Softmax function

The softmax function also referred to as softargmax:184 or normalized exponential, 198 may be a generalization of the logistic function to multiple dimensions.

Softmax function


$$value=1$$ $$value=2$$ $$value=3$$ $$value=4$$ $$value=5$$


$$0.01165623095604$$ $$0.031684920796124$$ $$0.086128544436269$$ $$0.23412165725274$$ $$0.63640864655883$$


$$\normalsize Softmax\ function\ \sigma (\bf z \rm)_{j}$$ $$\\ \sigma (\bf z \rm)_{j} = \Large \frac{e^{z_{j}} }{\displaystyle \sum_{k=1}^{K} e^{z_{k}} } \normalsize \hspace{3px} for\ j = 1,\cdots, K.$$

What is softmax function?

The softmax function also referred to as softargmax:184 or normalized exponential, 198 may be a generalization of the logistic function to multiple dimensions. It utilized in multinomial logistic regression and is usually used because the last activation function of a neural network to normalize the output of a network to a probability distribution over predicted output classes supported Luce's choice axiom.

The softmax function takes as input a vector z of K real numbers and normalizes it into a probability distribution consisting of K probabilities proportional to the exponentials of the input numbers. That is, before applying softmax, some vector components might be negative, or greater than one; and won't sum to 1; but after applying softmax, each component are going to be within the interval {\displaystyle (0,1)}(0,1), and therefore the components will add up to 1, in order that they will be interpreted as probabilities. Furthermore, the larger input components will correspond to larger probabilities.

The standard (unit) softmax function is defined by the formula

In words: we apply the quality exponential to every element of the input vector and normalize these values by dividing by the sum of these exponentials; this normalization ensures that the sum of the components of the output vector is 1.

Instead of e, a special base b > 0 is often used; choosing a bigger value of b will create a probability distribution that's more concentrated around the positions of the most important input values. Writing  yields the expressions:

In some fields, the bottom is fixed, like a hard and fast scale, while in others the parameter β is varied.

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