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):
  """
  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: 2.327451
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 There are 10 classes, so the accuray of guessing is around 10%, giving loss -log(0.1).


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: 2.937596 analytic: 2.937596, relative error: 6.572934e-09
numerical: -0.561595 analytic: -0.561595, relative error: 1.131728e-10
numerical: 0.402317 analytic: 0.402317, relative error: 1.679271e-08
numerical: 1.908357 analytic: 1.908357, relative error: 3.028894e-09
numerical: 0.956833 analytic: 0.956833, relative error: 1.295795e-08
numerical: 1.042255 analytic: 1.042255, relative error: 8.474881e-08
numerical: -0.014579 analytic: -0.014579, relative error: 6.439241e-07
numerical: -0.212309 analytic: -0.212309, relative error: 7.422132e-08
numerical: 0.535360 analytic: 0.535360, relative error: 2.296886e-08
numerical: -0.688214 analytic: -0.688214, relative error: 7.154233e-09

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: 2.327451e+00 computed in 4.924036s
vectorized loss: 2.327451e+00 computed in 0.206513s
Loss difference: 0.000000
Gradient difference: 0.000000

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

################################################################################
# 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()
        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
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


lr 1.000000e-07 reg 1.000000e-05 train accuracy: 0.251755 val accuracy: 0.267000
lr 1.000000e-07 reg 5.000000e-05 train accuracy: 0.252510 val accuracy: 0.253000
lr 1.000000e-07 reg 1.000000e-04 train accuracy: 0.252306 val accuracy: 0.253000
lr 1.000000e-07 reg 5.000000e-04 train accuracy: 0.247082 val accuracy: 0.225000
lr 1.000000e-07 reg 5.000000e+04 train accuracy: 0.325816 val accuracy: 0.345000
lr 5.000000e-07 reg 1.000000e-05 train accuracy: 0.317918 val accuracy: 0.333000
lr 5.000000e-07 reg 5.000000e-05 train accuracy: 0.321204 val accuracy: 0.308000
lr 5.000000e-07 reg 1.000000e-04 train accuracy: 0.313551 val accuracy: 0.304000
lr 5.000000e-07 reg 5.000000e-04 train accuracy: 0.315694 val accuracy: 0.301000
lr 5.000000e-07 reg 5.000000e+04 train accuracy: 0.328531 val accuracy: 0.330000
lr 1.000000e-06 reg 1.000000e-05 train accuracy: 0.346776 val accuracy: 0.363000
lr 1.000000e-06 reg 5.000000e-05 train accuracy: 0.350837 val accuracy: 0.364000
lr 1.000000e-06 reg 1.000000e-04 train accuracy: 0.347327 val accuracy: 0.356000
lr 1.000000e-06 reg 5.000000e-04 train accuracy: 0.348327 val accuracy: 0.356000
lr 1.000000e-06 reg 5.000000e+04 train accuracy: 0.314571 val accuracy: 0.317000
lr 5.000000e-06 reg 1.000000e-05 train accuracy: 0.394204 val accuracy: 0.357000
lr 5.000000e-06 reg 5.000000e-05 train accuracy: 0.383408 val accuracy: 0.345000
lr 5.000000e-06 reg 1.000000e-04 train accuracy: 0.390163 val accuracy: 0.380000
lr 5.000000e-06 reg 5.000000e-04 train accuracy: 0.386490 val accuracy: 0.358000
lr 5.000000e-06 reg 5.000000e+04 train accuracy: 0.205755 val accuracy: 0.222000
best validation accuracy achieved during cross-validation: 0.380000

In [7]:
# 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: 0.340000

In [8]:
# 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])