two_layer_net

two_layer_net原理

首先该模型的主要公式为 $f=W_2max(0,W_1x+b_1)+b_2$ , 在计算梯度时，首先计算 $W_2$ 的梯度，然后计算 $W_1$ 的梯度

Propagation:
$FC_1out=X.W_1+b_1$

$H_{out}=maximum(0,FC_1out)$

$FC_2out=H_{out}.W_2+b_2$

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Backpropogation:

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$KaTeX parse error: Expected '}', got '_' at position 113: …C_2out}=\text{H_̲out}^T.\frac{\p…$

$\frac{\partial L}{\partial b_{2}}=\frac{\partial F C 2_{-} o u t}{\partial b_{2}} \frac{\partial L}{\partial F C 2_{-} o u t}=[1 \ldots 1]_{[1 \times H]} \cdot \frac{\partial L}{\partial F C 2_{-} o u t}$

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$\frac{\partial L}{\partial W_{1}}=\frac{\partial H_{-} c o u t}{\partial W_{1}} \cdot \frac{\partial L}{\partial H_{-} o u t}=X^{T} \cdot \frac{\partial L}{\partial H_{-} o u t}$

$\frac{\partial L}{\partial b_{1}}=\frac{\partial H_{-} c o u t}{\partial b_{1}} \cdot \frac{\partial L}{\partial H_{-} o u t}=[1 \ldots 1]_{[1 \times N]} \cdot \frac{\partial L}{\partial H_{-} o u t}$

作业实现

neural_net.py

loss函数实现

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). #
    #############################################################################
    Z1 = X.dot(W1)+b1
    A1 = np.maximum(0,Z1)
    scores = A1.dot(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. #
    #############################################################################
    scores -= np.max(scores, axis=1, keepdims=True)
    exp_scores = np.exp(scores)
    probs = exp_scores/np.sum(exp_scores,axis=1,keepdims=True)
    y_label = np.zeros((N,probs.shape[1]))
    y_label[np.arange(N),y] = 1
    loss = (-1)*np.sum(np.multiply(np.log(probs),y_label))/N
    loss += reg*(np.sum(W1*W1)+np.sum(W2*W2))
    #############################################################################
    # 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 #
    #############################################################################
    dZ2 = probs - y_label
    dW2 = A1.T.dot(dZ2)/N + 2*reg*W2
    db2 = np.sum(dZ2,axis=0)/N
    dZ1 = (dZ2).dot(W2.T)*(A1>0)
    dW1 = X.T.dot(dZ1)/N + 2*reg*W1
    db1 = np.sum(dZ1,axis=0)/N
    grads['W2'] = dW2
    grads['b2'] = db2
    grads['W1'] = dW1
    grads['b1'] = db1
    
    #############################################################################
    # END OF YOUR CODE #
    #############################################################################

    return loss, grads

train函数实现

def train(self, X, y, X_val, y_val,
            learning_rate=1e-3, learning_rate_decay=0.95,
            reg=5e-6, 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. #
      #########################################################################
      batch_inx = np.random.choice(num_train,batch_size)
      X_batch = X[batch_inx,:]
      y_batch = y[batch_inx]
    
      #########################################################################
      # 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. #
      #########################################################################
      self.params['W1'] -= learning_rate * grads['W1']
      self.params['b1'] -= learning_rate * grads['b1']
      self.params['W2'] -= learning_rate * grads['W2']
      self.params['b2'] -= learning_rate * grads['b2']
      #########################################################################
      # 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,
    }

predict函数

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! #
    ###########################################################################
    score = self.loss(X)
    y_pred = np.argmax(score, axis=1)
    ###########################################################################
    # END OF YOUR CODE #
    ###########################################################################

    return y_pred

two_layer_net.ipynb

计算分数

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计算损失的差异

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反向传播计算梯度差异

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训练网络

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训练超参数

best_net = None # store the best model into this 

#################################################################################
# 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 = [100, 200, 250]
num_classes = 10
reg = [0.03,0.05, 0.09]
learning_rate = [1e-3, 1e-4]
best_acc = 0
for hs in hidden_size:
    net = TwoLayerNet(input_size, hs, num_classes)
    
    for r in reg:
        for lr in learning_rate:
            stats = net.train(X_train, y_train, X_val, y_val, num_iters=3000, batch_size=400,
                             learning_rate=lr, learning_rate_decay=0.95, reg=r,verbose=False)
            val_acc = (net.predict(X_val)==y_val).mean()
            print('hidden_size:%f, reg:%f, learning_rate:%f'%(hs,r,lr))
            print('val accuracy:', val_acc)
            if (val_acc > best_acc):
                best_acc = val_acc
                best_net = net
#################################################################################
# END OF YOUR CODE #
#################################################################################

最后训练结果如图

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最后在测试集的准确率结果为：

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cs231n_two_layer_net

two_layer_net

two_layer_net原理

作业实现

neural_net.py

two_layer_net.ipynb