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

In [1]:
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

In [2]:
def get_CIFAR10_data(num_training=49000, num_validation=1000, num_test=1000):
  """
  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]
  
  # 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))
  
  # 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
  
  # add bias dimension and transform into columns
  X_train = np.hstack([X_train, np.ones((X_train.shape[0], 1))]).T
  X_val = np.hstack([X_val, np.ones((X_val.shape[0], 1))]).T
  X_test = np.hstack([X_test, np.ones((X_test.shape[0], 1))]).T
  
  return X_train, y_train, X_val, y_val, X_test, y_test


# Invoke the above function to get our data.
X_train, y_train, X_val, y_val, X_test, y_test = 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


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

Softmax Classifier

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


In [3]:
# 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(10, 3073) * 0.0001
loss, grad = softmax_loss_naive(W, X_train, y_train, 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: 2.377014
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 we only have 10 classes, with current setting, the softmax result should be similar to random guess a class in those 10 classes . And this turns out to be near 0.10

In [4]:
# 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_train, y_train, 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_train, y_train, 0.0)[0]
grad_numerical = grad_check_sparse(f, W, grad, 10)


numerical: 1.612066 analytic: 1.612066, relative error: 0.000
numerical: -2.279963 analytic: -2.279963, relative error: 0.000
numerical: -1.628069 analytic: -1.628069, relative error: 0.000
numerical: -0.540226 analytic: -0.540226, relative error: 0.000
numerical: -0.857418 analytic: -0.857418, relative error: 0.000
numerical: 0.868607 analytic: 0.868607, relative error: 0.000
numerical: 2.001366 analytic: 2.001366, relative error: 0.000
numerical: -0.219715 analytic: -0.219715, relative error: 0.000
numerical: -0.871933 analytic: -0.871932, relative error: 0.000
numerical: -3.223025 analytic: -3.223025, relative error: 0.000

In [5]:
# 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_train, y_train, 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_train, y_train, 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: 2.377014e+00 computed in 5.644318s
vectorized loss: 2.377014e+00 computed in 0.620703s
Loss difference: 0.000000
Gradient difference: 0.000000

In [7]:
# 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 = [5e-7, 1e-7, 5e-6, 1e-6]
regularization_strengths = [5e4, 1e5]

################################################################################
# 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.                          #
################################################################################
import sys
verbose = True
for lr in learning_rates:
    for reg in regularization_strengths:
        if verbose: sys.stdout.write("Training with hyper parameter learning rate: %e, regularization: %e\n" 
                                     % ( lr, reg ))
        softmax = Softmax()
        loss_hist = softmax.train(X_train, y_train, learning_rate=lr, reg=reg,
                      num_iters=1500, verbose=False)
        
        y_train_pred = softmax.predict(X_train)
        training_accuracy = np.mean(y_train == y_train_pred)
        
        y_val_pred = softmax.predict(X_val)
        val_accuracy = np.mean(y_val == y_val_pred)
        
        results[lr, reg] = (training_accuracy, val_accuracy)
        if val_accuracy > best_val:
            best_val = val_accuracy
            best_softmax = softmax
                
# 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


Training with hyper parameter learning rate: 5.000000e-07, regularization: 5.000000e+04
Training with hyper parameter learning rate: 5.000000e-07, regularization: 1.000000e+05
Training with hyper parameter learning rate: 1.000000e-07, regularization: 5.000000e+04
Training with hyper parameter learning rate: 1.000000e-07, regularization: 1.000000e+05
Training with hyper parameter learning rate: 5.000000e-06, regularization: 5.000000e+04
Training with hyper parameter learning rate: 5.000000e-06, regularization: 1.000000e+05
Training with hyper parameter learning rate: 1.000000e-06, regularization: 5.000000e+04
Training with hyper parameter learning rate: 1.000000e-06, regularization: 1.000000e+05
cs231n/classifiers/softmax.py:103: RuntimeWarning: overflow encountered in double_scalars
  loss += 0.5 * reg * np.sum( W ** 2)
cs231n/classifiers/softmax.py:103: RuntimeWarning: overflow encountered in square
  loss += 0.5 * reg * np.sum( W ** 2)
Training with hyper parameter learning rate: 5.000000e-05, regularization: 5.000000e+04
Training with hyper parameter learning rate: 5.000000e-05, regularization: 1.000000e+05
lr 1.000000e-07 reg 5.000000e+04 train accuracy: 0.326816 val accuracy: 0.350000
lr 1.000000e-07 reg 1.000000e+05 train accuracy: 0.301184 val accuracy: 0.312000
lr 5.000000e-07 reg 5.000000e+04 train accuracy: 0.317776 val accuracy: 0.332000
lr 5.000000e-07 reg 1.000000e+05 train accuracy: 0.310327 val accuracy: 0.325000
lr 1.000000e-06 reg 5.000000e+04 train accuracy: 0.304327 val accuracy: 0.305000
lr 1.000000e-06 reg 1.000000e+05 train accuracy: 0.301082 val accuracy: 0.311000
lr 5.000000e-06 reg 5.000000e+04 train accuracy: 0.215204 val accuracy: 0.221000
lr 5.000000e-06 reg 1.000000e+05 train accuracy: 0.148796 val accuracy: 0.169000
lr 5.000000e-05 reg 5.000000e+04 train accuracy: 0.056082 val accuracy: 0.058000
lr 5.000000e-05 reg 1.000000e+05 train accuracy: 0.100265 val accuracy: 0.087000
best validation accuracy achieved during cross-validation: 0.350000
cs231n/classifiers/softmax.py:104: RuntimeWarning: overflow encountered in multiply
  dW += reg * W

In [10]:
# evaluate on test set
# Evaluate the best svm 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: 0.342000

In [11]:
# Visualize the learned weights for each class
w = best_softmax.W[:,:-1] # strip out the bias
w = w.reshape(10, 32, 32, 3)

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])



In [ ]: