Softmax Function

Softmax函数是一种常用于机器学习中的激活函数，它将一个由K个实数组成的向量标准化成由K个概率组成的概率分布。具体来说，对于一个由K个实数组成的向量 ~z = (z1, . . . , zK)，softmax函数的第i个输出为：σ(~z)i = e βzi / Σj=1 to K e βzj，其中i = 1, . . . ,K，β ∈ R是一个参数。softmax函数的主要作用是将实数向量映射为概率分布，使得每个元素都落在[0,1]的区间内，且所有元素之和为1，因此softmax函数常用于分类问题的输出层。在训练神经网络的过程中，softmax函数的输出可以用来计算损失函数，从而进行反向传播更新参数。

相关问题

what is softmax function

The softmax function is a mathematical function that converts a vector of real numbers into a probability distribution. It is commonly used in machine learning and deep learning algorithms for classification tasks. The softmax function takes an input vector and applies the exponential function to each element of the vector, then normalizes the resulting vector by dividing each element by the sum of all exponential values. This results in a vector of probabilities that add up to 1, where each element represents the probability of the input belonging to a particular class. The softmax function is defined as follows:

softmax(x_i) = e^(x_i) / sum(e^(x_j)) for all j in the input vector x.

class Softmax(Activation): ''' softmax nonlinear function. ''' def __init__(self): ''' There are no parameters in softmax function. ''' super(Softmax, self).__init__() def value(self, x: np.ndarray) -> np.ndarray: ''' Parameters: x is the input to the softmax function. x is a two dimensional numpy array. Each row is the input to the softmax function Returns: output of the softmax function. The returned value is with the same shape as that of x. ''' #### write your code below #### return def derivative(self, x: np.ndarray) -> np.ndarray: ''' Parameters: x is the input to the softmax function. x is a two dimensional numpy array. Returns: a two dimensional array representing the derivative of softmax function w.r.t. x. ''' #### write your code below ####

下面是对 `Softmax` 激活函数的实现，你可以将其加入到代码中：

class Softmax(Activation):
    '''
    softmax nonlinear function.
    '''
    def __init__(self):
        '''
        There are no parameters in softmax function.
        '''
        super(Softmax, self).__init__()

    def value(self, x: np.ndarray) -> np.ndarray:
        '''
        Parameters:
          x is the input to the softmax function. x is a two dimensional numpy array. Each row is the input to the softmax function
        Returns:
          output of the softmax function. The returned value is with the same shape as that of x.
        '''
        exp_scores = np.exp(x)
        return exp_scores / np.sum(exp_scores, axis=1, keepdims=True)

    def derivative(self, x: np.ndarray) -> np.ndarray:
        '''
        Parameters:
          x is the input to the softmax function. x is a two dimensional numpy array.
        Returns:
          a two dimensional array representing the derivative of softmax function w.r.t. x.
        '''
        softmax_output = self.value(x)
        return softmax_output * (1 - softmax_output)