softmax


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, 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.


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(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: 2.393762
sanity check: 2.302585

Inline Question 1:

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

Your answer: Fill this in


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_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: -0.269474 analytic: -0.269474, relative error: 1.144656e-07
numerical: -6.901038 analytic: -6.901038, relative error: 1.506834e-08
numerical: 0.239691 analytic: 0.239691, relative error: 1.228082e-07
numerical: 1.911780 analytic: 1.911779, relative error: 3.841555e-08
numerical: 0.014523 analytic: 0.014522, relative error: 2.767853e-06
numerical: -6.706081 analytic: -6.706081, relative error: 2.280536e-09
numerical: -0.046707 analytic: -0.046707, relative error: 1.778289e-07
numerical: -2.889523 analytic: -2.889523, relative error: 9.952746e-09
numerical: -0.160093 analytic: -0.160093, relative error: 1.150033e-07
numerical: 1.881268 analytic: 1.881268, relative error: 4.187309e-08
numerical: -0.665926 analytic: -0.665926, relative error: 1.083845e-07
numerical: -0.087886 analytic: -0.087886, relative error: 3.073739e-07
numerical: -0.408627 analytic: -0.408627, relative error: 6.285703e-08
numerical: -4.612586 analytic: -4.612586, relative error: 9.492958e-09
numerical: 2.369024 analytic: 2.369024, relative error: 2.706951e-09
numerical: -0.216228 analytic: -0.216228, relative error: 1.160926e-07
numerical: 1.589667 analytic: 1.589667, relative error: 1.422396e-08
numerical: 1.655432 analytic: 1.655432, relative error: 5.379646e-09
numerical: 2.245567 analytic: 2.245567, relative error: 1.167097e-08
numerical: -2.964640 analytic: -2.964640, relative error: 1.089711e-08

In [ ]:
# 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: 2.393762e+00 computed in 0.055214s
vectorized loss: 2.393762e+00 computed in 0.004401s
Loss difference: 0.000000
Gradient difference: 0.000000

In [ ]:
# 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, 5e-6, 1e-6]
regularization_strengths = [5e4, 1e-4, 1e-5]

################################################################################
# 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.                          #
################################################################################
for learning_rate in learning_rates:
    for reg in regularization_strengths:
        classifier = Softmax()
        try:
            classifier.train(X_train, y_train, learning_rate=learning_rate, reg=reg, num_iters=1500,
                            batch_size=200, verbose=False)
            y_train_predict = classifier.predict(X_train)
            y_val_predict = classifier.predict(X_val)
            train_accuracy = np.mean(y_train==y_train_predict)
            val_accuracy = np.mean(y_val==y_val_predict)
            results[(learning_rate, reg,)] = (train_accuracy, val_accuracy)
            if val_accuracy > best_val:
                best_val = val_accuracy
                best_softmax = classifier
        except Exception as e:
            print(e)
            pass
pass
################################################################################
#                              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

In [ ]:
# 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, )

In [ ]:
# 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])