y_train == y_train_pred)
        accuracy_test = np.mean(y_val == y_val_pred)
        if best_val < accuracy:
            best_val = accuracy
            best_svm = svm
        results[(rate,regular)]=(accuracy_train,accuracy_test)

################################################################################
#                              END OF YOUR CODE                                #
################################################################################
    
# Print out results.
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)

最后这部分代码很有意思，它显示表示了W的实际意义，某块的颜色更深，不同类的W要去寻找图片的模板，比如说horse的W就是去寻找一个类似于马形状的图片，而且左右两边都有头。

# Visualize the learned weights for each class.
# Depending on your choice of learning rate and regularization strength, these may
# or may not be nice to look at.
w = best_svm.W[:-1,:] # strip out the 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])

第二次作业写了挺久，主要是gradient那里，还重新去看了好几遍书和知乎，最后终于搞懂了，撒花！！！