从Udacity的深度学习课程中，y_i的softmax仅仅是指数除以整个Y向量的指数之和:
其中S(y_i)是y_i的软最大函数e是指数函数j是no。输入向量Y中的列。
我试过以下几种方法:
import numpy as np
def softmax(x):
"""Compute softmax values for each sets of scores in x."""
e_x = np.exp(x - np.max(x))
return e_x / e_x.sum()
scores = [3.0, 1.0, 0.2]
print(softmax(scores))
返回:
[ 0.8360188 0.11314284 0.05083836]
但建议的解决方案是:
def softmax(x):
"""Compute softmax values for each sets of scores in x."""
return np.exp(x) / np.sum(np.exp(x), axis=0)
它产生与第一个实现相同的输出，尽管第一个实现显式地取每列与Max的差值，然后除以和。
有人能用数学方法解释一下吗?一个是对的，另一个是错的?
实现在代码和时间复杂度方面是否相似?哪个更有效率?
The purpose of the softmax function is to preserve the ratio of the vectors as opposed to squashing the end-points with a sigmoid as the values saturate (i.e. tend to +/- 1 (tanh) or from 0 to 1 (logistical)). This is because it preserves more information about the rate of change at the end-points and thus is more applicable to neural nets with 1-of-N Output Encoding (i.e. if we squashed the end-points it would be harder to differentiate the 1-of-N output class because we can't tell which one is the "biggest" or "smallest" because they got squished.); also it makes the total output sum to 1, and the clear winner will be closer to 1 while other numbers that are close to each other will sum to 1/p, where p is the number of output neurons with similar values.
从向量中减去最大值的目的是，当你计算e^y指数时，你可能会得到非常高的值，将浮点数夹在最大值处，导致平局，但在这个例子中不是这样。如果你减去最大值得到一个负数，那么就会出现一个大问题，然后你就会得到一个负指数，它会迅速缩小数值，改变比率，这就是在海报上的问题中发生的情况，并得到错误的答案。
Udacity提供的答案效率低得可怕。我们需要做的第一件事是计算所有向量分量的e^y_j, KEEP这些值，然后求和，然后除。Udacity搞砸的地方是他们计算了两次e^y_j !!正确答案如下:
def softmax(y):
e_to_the_y_j = np.exp(y)
return e_to_the_y_j / np.sum(e_to_the_y_j, axis=0)
从数学的角度看，两边是相等的。
这很容易证明。m = max (x)。现在你的函数softmax返回一个向量，它的第i个坐标等于
注意，这适用于任何m，因为对于所有(甚至是复数)数e^m != 0
from computational complexity point of view they are also equivalent and both run in O(n) time, where n is the size of a vector.
from numerical stability point of view, the first solution is preferred, because e^x grows very fast and even for pretty small values of x it will overflow. Subtracting the maximum value allows to get rid of this overflow. To practically experience the stuff I was talking about try to feed x = np.array([1000, 5]) into both of your functions. One will return correct probability, the second will overflow with nan
your solution works only for vectors (Udacity quiz wants you to calculate it for matrices as well). In order to fix it you need to use sum(axis=0)
