python - 具有 'tanh' 作为激活和 'cross-entropy' 作为成本函数的神经网络不起作用

Question

我实现了一个简单的神经网络。它适用于“sigmoid + 交叉熵”、“sigmoid + 二次成本”和“tanh + 二次成本”，但不适用于“tanh + 交叉熵”(不比随机猜测好)。谁能帮我弄清楚为什么？看看FullConnectedLayer的代码:

class FullConnectedLayer(BaseLayer):
    """
    FullConnectedLayer
    ~~~~~~~~~~~~~~~~~~~~
    Data members: 
    sizes       ---- <type list> sizes of the network
    n_layers    ---- <type int> number of sublayers
    activation  ---- <type Activation> activation function for neurons
    weights     ---- <type list> to store weights
    biases      ---- <type list> to store biases
    neurons     ---- <type list> to store states (outputs) of neurons
    zs          ---- <type list> to store weighted inputs to neurons
    grad_w      ---- <type list> to store gradient of Cost w.r.t weights
    grad_b      ---- <type list> to store gradient of Cost w.r.t biases
    ---------------------
    Methods:
    __init__(self, sizes, activation = Sigmoid())
    size(self)
    model(self)
    feedforward(self, a)
    backprop(self, C_p)
    update(self, eta, lmbda, batch_size, n)
    """

    def __init__(self, sizes, activation = Sigmoid(), normal_initialization = False):
        """
        The list ''sizes'' contains the number of neurons in repective layers
        of the network. For example, sizes = [2, 3, 2] represents 3 layers, with
        the first layer having 2 neurons, the second 3 neurons, and the third 2 
        neurons.

        Note that the input layer may be passed by other layer of another type 
        when connected after the layer, and we don't set biases for this layer.
        Also note that the output layer my be passed to other layer if connected
        before the layer, in this case, just assign the outputs to its inputs.
        For examle, Layer1([3, 2, 4])->Layer2([4, 6, 3])->Layer3([3, 2]). Just
        assign the output of Layer1 to the input Layer2, it will be safe.
        """

        BaseLayer.__init__(self, sizes, activation)

        if normal_initialization:
            self.weights = [np.random.randn(j, i)
                    for i, j in zip(sizes[:-1], sizes[1:])]
        else:
            self.weights = [np.random.randn(j, i) / np.sqrt(i)
                    for i, j in zip(sizes[:-1], sizes[1:])]
        self.biases = [np.random.randn(j, 1) for j in sizes[1:]]

        self.grad_w = [np.zeros(w.shape) for w in self.weights]
        self.grad_b = [np.zeros(b.shape) for b in self.biases]

    def feedforward(self, a):
        """
        Return output of the network if ''a'' is input.
        """
        self.neurons = [a] # to store activations (outputs) of all layers
        self.zs = []
        for w, b in zip(self.weights, self.biases):
            z = np.dot(w, self.neurons[-1]) + b
            self.zs.append(z)
            self.neurons.append(self.activation.func(z))
        return self.neurons[-1]


    def backprop(self, Cp_a):
        """
        Backpropagate the delta error.
        ------------------------------
        Return a tuple whose first component is a list of the gradients of 
        weights and biases, whose second component is the backpropagated delta.
        Cp_a, dC/da: derivative of cost function w.r.t a, output of neurons. 
        """
        # The last layer
        delta = Cp_a * self.activation.prime(self.zs[-1])
        self.grad_b[-1] += delta
        self.grad_w[-1] += np.dot(delta, self.neurons[-2].transpose()) 

        for l in range(2, self.n_layers):
            sp = self.activation.prime(self.zs[-l])  # a.prime(z)
            delta = np.dot(self.weights[-l + 1].transpose(), delta) * sp  
            self.grad_b[-l] += delta
            self.grad_w[-l] += np.dot(delta, self.neurons[-l - 1].transpose())

        Cp_a_out = np.dot(self.weights[0].transpose(), delta)

        return Cp_a_out

    def update(self, eta, lmbda, batch_size, n):
        """
        Update the network's weights and biases by applying gradient descent
        algorithm.
        ''eta'' is the learning rate
        ''lmbda'' is the regularization parameter
        ''n'' is the total size of the training data set
        """
        self.weights = [(1 - eta * (lmbda/n)) * w - (eta/batch_size) * delta_w\
                for w, delta_w in zip(self.weights, self.grad_w)]
        self.biases = [ b - (eta / batch_size) * delta_b\
                for b, delta_b in zip(self.biases, self.grad_b)]

        # Clear ''grad_w'' and ''grad_b'' so that they are not added to the 
        # next update pass
        for dw, db in zip(self.grad_w, self.grad_b):
            dw.fill(0)
            db.fill(0)

这里是tanh函数的代码:

class Tanh(Activation):

    @staticmethod
    def func(z):
        """ The functionality. """
        return (np.exp(z) - np.exp(-z)) / (np.exp(z) + np.exp(-z))

    @staticmethod
    def prime(z):
        """ The derivative. """
        return 1. - Tanh.func(z) ** 2

下面是交叉熵类的代码:

class CrossEntropyCost(Cost):

    @staticmethod
    def func(a, y):
        """
        Return the cost associated with an output ''a'' and desired output
        ''y''. 
        Note that np.nan_to_num is used to ensure numerical stability. In
        particular, if both ''a'' and ''y'' have a 1.0 in the same slot, 
        then the expression (1-y) * np.log(1-a) returns nan. The np.nan_to_num
        ensures that that is converted to the correct value(0.0).
        """
        for ai in a:
            if ai < 0:
                print("in CrossEntropyCost.func(a, y)... require a_i > 0, a_i belong to a.")
                exit(1)

        return np.sum(np.nan_to_num(-y * np.log(a) - (1-y) * np.log(1-a)))

    @staticmethod
    def Cp_a(a, y):
        """
        Cp_a, dC/da: the derivative of C w.r.t a
        ''a'' is the output of neurons
        ''y'' is the expected output of neurons
        """
        #return (a - y) # delta
        return (a - y) / (a * (1 - a))

---

编辑: 似乎问题在于tanh的范围是-1到+1，这对于交叉熵来说是非法的。但是如果我只想要一个 tanh 激活和一个交叉熵成本，我应该如何处理呢？

Answer 1

回答晚了，但我认为值得一提。

如果您想使用 tanh 激活函数，而不是使用交叉熵成本函数，您可以修改它以提供介于 -1 和 1 之间的输出。

同样的看起来像:

((1 + y)/2 * log(a)) + ((1-y)/2 * log(1-a))

将其用作成本函数将使您可以使用 tanh 激活。