我实现交叉熵函数有什么问题？

Question

我正在学习神经网络，我想在python中写一个函数cross_entropy。其中定义为

其中N是样本数，k是class个数，log是自然对数，如果样本 i 在 class j 中，t_i,j 为 1，否则为 0，p_i,j 是样本 i 在 class j 中。为避免对数出现数值问题，请将预测限制在 [10^{−12}, 1 − 10^{−12}] 范围内。

根据上面的描述，我通过将预测剪裁到 [epsilon, 1 − epsilon] 范围内，然后根据上面的公式计算 cross_entropy 来写下代码。

def cross_entropy(predictions, targets, epsilon=1e-12):
    """
    Computes cross entropy between targets (encoded as one-hot vectors)
    and predictions. 
    Input: predictions (N, k) ndarray
           targets (N, k) ndarray        
    Returns: scalar
    """
    predictions = np.clip(predictions, epsilon, 1. - epsilon)
    ce = - np.mean(np.log(predictions) * targets) 
    return ce

下面的代码将用于检查函数 cross_entropy 是否正确。

predictions = np.array([[0.25,0.25,0.25,0.25],
                        [0.01,0.01,0.01,0.96]])
targets = np.array([[0,0,0,1],
                  [0,0,0,1]])
ans = 0.71355817782  #Correct answer
x = cross_entropy(predictions, targets)
print(np.isclose(x,ans))

以上代码的输出为False，也就是说我定义函数cross_entropy的代码不正确。然后我打印 cross_entropy(predictions, targets) 的结果。它给出了0.178389544455，正确的结果应该是ans = 0.71355817782。谁能帮我检查一下我的代码有什么问题吗？

Answer 1

您离得并不远，但请记住，您取的是 N 个总和的平均值，其中 N = 2（在本例中）。所以你的代码可以是：

def cross_entropy(predictions, targets, epsilon=1e-12):
    """
    Computes cross entropy between targets (encoded as one-hot vectors)
    and predictions. 
    Input: predictions (N, k) ndarray
           targets (N, k) ndarray        
    Returns: scalar
    """
    predictions = np.clip(predictions, epsilon, 1. - epsilon)
    N = predictions.shape[0]
    ce = -np.sum(targets*np.log(predictions+1e-9))/N
    return ce

predictions = np.array([[0.25,0.25,0.25,0.25],
                        [0.01,0.01,0.01,0.96]])
targets = np.array([[0,0,0,1],
                   [0,0,0,1]])
ans = 0.71355817782  #Correct answer
x = cross_entropy(predictions, targets)
print(np.isclose(x,ans))

在这里，我认为如果你坚持使用 np.sum() 会更清楚一些。此外，我将 1e-9 添加到 np.log() 中以避免在计算中出现 log(0) 的可能性。希望这对您有所帮助！

注意：根据@Peter 的评论，如果您的 epsilon 值大于 0，1e-9 的偏移量确实是多余的。

Answer 2

def cross_entropy(x, y):
    """ Computes cross entropy between two distributions.
    Input: x: iterabale of N non-negative values
           y: iterabale of N non-negative values
    Returns: scalar
    """

    if np.any(x < 0) or np.any(y < 0):
        raise ValueError('Negative values exist.')

    # Force to proper probability mass function.
    x = np.array(x, dtype=np.float)
    y = np.array(y, dtype=np.float)
    x /= np.sum(x)
    y /= np.sum(y)

    # Ignore zero 'y' elements.
    mask = y > 0
    x = x[mask]
    y = y[mask]    
    ce = -np.sum(x * np.log(y)) 
    return ce

def cross_entropy_via_scipy(x, y):
        ''' SEE: https://en.wikipedia.org/wiki/Cross_entropy'''
        return  entropy(x) + entropy(x, y)

from scipy.stats import entropy, truncnorm

x = truncnorm.rvs(0.1, 2, size=100)
y = truncnorm.rvs(0.1, 2, size=100)
print np.isclose(cross_entropy(x, y), cross_entropy_via_scipy(x, y))

我实现交叉熵函数有什么问题？

What is the problem with my implementation of the cross-entropy function?

python

machine-learning

neural-network

cross-entropy