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

from __future__ import print_function

%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 = list(range(num_training, num_training + num_validation))
    X_val = X_train[mask]
    y_val = y_train[mask]
    mask = list(range(num_training))
    X_train = X_train[mask]
    y_train = y_train[mask]
    mask = list(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


# 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

# 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 [11]:
# 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: 2.355601
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 [13]:
# 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, 5e1)
f = lambda w: softmax_loss_naive(w, X_dev, y_dev, 5e1)[0]
grad_numerical = grad_check_sparse(f, W, grad, 10)


numerical: 0.604963 analytic: 0.788023, relative error: 1.314157e-01
numerical: 0.266510 analytic: 0.796899, relative error: 4.987634e-01
numerical: 0.071656 analytic: 0.480887, relative error: 7.406305e-01
numerical: 0.428493 analytic: -0.559722, relative error: 1.000000e+00
numerical: -1.077128 analytic: -0.172891, relative error: 7.233790e-01
numerical: -2.347162 analytic: -0.762564, relative error: 5.095621e-01
numerical: 0.177544 analytic: 0.122518, relative error: 1.833828e-01
numerical: 3.516772 analytic: 3.030395, relative error: 7.428813e-02
numerical: 0.061933 analytic: -0.530364, relative error: 1.000000e+00
numerical: 0.848428 analytic: 1.726193, relative error: 3.409299e-01
numerical: -1.001548 analytic: -0.984319, relative error: 8.676068e-03
numerical: 0.024697 analytic: 0.394355, relative error: 8.821275e-01
numerical: 1.742707 analytic: 0.807742, relative error: 3.665886e-01
numerical: 2.190660 analytic: 1.317826, relative error: 2.487779e-01
numerical: 2.584184 analytic: 1.238520, relative error: 3.520187e-01
numerical: 0.619409 analytic: 0.356417, relative error: 2.695067e-01
numerical: 2.353268 analytic: 1.180235, relative error: 3.319746e-01
numerical: 0.532905 analytic: -0.040667, relative error: 1.000000e+00
numerical: -0.860948 analytic: -0.207516, relative error: 6.115626e-01
numerical: 2.005497 analytic: 1.160688, relative error: 2.668224e-01

In [16]:
# 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, 0.000005)
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, 0.000005)
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: 2.355601e+00 computed in 0.024065s
vectorized loss: 2.355601e+00 computed in 0.019049s
Loss difference: 0.000000
Gradient difference: 0.000000

In [18]:
# 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 = [2.5e4, 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 rate in learning_rates:
    for regularization in regularization_strengths:
        softmax = Softmax()
        softmax.train(X_train, y_train, learning_rate=rate,
                      reg=regularization, num_iters=3000)
        y_train_pred = softmax.predict(X_train)
        y_val_pred = softmax.predict(X_val)
        train_acc = np.mean(y_train_pred == y_train)
        val_acc = np.mean(y_val_pred == y_val)
        results[(rate, regularization)] = (train_acc, val_acc)
        if(best_val < val_acc):
            best_val = val_acc
            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)


lr 1.000000e-07 reg 2.500000e+04 train accuracy: 0.243939 val accuracy: 0.260000
lr 1.000000e-07 reg 5.000000e+04 train accuracy: 0.244592 val accuracy: 0.254000
lr 5.000000e-07 reg 2.500000e+04 train accuracy: 0.251980 val accuracy: 0.260000
lr 5.000000e-07 reg 5.000000e+04 train accuracy: 0.244469 val accuracy: 0.263000
best validation accuracy achieved during cross-validation: 0.263000

In [19]:
# 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: 0.261000

Inline Question - True or False

It's possible to add a new datapoint to a training set that would leave the SVM loss unchanged, but this is not the case with the Softmax classifier loss.

Your answer:

Your explanation:


In [20]:
# 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 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])



In [ ]: