CS231n（Spring 2019）Assignment 1 - Softmax
Preprocessing
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'
# 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
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
# 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)
即将数据集分为 Training data， Validation data 和 Test data，数据规模如下所示：
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模块
（in cs231n/classifiers/softmax.py）
from builtins import range
import numpy as np
from random import shuffle
from past.builtins import xrange
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! #
#############################################################################
# *****START OF YOUR CODE (DO NOT DELETE/MODIFY THIS LINE)*****
num_train = X.shape[0] #N
num_classes = W.shape[1] #C
for i in range(num_train):
f_i = X[i].dot(W)
f_i -= np.max(f_i)
sum_j = np.sum(np.exp(f_i))
p = lambda k:np.exp(f_i[k]) / sum_j
loss += -np.log(p(y[i]))
for k in range(num_classes):
p_k = p(k)
dW[:,k] += (p_k - (k==y[i])) * X[i]
loss /= num_train
loss += 0.5 * reg * np.sum(W * W)
dW /= num_train
dW += reg*W
# *****END OF YOUR CODE (DO NOT DELETE/MODIFY THIS LINE)*****
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! #
#############################################################################
# *****START OF YOUR CODE (DO NOT DELETE/MODIFY THIS LINE)*****
num_train = X.shape[0] #N
f = X.dot(W)
#X(N,D) W(D,C) f(N,C)
f -= np.max(f,axis = 1,keepdims = True)
sum_f = np.sum(np.exp(f),axis = 1,keepdims = True)
p = np.exp(f)/sum_f
loss = np.sum(-np.log(p[np.arange(num_train),y]))
ind = np.zeros_like(p)
ind[np.arange(num_train),y] = 1
dW = X.T.dot(p - ind)
loss /= num_train
loss += 0.5 * reg * np.sum(W * W)
dW /= num_train
dW += reg*W
# *****END OF YOUR CODE (DO NOT DELETE/MODIFY THIS LINE)*****
return loss, dW
其中关于loss和dW的推导见博文：
《多分类器softmax——绝对简单易懂的梯度推导》- 月下花弄影
结果验证
# 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: 2.369896
sanity check: 2.302585
# 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: 1.160023 analytic: 1.160022, relative error: 3.931098e-08
numerical: 3.555074 analytic: 3.555074, relative error: 1.285894e-08
numerical: -0.345163 analytic: -0.345163, relative error: 1.247007e-07
numerical: 0.430499 analytic: 0.430499, relative error: 1.252352e-07
numerical: -0.476966 analytic: -0.476966, relative error: 6.130511e-08
numerical: 4.303385 analytic: 4.303385, relative error: 1.042744e-08
numerical: 1.248775 analytic: 1.248775, relative error: 7.940834e-08
numerical: -4.852993 analytic: -4.852993, relative error: 7.047699e-09
numerical: 0.661671 analytic: 0.661671, relative error: 7.513673e-08
numerical: 1.635486 analytic: 1.635486, relative error: 1.502169e-08
numerical: -0.625054 analytic: -0.625054, relative error: 1.526714e-08
numerical: 1.361989 analytic: 1.361989, relative error: 3.627339e-09
numerical: 0.280297 analytic: 0.280297, relative error: 2.871195e-07
numerical: -6.703095 analytic: -6.703095, relative error: 5.533269e-09
numerical: -0.310387 analytic: -0.310387, relative error: 1.759413e-07
numerical: -1.079213 analytic: -1.079213, relative error: 6.495149e-09
numerical: -2.684781 analytic: -2.684781, relative error: 5.743039e-09
numerical: 1.747734 analytic: 1.747734, relative error: 1.403972e-09
numerical: -2.146907 analytic: -2.146907, relative error: 2.798771e-09
numerical: -1.233420 analytic: -1.233420, relative error: 1.297245e-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, 0.000005)
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: 2.369896e+00 computed in 0.147392s
vectorized loss: 2.369896e+00 computed in 0.003993s
Loss difference: 0.000000
Gradient difference: 0.000000
调参部分
# 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. #
################################################################################
# *****START OF YOUR CODE (DO NOT DELETE/MODIFY THIS LINE)*****
for reg in regularization_strengths:
for lr in learning_rates:
svm = Softmax()
loss_hist = svm.train(X_train, y_train, lr, reg, num_iters=1500)
y_train_pred = svm.predict(X_train)
train_accuracy = np.mean(y_train == y_train_pred)
y_val_pred = svm.predict(X_val)
val_accuracy = np.mean(y_val == y_val_pred)
if val_accuracy > best_val:
best_val = val_accuracy
best_softmax = svm
results[(lr,reg)] = train_accuracy, val_accuracy
# *****END OF YOUR CODE (DO NOT DELETE/MODIFY THIS LINE)*****
# 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: 0.351061 val accuracy: 0.363000
lr 1.000000e-07 reg 5.000000e+04 train accuracy: 0.322796 val accuracy: 0.339000
lr 5.000000e-07 reg 2.500000e+04 train accuracy: 0.350510 val accuracy: 0.366000
lr 5.000000e-07 reg 5.000000e+04 train accuracy: 0.326204 val accuracy: 0.342000
best validation accuracy achieved during cross-validation: 0.366000
# 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: 0.363000
结果可视化
# 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])
Figure：W的可视化结果