[TOC]

Jupyter notebook softmax.ipynb 内容:

Softmax exercise

Complete and hand in this completed worksheet (including its outputs and any supporting code outside of the worksheet) with your assignment submission. For more details see the assignments page on the course website.

This exercise is analogous to the SVM exercise. You will:

• implement a fully-vectorized loss function for the Softmax classifier
• implement the fully-vectorized expression for its analytic gradient
• check your implementation with numerical gradient
• use a validation set to tune the learning rate and regularization strength
• optimize the loss function with SGD
• visualize the final learned weights

import random
import numpy as np
from cs231n.data_utils import load_CIFAR10
import matplotlib.pyplot as plt
%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'

# for auto-reloading extenrnal modules
# see http://stackoverflow.com/questions/1907993/autoreload-of-modules-in-ipython
%load_ext autoreload
%autoreload 2

def get_CIFAR10_data(num_training=49000, num_validation=1000, num_test=1000, num_dev=500):
  """
  Load the CIFAR-10 dataset from disk and perform preprocessing to prepare
  it for the linear classifier. These are the same steps as we used for the
  SVM, but condensed to a single function.  
  """
  # Load the raw CIFAR-10 data
  cifar10_dir = 'cs231n/datasets/cifar-10-batches-py'
  X_train, y_train, X_test, y_test = load_CIFAR10(cifar10_dir)
  
  # subsample the data
  mask = range(num_training, num_training + num_validation)
  X_val = X_train[mask]
  y_val = y_train[mask]
  mask = range(num_training)
  X_train = X_train[mask]
  y_train = y_train[mask]
  mask = range(num_test)
  X_test = X_test[mask]
  y_test = y_test[mask]
  mask = np.random.choice(num_training, num_dev, replace=False)
  X_dev = X_train[mask]
  y_dev = y_train[mask]
  
  # 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))
  
  # Normalize the data: subtract the mean image
  mean_image = np.mean(X_train, axis = 0)
  X_train -= mean_image
  X_val -= mean_image
  X_test -= mean_image
  X_dev -= mean_image
  
  # add bias dimension and transform into columns
  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))])
  
  return X_train, y_train, X_val, y_val, X_test, y_test, X_dev, y_dev


# Invoke the above function to get our data.
X_train, y_train, X_val, y_val, X_test, y_test, X_dev, y_dev = get_CIFAR10_data()
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
print 'dev data shape: ', X_dev.shape
print 'dev labels shape: ', y_dev.shape

Train data shape:  (49000, 3073)
Train labels shape:  (49000,)
Validation data shape:  (1000, 3073)
Validation labels shape:  (1000,)
Test data shape:  (1000, 3073)
Test labels shape:  (1000,)
dev data shape:  (500, 3073)
dev labels shape:  (500,)

Softmax Classifier

Your code for this section will all be written inside cs231n/classifiers/softmax.py.

# First implement the naive softmax loss function with nested loops.
# Open the file cs231n/classifiers/softmax.py and implement the
# softmax_loss_naive function.

from cs231n.classifiers.softmax import softmax_loss_naive
import time

# Generate a random softmax weight matrix and use it to compute the loss.
W = np.random.randn(3073, 10) * 0.0001
loss, grad = softmax_loss_naive(W, X_dev, y_dev, 0.0)

# As a rough sanity check, our loss should be something close to -log(0.1).
print 'loss: %f' % loss
print 'sanity check: %f' % (-np.log(0.1))

loss: 2.382097
sanity check: 2.302585

Inline Question 1:

Why do we expect our loss to be close to -log(0.1)? Explain briefly.**

Your answer: because initialization is random and the sum of classes is 10， so the probably of predict correctly class number is 1/10， then loss would be -log（0.1）

# Complete the implementation of softmax_loss_naive and implement a (naive)
# version of the gradient that uses nested loops.
loss, grad = softmax_loss_naive(W, X_dev, y_dev, 0.0)

# As we did for the SVM, use numeric gradient checking as a debugging tool.
# The numeric gradient should be close to the analytic gradient.
from cs231n.gradient_check import grad_check_sparse
f = lambda w: softmax_loss_naive(w, X_dev, y_dev, 0.0)[0]
grad_numerical = grad_check_sparse(f, W, grad, 10)

# similar to SVM case, do another gradient check with regularization
loss, grad = softmax_loss_naive(W, X_dev, y_dev, 1e2)
f = lambda w: softmax_loss_naive(w, X_dev, y_dev, 1e2)[0]
grad_numerical = grad_check_sparse(f, W, grad, 10)

numerical: -2.808017 analytic: -2.808017, relative error: 2.427338e-08
numerical: 2.623199 analytic: 2.623199, relative error: 1.099454e-08
numerical: 2.103685 analytic: 2.103685, relative error: 2.239864e-09
numerical: 0.707572 analytic: 0.707572, relative error: 1.118403e-08
numerical: 0.075667 analytic: 0.075667, relative error: 1.060205e-07
numerical: 0.518481 analytic: 0.518480, relative error: 6.611405e-08
numerical: -0.330835 analytic: -0.330835, relative error: 7.891339e-08
numerical: -0.231122 analytic: -0.231122, relative error: 2.356736e-07
numerical: -3.721387 analytic: -3.721387, relative error: 2.974035e-09
numerical: 3.969571 analytic: 3.969571, relative error: 2.303792e-08
numerical: -2.865714 analytic: -2.865714, relative error: 2.069379e-08
numerical: -0.233447 analytic: -0.233447, relative error: 5.429056e-09
numerical: -2.300726 analytic: -2.300726, relative error: 1.802526e-08
numerical: -4.972360 analytic: -4.972360, relative error: 9.030189e-09
numerical: 1.826103 analytic: 1.826103, relative error: 2.318045e-08
numerical: 2.912138 analytic: 2.912138, relative error: 2.626010e-08
numerical: 4.397912 analytic: 4.397911, relative error: 2.148299e-08
numerical: 1.548278 analytic: 1.548278, relative error: 4.536950e-08
numerical: -1.672722 analytic: -1.672722, relative error: 8.208277e-09
numerical: 1.472942 analytic: 1.472942, relative error: 3.173142e-09

# Now that we have a naive implementation of the softmax loss function and its gradient,
# implement a vectorized version in softmax_loss_vectorized.
# The two versions should compute the same results, but the vectorized version should be
# much faster.
tic = time.time()
loss_naive, grad_naive = softmax_loss_naive(W, X_dev, y_dev, 0.00001)
toc = time.time()
print 'naive loss: %e computed in %fs' % (loss_naive, toc - tic)

from cs231n.classifiers.softmax import softmax_loss_vectorized
tic = time.time()
loss_vectorized, grad_vectorized = softmax_loss_vectorized(W, X_dev, y_dev, 0.00001)
toc = time.time()
print 'vectorized loss: %e computed in %fs' % (loss_vectorized, toc - tic)

# As we did for the SVM, we use the Frobenius norm to compare the two versions
# of the gradient.
grad_difference = np.linalg.norm(grad_naive - grad_vectorized, ord='fro')
print 'Loss difference: %f' % np.abs(loss_naive - loss_vectorized)
print 'Gradient difference: %f' % grad_difference

naive loss: 2.382097e+00 computed in 0.143677s
vectorized loss: 9.704155e-02 computed in 0.063678s
Loss difference: 2.285055
Gradient difference: 0.000000

# 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 over 0.35 on the validation set.
from cs231n.classifiers import Softmax
results = {}
best_val = -1
best_softmax = None
learning_rates = [1e-7, 5e-7]
regularization_strengths = [5e4, 1e8]

################################################################################
# TODO:                                                                        #
# Use the validation set to set the learning rate and regularization strength. #
# This should be identical to the validation that you did for the SVM; save    #
# the best trained softmax classifer in best_softmax.                          #
################################################################################
num_splt_lr = 3
num_splt_rs = 8
for i in xrange(num_splt_lr):
    for j in xrange(num_splt_rs):
        learning_rate_ij = learning_rates[0] + i * (learning_rates[1] - learning_rates[0]) / num_splt_lr
        reg_ij = regularization_strengths[0] + j * (regularization_strengths[1] - regularization_strengths[0])/ num_splt_rs
        softmax = Softmax()
        loss_hist = softmax.train(X_train, y_train, learning_rate=learning_rate_ij, reg=reg_ij,
                              num_iters=1500, verbose=False)

        y_train_pred = softmax.predict(X_train)
        accuracy_train = np.mean(y_train == y_train_pred)
        y_val_pred = softmax.predict(X_val)
        accuracy_val = np.mean(y_val == y_val_pred)

        results[(learning_rate_ij, reg_ij)] = (accuracy_train, accuracy_val)

        if accuracy_val > best_val:
            best_val = accuracy_val
            best_softmax = softmax
################################################################################
#                              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

cs231n/classifiers/softmax.py:79: RuntimeWarning: overflow encountered in exp
  exp_scores = np.exp(scores)
cs231n/classifiers/softmax.py:84: RuntimeWarning: invalid value encountered in divide
  norm_exp_scores = exp_scores / row_sum


lr 1.000000e-07 reg 5.000000e+04 train accuracy: 0.324122 val accuracy: 0.340000
lr 1.000000e-07 reg 1.254375e+07 train accuracy: 0.210694 val accuracy: 0.220000
lr 1.000000e-07 reg 2.503750e+07 train accuracy: 0.100265 val accuracy: 0.087000
lr 1.000000e-07 reg 3.753125e+07 train accuracy: 0.100265 val accuracy: 0.087000
lr 1.000000e-07 reg 5.002500e+07 train accuracy: 0.100265 val accuracy: 0.087000
lr 1.000000e-07 reg 6.251875e+07 train accuracy: 0.100265 val accuracy: 0.087000
lr 1.000000e-07 reg 7.501250e+07 train accuracy: 0.100265 val accuracy: 0.087000
lr 1.000000e-07 reg 8.750625e+07 train accuracy: 0.100265 val accuracy: 0.087000
lr 2.333333e-07 reg 5.000000e+04 train accuracy: 0.333796 val accuracy: 0.355000
lr 2.333333e-07 reg 1.254375e+07 train accuracy: 0.100265 val accuracy: 0.087000
lr 2.333333e-07 reg 2.503750e+07 train accuracy: 0.100265 val accuracy: 0.087000
lr 2.333333e-07 reg 3.753125e+07 train accuracy: 0.100265 val accuracy: 0.087000
lr 2.333333e-07 reg 5.002500e+07 train accuracy: 0.100265 val accuracy: 0.087000
lr 2.333333e-07 reg 6.251875e+07 train accuracy: 0.100265 val accuracy: 0.087000
lr 2.333333e-07 reg 7.501250e+07 train accuracy: 0.100265 val accuracy: 0.087000
lr 2.333333e-07 reg 8.750625e+07 train accuracy: 0.100265 val accuracy: 0.087000
lr 3.666667e-07 reg 5.000000e+04 train accuracy: 0.323347 val accuracy: 0.345000
lr 3.666667e-07 reg 1.254375e+07 train accuracy: 0.100265 val accuracy: 0.087000
lr 3.666667e-07 reg 2.503750e+07 train accuracy: 0.100265 val accuracy: 0.087000
lr 3.666667e-07 reg 3.753125e+07 train accuracy: 0.100265 val accuracy: 0.087000
lr 3.666667e-07 reg 5.002500e+07 train accuracy: 0.100265 val accuracy: 0.087000
lr 3.666667e-07 reg 6.251875e+07 train accuracy: 0.100265 val accuracy: 0.087000
lr 3.666667e-07 reg 7.501250e+07 train accuracy: 0.100265 val accuracy: 0.087000
lr 3.666667e-07 reg 8.750625e+07 train accuracy: 0.100265 val accuracy: 0.087000
best validation accuracy achieved during cross-validation: 0.355000

# evaluate on test set
# Evaluate the best softmax on test set
y_test_pred = best_softmax.predict(X_test)
test_accuracy = np.mean(y_test == y_test_pred)
print 'softmax on raw pixels final test set accuracy: %f' % (test_accuracy, )

softmax on raw pixels final test set accuracy: 0.346000

# Visualize the learned weights for each class
w = best_softmax.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 xrange(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])

output_10

softmax.py 内容:

import numpy as np
from random import shuffle

def softmax_loss_naive(W, X, y, reg):
  """
  Softmax 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
  """
  # Initialize the loss and gradient to zero.
  loss = 0.0
  dW = np.zeros_like(W)

  #############################################################################
  # TODO: Compute the softmax loss and its gradient using explicit loops.     #
  # Store the loss in loss and the gradient in dW. If you are not careful     #
  # here, it is easy to run into numeric instability. Don't forget the        #
  # regularization!                                                           #
  #############################################################################
  num_classes = W.shape[1]
  num_train = X.shape[0]
  for i in xrange(num_train):
      scores = X[i].dot(W)
      correct_class = y[i]
      exp_scores = np.zeros_like(scores)
      row_sum = 0
      for j in xrange(num_classes):
          exp_scores[j] = np.exp(scores[j])
          row_sum += exp_scores[j]
      loss += -np.log(exp_scores[correct_class]/row_sum)
      #compute dW loops:
      for k in xrange(num_classes):
        if k != correct_class:
          dW[:,k] += exp_scores[k] / row_sum * X[i]
        else:
          dW[:,k] += (exp_scores[correct_class]/row_sum - 1) * X[i]
  loss /= num_train
  reg_loss = 0.5 * reg * np.sum(W**2)
  loss += reg_loss
  dW /= num_train
  dW += reg * W
  #############################################################################
  #                          END OF YOUR CODE                                 #
  #############################################################################

  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)

  #############################################################################
  # TODO: Compute the softmax loss and its gradient using no explicit loops.  #
  # Store the loss in loss and the gradient in dW. If you are not careful     #
  # here, it is easy to run into numeric instability. Don't forget the        #
  # regularization!                                                           #
  #############################################################################
  num_train = X.shape[0]
  scores = X.dot(W)
  exp_scores = np.exp(scores)
  row_sum = exp_scores.sum(axis=1)
  row_sum = row_sum.reshape((num_train, 1))

  #compute loss
  norm_exp_scores = exp_scores / row_sum
  row_index = np.arange(num_train)
  data_loss = norm_exp_scores[row_index, y].sum()
  loss = data_loss / num_train + 0.5 * reg * np.sum(W*W)
  norm_exp_scores[row_index, y] -= 1

  dW = X.T.dot(norm_exp_scores)
  dW = dW/num_train + reg * W
  #############################################################################
  #                          END OF YOUR CODE                                 #
  #############################################################################

  return loss, dW

linear_classifier.py 内容:

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:
      # 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)   #
      #                                                                       #
      #                $$this may be wrong, it shuould be (batch_size, dim)$$ #
      #                                                                       #
      # 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.              #
      #########################################################################
      batch_inx = np.random.choice(num_train, batch_size)
      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 = 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: D x N array of training data. Each column is a D-dimensional point.
          $ it should be X: N x 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_scores = np.dot(X, self.W)
    y_pred = np.argmax(y_scores, 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)