# Run some setup code for this notebook.import random
import numpy as np
from cs231n.data_utils import load_CIFAR10
import matplotlib.pyplot as plt
# This is a bit of magic to make matplotlib figures appear inline in the# notebook rather than in a new window.
%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'# Some more magic so that the notebook will reload external python modules;# see http://stackoverflow.com/questions/1907993/autoreload-of-modules-in-ipython
%load_ext autoreload
%autoreload 2
↑首先第一步，仍是导入一些必要的库，设置一下图像参数。
CIFAR-10 Data Loading and Preprocessing
# 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)
# As a sanity check, we print out the size of the training and test data.print('Training data shape: ', X_train.shape)
print('Training labels shape: ', y_train.shape)
print('Test data shape: ', X_test.shape)
print('Test labels shape: ', y_test.shape)
↑这儿加载了CIFAR-10的数据，展示了练习集和测验集的大小：
Training data shape: (50000, 32, 32, 3)
Training labels shape: (50000,)
Test data shape: (10000, 32, 32, 3)
Test labels shape: (10000,)
# Visualize some examples from the dataset.# We show a few examples of training images from each class.
classes = ['plane', 'car', 'bird', 'cat', 'deer', 'dog', 'frog', 'horse', 'ship', 'truck']
num_classes = len(classes)
samples_per_class = 7for y, cls inenumerate(classes):
idxs = np.flatnonzero(y_train == y)
idxs = np.random.choice(idxs, samples_per_class, replace=False)
for i, idx inenumerate(idxs):
plt_idx = i * num_classes + y + 1
plt.subplot(samples_per_class, num_classes, plt_idx)
plt.imshow(X_train[idx].astype('uint8'))
plt.axis('off')
if i == 0:
plt.title(cls)
plt.show()
↑每一类选出7张图，并进行可视化，操作成果如下：
# Split the data into train, val, and test sets. In addition we will# create a small development set as a subset of the training data;# we can use this for development so our code runs faster.
num_training = 49000
num_validation = 1000
num_test = 1000
num_dev = 500# Our validation set will be num_validation points from the original# training set.
mask = range(num_training, num_training + num_validation)
X_val = X_train[mask]
y_val = y_train[mask]
# Our training set will be the first num_train points from the original# training set.
mask = range(num_training)
X_train = X_train[mask]
y_train = y_train[mask]
# We will also make a development set, which is a small subset of# the training set.
mask = np.random.choice(num_training, num_dev, replace=False)
X_dev = X_train[mask]
y_dev = y_train[mask]
# We use the first num_test points of the original test set as our# test set.
mask = range(num_test)
X_test = X_test[mask]
y_test = y_test[mask]
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: (49000, 32, 32, 3)
Train labels shape: (49000,)
Validation data shape: (1000, 32, 32, 3)
Validation labels shape: (1000,)
Test data shape: (1000, 32, 32, 3)
Test labels shape: (1000,)
# 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))
# As a sanity check, print out the shapes of the dataprint('Training data shape: ', X_train.shape)
print('Validation data shape: ', X_val.shape)
print('Test data shape: ', X_test.shape)
print('dev data shape: ', X_dev.shape)
↑将矩阵展开成向量：
Training data shape: (49000, 3072)
Validation data shape: (1000, 3072)
Test data shape: (1000, 3072)
dev data shape: (500, 3072)
# Preprocessing: subtract the mean image# first: compute the image mean based on the training data
mean_image = np.mean(X_train, axis=0)
print(mean_image[:10]) # print a few of the elements
plt.figure(figsize=(4,4))
plt.imshow(mean_image.reshape((32,32,3)).astype('uint8')) # visualize the mean image
plt.show()
# second: subtract the mean image from train and test data
X_train -= mean_image
X_val -= mean_image
X_test -= mean_image
X_dev -= mean_image
# third: append the bias dimension of ones (i.e. bias trick) so that our SVM# only has to worry about optimizing a single weight matrix W.
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))])
print(X_train.shape, X_val.shape, X_test.shape, X_dev.shape)
SVM Classifier Your code for this section will all be written inside cs231n/classifiers/linear_svm.py.
As you can see, we have prefilled the function svm_loss_naive which
uses for loops to evaluate the multiclass SVM loss function.
linear_svm.py
from builtins importrangeimport numpy as np
from random import shuffle
from past.builtins import xrange
defsvm_loss_naive(W, X, y, reg):
"""
Structured SVM 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
"""
dW = np.zeros(W.shape) # initialize the gradient as zero# compute the loss and the gradient
num_classes = W.shape[1] #将权重W的列数赋值给num_classes
num_train = X.shape[0] #将X的行数赋值给num_train
loss = 0.0for i inrange(num_train):
scores = X[i].dot(W) #每个样本Xi点乘W，成果大小为（1，C），也便是算出了这个样本每个分类的得分
correct_class_score = scores[y[i]] #正确的那个类别的得分for j inrange(num_classes):
if j == y[i]:
continue
margin = scores[j] - correct_class_score + 1# note delta = 1if margin > 0:
loss += margin #核算loss
dW [:,j] += X[i,:].T
dW [:,y[i]] += -X[i,:].T #更新梯度# Right now the loss is a sum over all training examples, but we want it# to be an average instead so we divide by num_train.
loss /= num_train
dW /= num_train #求平均值# Add regularization to the loss.
loss += reg * np.sum(W * W)
dW += reg * W #添加正则化############################################################################## TODO: ## Compute the gradient of the loss function and store it dW. ## Rather than first computing the loss and then computing the derivative, ## it may be simpler to compute the derivative at the same time that the ## loss is being computed. As a result you may need to modify some of the ## code above to compute the gradient. ############################################################################### *****START OF YOUR CODE (DO NOT DELETE/MODIFY THIS LINE)*****pass# *****END OF YOUR CODE (DO NOT DELETE/MODIFY THIS LINE)*****return loss, dW
defsvm_loss_vectorized(W, X, y, reg): #向量化"""
Structured SVM loss function, vectorized implementation.
Inputs and outputs are the same as svm_loss_naive.
"""
loss = 0.0
dW = np.zeros(W.shape) # initialize the gradient as zero############################################################################## TODO: ## Implement a vectorized version of the structured SVM loss, storing the ## result in loss. ############################################################################### *****START OF YOUR CODE (DO NOT DELETE/MODIFY THIS LINE)*****
num_train = X.shape[0] #500
scores = np.dot(X, W) #点乘,得到评分#print(scores.shape) #(500,10)
correct_class_scores = scores[np.arange(num_train), y] #变成了 (num_train,y)的矩阵
correct_class_scores = np.reshape(correct_class_scores, (num_train, -1))
#print(correct_class_scores.shape) # (500,1)
margin = scores - correct_class_scores + 1.0
margin[np.arange(num_train), y] = 0.0#把所有y的位置置0
margin[margin <= 0] = 0.0# max()公式的完成
loss += np.sum(margin) / num_train #核算loss
loss += 0.5 * reg * np.sum(W * W)
pass# *****END OF YOUR CODE (DO NOT DELETE/MODIFY THIS LINE)*****############################################################################## TODO: ## Implement a vectorized version of the gradient for the structured SVM ## loss, storing the result in dW. ## ## Hint: Instead of computing the gradient from scratch, it may be easier ## to reuse some of the intermediate values that you used to compute the ## loss. ############################################################################### *****START OF YOUR CODE (DO NOT DELETE/MODIFY THIS LINE)*****
margin[margin > 0] = 1.0
row_sum = np.sum(margin, axis=1)
margin[np.arange(num_train), y] = -row_sum
dW = 1.0 / num_train * np.dot(X.T, margin) + reg * W
pass# *****END OF YOUR CODE (DO NOT DELETE/MODIFY THIS LINE)*****return loss, dW
↑相关注释现已写在代码里了
svm.ipynb（中）
接下来咱们回到 svm.ipynb 持续完成剩余的作业
# Evaluate the naive implementation of the loss we provided for you:from cs231n.classifiers.linear_svm import svm_loss_naive
import time
# generate a random SVM weight matrix of small numbers
W = np.random.randn(3073, 10) * 0.0001
loss, grad = svm_loss_naive(W, X_dev, y_dev, 0.000005)
print('loss: %f' % (loss, ))
↑随机初始化权重W，核算loss，成果如下：
loss: 9.503077
上面的梯度全为0，下面用作业给出的函数核算并做梯度查看
The grad returned from the function above is right now all zero. Derive and implement the gradient for the SVM cost function and implement it inline inside the function svm_loss_naive. You will find it helpful to interleave your new code inside the existing function.
To check that you have correctly implemented the gradient correctly, you can numerically estimate the gradient of the loss function and compare the numeric estimate to the gradient that you computed. We have provided code that does this for you:
# Once you've implemented the gradient, recompute it with the code below# and gradient check it with the function we provided for you# Compute the loss and its gradient at W.
loss, grad = svm_loss_naive(W, X_dev, y_dev, 0.0)
# Numerically compute the gradient along several randomly chosen dimensions, and# compare them with your analytically computed gradient. The numbers should match# almost exactly along all dimensions.from cs231n.gradient_check import grad_check_sparse
f = lambda w: svm_loss_naive(w, X_dev, y_dev, 0.0)[0]
grad_numerical = grad_check_sparse(f, W, grad)
# do the gradient check once again with regularization turned on# you didn't forget the regularization gradient did you?
loss, grad = svm_loss_naive(W, X_dev, y_dev, 5e1)
f = lambda w: svm_loss_naive(w, X_dev, y_dev, 5e1)[0]
grad_numerical = grad_check_sparse(f, W, grad)
Inline Question 1
It is possible that once in a while a dimension in the gradcheck will not match exactly. What could such a discrepancy be caused by? Is it a reason for concern? What is a simple example in one dimension where a gradient check could fail? How would change the margin affect of the frequency of this happening? Hint: the SVM loss function is not strictly speaking differentiable
# Next implement the function svm_loss_vectorized; for now only compute the loss;# we will implement the gradient in a moment.
tic = time.time()
loss_naive, grad_naive = svm_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.linear_svm import svm_loss_vectorized
tic = time.time()
loss_vectorized, _ = svm_loss_vectorized(W, X_dev, y_dev, 0.000005)
toc = time.time()
print('Vectorized loss: %e computed in %fs' % (loss_vectorized, toc - tic))
# The losses should match but your vectorized implementation should be much faster.print('difference: %f' % (loss_naive - loss_vectorized))
↑向量化核算，一起求出运转时刻：
Naive loss: 9.503077e+00 computed in 0.166552s
Vectorized loss: 9.503077e+00 computed in 0.004987s
difference: 0.000000
# Complete the implementation of svm_loss_vectorized, and compute the gradient# of the loss function in a vectorized way.# The naive implementation and the vectorized implementation should match, but# the vectorized version should still be much faster.
tic = time.time()
_, grad_naive = svm_loss_naive(W, X_dev, y_dev, 0.000005)
toc = time.time()
print('Naive loss and gradient: computed in %fs' % (toc - tic))
tic = time.time()
_, grad_vectorized = svm_loss_vectorized(W, X_dev, y_dev, 0.000005)
toc = time.time()
print('Vectorized loss and gradient: computed in %fs' % (toc - tic))
# The loss is a single number, so it is easy to compare the values computed# by the two implementations. The gradient on the other hand is a matrix, so# we use the Frobenius norm to compare them.
difference = np.linalg.norm(grad_naive - grad_vectorized, ord='fro')
print('difference: %f' % difference)
↑梯度核算向量化：
Naive loss and gradient: computed in 0.159082s
Vectorized loss and gradient: computed in 0.004987s
difference: 0.000000
随机梯度下降
Stochastic Gradient Descent
We now have vectorized and efficient expressions for the loss, the gradient and our gradient matches the numerical gradient. We are therefore ready to do SGD to minimize the loss. Your code for this part will be written inside cs231n/classifiers/linear_classifier.py.
from __future__ import print_function
from builtins importrangefrom builtins importobjectimport numpy as np
from cs231n.classifiers.linear_svm import *
from cs231n.classifiers.softmax import *
from past.builtins import xrange
classLinearClassifier(object):
def__init__(self):
self.W = Nonedeftrain(self, X, y, learning_rate=1e-3, reg=1e-5, num_iters=100,
batch_size=200, verbose=False):
"""
Train this linear classifier using stochastic gradient descent.
Inputs:
- X: A numpy array of shape (N, D) containing training data; there are N
training samples each of dimension D.
- y: A numpy array of shape (N,) containing training labels; y[i] = c
means that X[i] has label 0 <= c < C for C classes.
- learning_rate: (float) learning rate for optimization.
- reg: (float) regularization strength.
- num_iters: (integer) number of steps to take when optimizing
- batch_size: (integer) number of training examples to use at each step.
- verbose: (boolean) If true, print progress during optimization.
Outputs:
A list containing the value of the loss function at each training iteration.
"""
num_train, dim = X.shape
num_classes = np.max(y) + 1# assume y takes values 0...K-1 where K is number of classesif self.W isNone:
# lazily initialize W
self.W = 0.001 * np.random.randn(dim, num_classes)
# Run stochastic gradient descent to optimize W
loss_history = []
for it inrange(num_iters):
X_batch = None
y_batch = None########################################################################## TODO: ## Sample batch_size elements from the training data and their ## corresponding labels to use in this round of gradient descent. ## Store the data in X_batch and their corresponding labels in ## y_batch; after sampling X_batch should have shape (batch_size, dim) ## and y_batch should have shape (batch_size,) ## ## Hint: Use np.random.choice to generate indices. Sampling with ## replacement is faster than sampling without replacement. ########################################################################### *****START OF YOUR CODE (DO NOT DELETE/MODIFY THIS LINE)*****
sample_index = np.random.choice(num_train,batch_size,replace=False)
X_batch = X[sample_index,:]
y_batch = y[sample_index]
pass# *****END OF YOUR CODE (DO NOT DELETE/MODIFY THIS LINE)*****# evaluate loss and gradient
loss, grad = self.loss(X_batch, y_batch, reg)
loss_history.append(loss)
# perform parameter update########################################################################## TODO: ## Update the weights using the gradient and the learning rate. ########################################################################### *****START OF YOUR CODE (DO NOT DELETE/MODIFY THIS LINE)*****
self.W = self.W - learning_rate*grad
pass# *****END OF YOUR CODE (DO NOT DELETE/MODIFY THIS LINE)*****if verbose and it % 100 == 0:
print('iteration %d / %d: loss %f' % (it, num_iters, loss))
return loss_history
defpredict(self, X):
"""
Use the trained weights of this linear classifier to predict labels for
data points.
Inputs:
- X: A numpy array of shape (N, D) containing training data; there are N
training samples each of dimension D.
Returns:
- y_pred: Predicted labels for the data in X. y_pred is a 1-dimensional
array of length N, and each element is an integer giving the predicted
class.
"""
y_pred = np.zeros(X.shape[0])
############################################################################ TODO: ## Implement this method. Store the predicted labels in y_pred. ############################################################################# *****START OF YOUR CODE (DO NOT DELETE/MODIFY THIS LINE)*****
score = X.dot(self.W)
y_pred = np.argmax(score,axis=1)
pass# *****END OF YOUR CODE (DO NOT DELETE/MODIFY THIS LINE)*****return y_pred
defloss(self, X_batch, y_batch, reg):
"""
Compute the loss function and its derivative.
Subclasses will override this.
Inputs:
- X_batch: A numpy array of shape (N, D) containing a minibatch of N
data points; each point has dimension D.
- y_batch: A numpy array of shape (N,) containing labels for the minibatch.
- reg: (float) regularization strength.
Returns: A tuple containing:
- loss as a single float
- gradient with respect to self.W; an array of the same shape as W
"""passclassLinearSVM(LinearClassifier):
""" A subclass that uses the Multiclass SVM loss function """defloss(self, X_batch, y_batch, reg):
return svm_loss_vectorized(self.W, X_batch, y_batch, reg)
classSoftmax(LinearClassifier):
""" A subclass that uses the Softmax + Cross-entropy loss function """defloss(self, X_batch, y_batch, reg):
return softmax_loss_vectorized(self.W, X_batch, y_batch, reg)
svm.ipynb（下）
咱们持续回到SVM作业
# In the file linear_classifier.py, implement SGD in the function# LinearClassifier.train() and then run it with the code below.from cs231n.classifiers import LinearSVM
svm = LinearSVM()
tic = time.time()
loss_hist = svm.train(X_train, y_train, learning_rate=1e-7, reg=2.5e4,
num_iters=1500, verbose=True)
toc = time.time()
print('That took %fs' % (toc - tic))
↑进行梯度下降：
iteration 0 / 1500: loss 404.803504
iteration 100 / 1500: loss 240.999746
iteration 200 / 1500: loss 147.223258
iteration 300 / 1500: loss 90.523987
iteration 400 / 1500: loss 57.061188
iteration 500 / 1500: loss 34.765473
iteration 600 / 1500: loss 23.564806
iteration 700 / 1500: loss 16.603101
iteration 800 / 1500: loss 11.401976
iteration 900 / 1500: loss 9.060058
iteration 1000 / 1500: loss 7.689659
iteration 1100 / 1500: loss 6.790426
iteration 1200 / 1500: loss 5.630356
iteration 1300 / 1500: loss 5.295763
iteration 1400 / 1500: loss 5.502182
That took 12.969263s
# A useful debugging strategy is to plot the loss as a function of# iteration number:
plt.plot(loss_hist)
plt.xlabel('Iteration number')
plt.ylabel('Loss value')
plt.show()
画出图表：
# Write the LinearSVM.predict function and evaluate the performance on both the# training and validation set
y_train_pred = svm.predict(X_train)
print('training accuracy: %f' % (np.mean(y_train == y_train_pred), ))
y_val_pred = svm.predict(X_val)
print('validation accuracy: %f' % (np.mean(y_val == y_val_pred), ))
↑预测函数，并评价准确率：
training accuracy: 0.379204
validation accuracy: 0.388000
# 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 about 0.39 on the validation set.# Note: you may see runtime/overflow warnings during hyper-parameter search. # This may be caused by extreme values, and is not a bug.# results is dictionary mapping tuples of the form# (learning_rate, regularization_strength) to tuples of the form# (training_accuracy, validation_accuracy). The accuracy is simply the fraction# of data points that are correctly classified.
results = {}
best_val = -1# The highest validation accuracy that we have seen so far.
best_svm = None# The LinearSVM object that achieved the highest validation rate.################################################################################# TODO: ## Write code that chooses the best hyperparameters by tuning on the validation ## set. For each combination of hyperparameters, train a linear SVM on the ## training set, compute its accuracy on the training and validation sets, and ## store these numbers in the results dictionary. In addition, store the best ## validation accuracy in best_val and the LinearSVM object that achieves this ## accuracy in best_svm. ## ## Hint: You should use a small value for num_iters as you develop your ## validation code so that the SVMs don't take much time to train; once you are ## confident that your validation code works, you should rerun the validation ## code with a larger value for num_iters. ################################################################################## Provided as a reference. You may or may not want to change these hyperparameters
learning_rates = [1e-7, 5e-5]
regularization_strengths = [2.5e4, 5e4]
# *****START OF YOUR CODE (DO NOT DELETE/MODIFY THIS LINE)*****for rs in regularization_strengths:
for lr in learning_rates:
svm = LinearSVM()
loss_hist = svm.train(X_train, y_train, lr, rs, num_iters=3000)
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_svm = svm
results[(lr,rs)] = train_accuracy, val_accuracy
pass# *****END OF YOUR CODE (DO NOT DELETE/MODIFY THIS LINE)*****# Print out results.for lr, reg insorted(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)
↑挑选最佳的学习率和正则化参数：
lr 1.000000e-07 reg 2.500000e+04 train accuracy: 0.380592 val accuracy: 0.384000
lr 1.000000e-07 reg 5.000000e+04 train accuracy: 0.367898 val accuracy: 0.375000
lr 5.000000e-05 reg 2.500000e+04 train accuracy: 0.146327 val accuracy: 0.142000
lr 5.000000e-05 reg 5.000000e+04 train accuracy: 0.100265 val accuracy: 0.087000
best validation accuracy achieved during cross-validation: 0.384000
# Visualize the cross-validation resultsimport math
import pdb
# pdb.set_trace()
x_scatter = [math.log10(x[0]) for x in results]
y_scatter = [math.log10(x[1]) for x in results]
# plot training accuracy
marker_size = 100
colors = [results[x][0] for x in results]
plt.subplot(2, 1, 1)
plt.tight_layout(pad=3)
plt.scatter(x_scatter, y_scatter, marker_size, c=colors, cmap=plt.cm.coolwarm)
plt.colorbar()
plt.xlabel('log learning rate')
plt.ylabel('log regularization strength')
plt.title('CIFAR-10 training accuracy')
# plot validation accuracy
colors = [results[x][1] for x in results] # default size of markers is 20
plt.subplot(2, 1, 2)
plt.scatter(x_scatter, y_scatter, marker_size, c=colors, cmap=plt.cm.coolwarm)
plt.colorbar()
plt.xlabel('log learning rate')
plt.ylabel('log regularization strength')
plt.title('CIFAR-10 validation accuracy')
plt.show()
↑可视化：
# Evaluate the best svm on test set
y_test_pred = best_svm.predict(X_test)
test_accuracy = np.mean(y_test == y_test_pred)
print('linear SVM on raw pixels final test set accuracy: %f' % test_accuracy)
↑终究准确率：
linear SVM on raw pixels final test set accuracy: 0.368000
# Visualize the learned weights for each class.# Depending on your choice of learning rate and regularization strength, these may# or may not be nice to look at.
w = best_svm.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 inrange(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])
