# cs231n - assignment1 - neural net 梯度推导

## Implementing a Neural Network

In this exercise we will develop a neural network with fully-connected layers to perform classification, and test it out on the CIFAR-10 dataset.

Propagation:

FC1_out=X?W1+b1???(1) FC1\_out = X \cdot W_1+b_1 ---(1)

H_out=maximum(0,FC1_out)???(2) H\_out = maximum(0, FC1\_out)---(2)

FC2_out=H_out?W2+b2???(3) FC2\_out = H\_out \cdot W_2+b_2---(3)

final_output=softmax(FC2_out)???(4) final\_output = softmax(FC2\_out) ---(4)

Backpropogation

?L?FC2_out=final_output[N×C]?MaskMat[N×C]???（5) \frac{\partial L}{\partial FC2\_out } = final\_output_{[N \times C]} - MaskMat_{[N \times C] } ---（5)

?L?W2=?FC2_out?W2?L?FC2_out=H_outT??L?FC2_out???(6) {\frac{\partial L}{\partial W_2 }} = \frac{\partial FC2\_out}{\partial W_2 } \frac{\partial L}{\partial FC2\_out } =H\_out^T \cdot \frac{\partial L}{\partial FC2\_out } ---(6)

?L?b2=?FC2_out?b2?L?FC2_out=[1...1][1×H]??L?FC2_out???(7) \frac{\partial L}{\partial b_2 } = \frac{\partial FC2\_out}{\partial b_2 } \frac{\partial L}{\partial FC2\_out } = [1...1]_{ [1 \times H]} \cdot \frac{\partial L}{\partial FC2\_out } ---(7)

?L?H_out=?L?FC2_out?FC2_out?H_out=?L?FC2_outWT2,?L?H_out=maxmium(?L?H_out,0)???(8) \frac{\partial L}{\partial H\_out}= \frac{\partial L}{\partial FC2\_out } \frac{\partial FC2\_out}{\partial H\_out } = \frac{\partial L}{\partial FC2\_out }W_2^T , \frac{\partial L}{\partial H\_out} = maxmium( \frac{\partial L}{\partial H\_out}, 0)--- (8)

?L?W1=?H_cout?W1??L?H_out=XT??L?H_out???(9) {\frac{\partial L}{\partial W_1 }} =\frac{\partial H\_cout}{\partial W_1} \cdot \frac{\partial L}{\partial H\_out}=X^T \cdot \frac{\partial L}{\partial H\_out}---(9)

?L?b1=?H_cout?b1??L?H_out=[1...1][1×N]??L?H_out???(10) {\frac{\partial L}{\partial b_1 }} =\frac{\partial H\_cout}{\partial b_1} \cdot \frac{\partial L}{\partial H\_out}=[1...1]_{[1 \times N]} \cdot \frac{\partial L}{\partial H\_out}---(10)

# neural_net.py import numpy as np import matplotlib.pyplot as plt class TwoLayerNet(object): """ A two-layer fully-connected neural network. The net has an input dimension of N, a hidden layer dimension of H, and performs classification over C classes. We train the network with a softmax loss function and L2 regularization on the weight matrices. The network uses a ReLU nonlinearity after the first fully connected layer. In other words, the network has the following architecture: input - fully connected layer - ReLU - fully connected layer - softmax The outputs of the second fully-connected layer are the scores for each class. """ def __init__(self, input_size, hidden_size, output_size, std=1e-4): """ Initialize the model. Weights are initialized to small random values and biases are initialized to zero. Weights and biases are stored in the variable self.params, which is a dictionary with the following keys: W1: First layer weights; has shape (D, H) b1: First layer biases; has shape (H,) W2: Second layer weights; has shape (H, C) b2: Second layer biases; has shape (C,) Inputs: - input_size: The dimension D of the input data. - hidden_size: The number of neurons H in the hidden layer. - output_size: The number of classes C. """ self.params = {} self.params['W1'] = std * np.random.randn(input_size, hidden_size) self.params['b1'] = np.zeros(hidden_size) self.params['W2'] = std * np.random.randn(hidden_size, output_size) self.params['b2'] = np.zeros(output_size) def loss(self, X, y=None, reg=0.0): """ Compute the loss and gradients for a two layer fully connected neural network. Inputs: - X: Input data of shape (N, D). Each X[i] is a training sample. - y: Vector of training labels. y[i] is the label for X[i], and each y[i] is an integer in the range 0 <= y[i] < C. This parameter is optional; if it is not passed then we only return scores, and if it is passed then we instead return the loss and gradients. - reg: Regularization strength. Returns: If y is None, return a matrix scores of shape (N, C) where scores[i, c] is the score for class c on input X[i]. If y is not None, instead return a tuple of: - loss: Loss (data loss and regularization loss) for this batch of training samples. - grads: Dictionary mapping parameter names to gradients of those parameters with respect to the loss function; has the same keys as self.params. """ # Unpack variables from the params dictionary W1, b1 = self.params['W1'], self.params['b1'] W2, b2 = self.params['W2'], self.params['b2'] N, D = X.shape # Compute the forward pass scores = None ############################################################################# # TODO: Perform the forward pass, computing the class scores for the input. # # Store the result in the scores variable, which should be an array of # # shape (N, C). # ############################################################################# # evaluate class scores, [N x K] hidden_layer = np.maximum(0, np.dot(X,W1)+b1) # ReLU activation scores = np.dot(hidden_layer, W2)+b2 ############################################################################# # END OF YOUR CODE # ############################################################################# # If the targets are not given then jump out, we're done if y is None: return scores # Compute the loss loss = None ############################################################################# # TODO: Finish the forward pass, and compute the loss. This should include # # both the data loss and L2 regularization for W1 and W2. Store the result # # in the variable loss, which should be a scalar. Use the Softmax # # classifier loss. So that your results match ours, multiply the # # regularization loss by 0.5 # ############################################################################# # compute the class probabilities #scores -= np.max(scores, axis = 1)[:, np.newaxis] #exp_scores = np.exp(scores) exp_scores = np.exp(scores-np.max(scores, axis=1, keepdims=True)) probs = exp_scores/np.sum(exp_scores, axis=1, keepdims=True) #[N X C] correct_logprobs = -np.log(probs[range(N),y]) data_loss = np.sum(correct_logprobs)/N reg_loss = 0.5 * reg * ( np.sum(W1*W1) + np.sum(W2*W2) ) loss = data_loss + reg_loss ############################################################################# # END OF YOUR CODE # ############################################################################# # Backward pass: compute gradients grads = {} ############################################################################# # TODO: Compute the backward pass, computing the derivatives of the weights # # and biases. Store the results in the grads dictionary. For example, # # grads['W1'] should store the gradient on W1, and be a matrix of same size # ############################################################################# # compute the gradient on scores dscores = probs dscores[range(N),y] -= 1 dscores /= N # backpropate the gradient to the parameters # first backprop into parameters W2 and b2 dW2 = np.dot(hidden_layer.T, dscores) db2 = np.sum(dscores, axis=0, keepdims=False) # next backprop into hidden layer dhidden = np.dot(dscores, W2.T) # backprop the ReLU non-linearity dhidden[hidden_layer <= 0] = 0 # finally into W,b dW1 = np.dot(X.T, dhidden) db1 = np.sum(dhidden, axis=0, keepdims=False) # add regularization gradient contribution dW2 += reg * W2 dW1 += reg * W1 grads['W1'] = dW1 grads['W2'] = dW2 grads['b1'] = db1 grads['b2'] = db2 #print dW1.shape, dW2.shape, db1.shape, db2.shape ############################################################################# # END OF YOUR CODE # ############################################################################# return loss, grads def train(self, X, y, X_val, y_val, learning_rate=1e-3, learning_rate_decay=0.95, reg=1e-5, num_iters=100, batch_size=200, verbose=False): """ Train this neural network using stochastic gradient descent. Inputs: - X: A numpy array of shape (N, D) giving training data. - y: A numpy array f shape (N,) giving training labels; y[i] = c means that X[i] has label c, where 0 <= c < C. - X_val: A numpy array of shape (N_val, D) giving validation data. - y_val: A numpy array of shape (N_val,) giving validation labels. - learning_rate: Scalar giving learning rate for optimization. - learning_rate_decay: Scalar giving factor used to decay the learning rate after each epoch. - reg: Scalar giving regularization strength. - num_iters: Number of steps to take when optimizing. - batch_size: Number of training examples to use per step. - verbose: boolean; if true print progress during optimization. """ num_train = X.shape[0] iterations_per_epoch = max(num_train / batch_size, 1) # Use SGD to optimize the parameters in self.model loss_history = [] train_acc_history = [] val_acc_history = [] for it in xrange(num_iters): X_batch = None y_batch = None ######################################################################### # TODO: Create a random minibatch of training data and labels, storing # # them in X_batch and y_batch respectively. # ######################################################################### sample_index = np.random.choice(num_train, batch_size, replace=True) X_batch = X[sample_index, :] y_batch = y[sample_index] ######################################################################### # END OF YOUR CODE # ######################################################################### # Compute loss and gradients using the current minibatch loss, grads = self.loss(X_batch, y=y_batch, reg=reg) loss_history.append(loss) ######################################################################### # TODO: Use the gradients in the grads dictionary to update the # # parameters of the network (stored in the dictionary self.params) # # using stochastic gradient descent. You'll need to use the gradients # # stored in the grads dictionary defined above. # ######################################################################### dW1 = grads['W1'] dW2 = grads['W2'] db1 = grads['b1'] db2 = grads['b2'] self.params['W1'] -= learning_rate*dW1 self.params['W2'] -= learning_rate*dW2 self.params['b1'] -= learning_rate*db1 self.params['b2'] -= learning_rate*db2 ######################################################################### # END OF YOUR CODE # ######################################################################### if verbose and it % 100 == 0: print 'iteration %d / %d: loss %f' % (it, num_iters, loss) # Every epoch, check train and val accuracy and decay learning rate. if it % iterations_per_epoch == 0: # Check accuracy train_acc = (self.predict(X_batch) == y_batch).mean() val_acc = (self.predict(X_val) == y_val).mean() train_acc_history.append(train_acc) val_acc_history.append(val_acc) # Decay learning rate learning_rate *= learning_rate_decay return { 'loss_history': loss_history, 'train_acc_history': train_acc_history, 'val_acc_history': val_acc_history, } def predict(self, X): """ Use the trained weights of this two-layer network to predict labels for data points. For each data point we predict scores for each of the C classes, and assign each data point to the class with the highest score. Inputs: - X: A numpy array of shape (N, D) giving N D-dimensional data points to classify. Returns: - y_pred: A numpy array of shape (N,) giving predicted labels for each of the elements of X. For all i, y_pred[i] = c means that X[i] is predicted to have class c, where 0 <= c < C. """ y_pred = None ########################################################################### # TODO: Implement this function; it should be VERY simple! # ########################################################################### hidden_lay = np.maximum(0, np.dot(X,self.params['W1'])+self.params['b1']) y_pred = np.argmax( np.dot(hidden_lay, self.params['W2']), axis=1) ########################################################################### # END OF YOUR CODE # ########################################################################### return y_pred 

What’s wrong?. Looking at the visualizations above, we see that the loss is decreasing more or less linearly, which seems to suggest that the learning rate may be too low. Moreover, there is no gap between the training and validation accuracy, suggesting that the model we used has low capacity, and that we should increase its size. On the other hand, with a very large model we would expect to see more overfitting, which would manifest itself as a very large gap between the training and validation accuracy.

Tuning. Tuning the hyperparameters and developing intuition for how they affect the final performance is a large part of using Neural Networks, so we want you to get a lot of practice. Below, you should experiment with different values of the various hyperparameters, including hidden layer size, learning rate, numer of training epochs, and regularization strength. You might also consider tuning the learning rate decay, but you should be able to get good performance using the default value.

Approximate results. You should be aim to achieve a classification accuracy of greater than 48% on the validation set. Our best network gets over 52% on the validation set.

Experiment: You goal in this exercise is to get as good of a result on CIFAR-10 as you can, with a fully-connected Neural Network. For every 1% above 52% on the Test set we will award you with one extra bonus point. Feel free implement your own techniques (e.g. PCA to reduce dimensionality, or adding dropout, or adding features to the solver, etc.).

# two_layer_net.ipynb best_net = None # store the best model into this best_stats = None ################################################################################# # TODO: Tune hyperparameters using the validation set. Store your best trained # # model in best_net. # # # # To help debug your network, it may help to use visualizations similar to the # # ones we used above; these visualizations will have significant qualitative # # differences from the ones we saw above for the poorly tuned network. # # # # Tweaking hyperparameters by hand can be fun, but you might find it useful to # # write code to sweep through possible combinations of hyperparameters # # automatically like we did on the previous exercises. # ################################################################################# input_size = 32 * 32 * 3 hidden_size = 300 num_classes = 10 results = {} best_val = -1 learning_rates = [1e-3, 1.2e-3, 1.4e-3, 1.6e-3, 1.8e-3] regularization_strengths = [1e-4, 1e-3, 1e-2] params = [(x,y) for x in learning_rates for y in regularization_strengths ] for lrate, regular in params: net = TwoLayerNet(input_size, hidden_size, num_classes) # Train the network stats = net.train(X_train, y_train, X_val, y_val, num_iters=1600, batch_size=400, learning_rate=lrate, learning_rate_decay=0.90, reg=regular, verbose=False) # Predict on the validation set accuracy_train = (net.predict(X_train) == y_train).mean() accuracy_val = (net.predict(X_val) == y_val).mean() results[(lrate, regular)] = (accuracy_train, accuracy_val) if( best_val < accuracy_val ): best_val = accuracy_val best_net = net best_stats = stats # 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 # Plot the loss function and train / validation accuracies plt.subplot(2, 1, 1) plt.plot(best_stats['loss_history']) plt.title('Loss history') plt.xlabel('Iteration') plt.ylabel('Loss') plt.subplot(2, 1, 2) plt.plot(best_stats['train_acc_history'], label='train',color='r') plt.plot(best_stats['val_acc_history'], label='val',color='g') plt.title('Classification accuracy history') plt.xlabel('Epoch') plt.ylabel('Clasification accuracy') plt.show() ################################################################################# # END OF YOUR CODE # #################################################################################

lr 1.000000e-03 reg 1.000000e-04 train accuracy: 0.541551 val accuracy: 0.499000

lr 1.000000e-03 reg 1.000000e-03 train accuracy: 0.541694 val accuracy: 0.511000

lr 1.000000e-03 reg 1.000000e-02 train accuracy: 0.540898 val accuracy: 0.490000

lr 1.200000e-03 reg 1.000000e-04 train accuracy: 0.562041 val accuracy: 0.528000

lr 1.200000e-03 reg 1.000000e-03 train accuracy: 0.563653 val accuracy: 0.507000

lr 1.200000e-03 reg 1.000000e-02 train accuracy: 0.564184 val accuracy: 0.512000

lr 1.400000e-03 reg 1.000000e-04 train accuracy: 0.580857 val accuracy: 0.532000

lr 1.400000e-03 reg 1.000000e-03 train accuracy: 0.580857 val accuracy: 0.513000

lr 1.400000e-03 reg 1.000000e-02 train accuracy: 0.575245 val accuracy: 0.534000

lr 1.600000e-03 reg 1.000000e-04 train accuracy: 0.593347 val accuracy: 0.529000

lr 1.600000e-03 reg 1.000000e-03 train accuracy: 0.594857 val accuracy: 0.548000

lr 1.600000e-03 reg 1.000000e-02 train accuracy: 0.593878 val accuracy: 0.551000

lr 1.800000e-03 reg 1.000000e-04 train accuracy: 0.605306 val accuracy: 0.537000

lr 1.800000e-03 reg 1.000000e-03 train accuracy: 0.610000 val accuracy: 0.533000

lr 1.800000e-03 reg 1.000000e-02 train accuracy: 0.603204 val accuracy: 0.546000

best validation accuracy achieved during cross-validation: 0.551000

Test accuracy: 0.542

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在游戏商店里在金币左面 有输入礼券号码，就是那个。

输入后要等很久才能给你礼物。完毕~

cs231n作业中的一句代码prob[range(num_train), y]，不懂这里什么意思

类似的还有scores[range(num_train), y]) ， num_train=500，y是一个向量 （500，1） scores大小是（500,10） 然后求np.sum(scores[range(num_train), y])) 不是很清楚scores[range(num_train), y])这个表达式是什么意思?

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