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Wikipedia / Softmax function

The softmax function, also known as softargmax or normalized exponential function, converts a tuple of real numbers into a probability distribution over possible outcomes. It is a generalization of the logistic function to multiple dimensions, and is used in multinomial logistic regression. The softmax function is often used as the last activation function of a neural network to normalize the output of a network to a probability distribution over predicted output classes.

Definition

The softmax function takes as input a tuple of real numbers, and normalizes it into a probability distribution consisting of probabilities proportional to the exponentials of the input numbers. That is, prior to applying softmax, some tuple components could be negative, or greater than one; and might not sum to 1; but after applying softmax, each component will be in the interval, and the components will add up to 1, so that they can be interpreted as probabilities. Furthermore, the larger input components will correspond to larger probabilities.

, where with

In words, the softmax applies the standard exponential function, which amounts to assigning almost all of the total unit weight in the result to the position of the tuple's maximal element (of 8).

In general, instead of a different base can be used. As above, if then larger input components will result in larger output probabilities, and increasing the value of will create probability distributions that are more concentrated around the positions of the largest input values. Conversely, if then smaller input components will result in larger output probabilities, and decreasing the value of (for real ) yields the expressions:

A value proportional to the reciprocal of is sometimes referred to as the temperature, where is typically 1 or the Boltzmann constant and is the temperature. A higher temperature results in a more uniform output distribution (i.e. with higher entropy; it is "more random"), while a lower temperature results in a sharper output distribution, usually with one value dominating.

In some fields, the base is fixed, corresponding to a fixed scale, while in others the parameter (or ) is varied.

The softmax function is a multiple-variable generalization of the logistic function.

Interpretations

Smooth arg max

The Softmax function is a smooth approximation to the arg max function: the function whose value is the index of a tuple's largest element. The name "softmax" may be misleading. Softmax is not a smooth maximum (that is, a smooth approximation to the maximum function). The term "softmax" is also used for the closely related LogSumExp function, which is a smooth maximum. For this reason, some prefer the more accurate term "softargmax", though the term "softmax" is conventional in machine learning. This section uses the term "softargmax" for clarity.

(corresponding to the index), consider the arg max function with one-hot since the third argument is the maximum.

being the maximum) by dividing the 1 between all max args; formally where Points with multiple arg max values are singular points (or singularities, and form the singular set) – these are the points where arg max is discontinuous (with a jump discontinuity) – while points with a single arg max are known as non-singular or regular points.

With the last expression given in the introduction, softargmax is now a smooth approximation of arg max: as , softargmax converges to arg max. There are various notions of convergence of a function; softargmax converges to arg max pointwise, meaning for each fixed input as However, softargmax does not converge uniformly, the slower they converge. However, softargmax does converge compactly on the non-singular set.

Conversely, as , softargmax converges to arg min in the same way, where here the singular set is points with two arg min values. In the language of tropical analysis, the softmax is a deformation or "quantization" of arg max and arg min, corresponding to using the log semiring instead of the max-plus semiring (respectively min-plus semiring), and recovering the arg max or arg min by taking the limit is called "tropicalization" or "dequantization".

It is also the case that, for any fixed , if one input is much larger than the others relative As , so eventually all differences become large (relative to a shrinking temperature), which gives another interpretation for the limit behavior.

Statistical mechanics

In statistical mechanics, the softargmax function is known as the Boltzmann distribution (or Gibbs distribution): are the microstates are the energies of that state; the denominator is known as the partition function, often denoted by ; and the factor is called the coldness (or thermodynamic beta, or inverse temperature).

Applications

The softmax function is used in various multiclass classification methods, such as multinomial logistic regression (also known as softmax regression), multiclass linear discriminant analysis, naive Bayes classifiers, and artificial neural networks. Specifically, in multinomial logistic regression and linear discriminant analysis, the input to the function is the result of distinct linear functions, and the predicted probability for the th class given a sample tuple and a weighting vector is:

This can be seen as the composition of , thus transforming the original, probably highly-dimensional, input to vectors in a .

Neural networks

The standard softmax function is often used in the final layer of a neural network-based classifier. Such networks are commonly trained under a log loss (or cross-entropy) regime, giving a non-linear variant of multinomial logistic regression.

to a real value, the derivative needs to take the index into account:

and thus may also be expressed as

Here, the Kronecker delta is used for simplicity (cf. the derivative of a sigmoid function, being expressed via the function itself).

To ensure stable numerical computations subtracting the maximum value from the input tuple is common. This approach, while not altering the output or the derivative theoretically, enhances stability by directly controlling the maximum exponent value computed.

See multinomial logit for a probability model which uses the softmax activation function.

Reinforcement learning

In the field of reinforcement learning, a softmax function can be used to convert values into action probabilities. The function commonly used is:

is called a temperature parameter (in allusion to statistical mechanics), the probability of the action with the highest expected reward tends to 1.

Computational complexity and remedies

In neural network applications, the number of possible outcomes is often large, e.g. in case of neural language models that predict the most likely outcome out of a vocabulary which might contain millions of possible words., followed by the application of the softmax function itself) computationally expensive. What's more, the gradient descent backpropagation method for training such a neural network involves calculating the softmax for every training example, and the number of training examples can also become large. The computational effort for the softmax became a major limiting factor in the development of larger neural language models, motivating various remedies to reduce training times.

Approaches that reorganize the softmax layer for more efficient calculation include the hierarchical softmax and the differentiated softmax. The hierarchical softmax (introduced by Morin and Bengio in 2005) uses a binary tree structure where the outcomes (vocabulary words) are the leaves and the intermediate nodes are suitably selected "classes" of outcomes, forming latent variables. The desired probability (softmax value) of a leaf (outcome) can then be calculated as the product of the probabilities of all nodes on the path from the root to that leaf. Ideally, when the tree is balanced, this would reduce the computational complexity. In practice, results depend on choosing a good strategy for clustering the outcomes into classes. A Huffman tree was used for this in Google's word2vec models (introduced in 2013) to achieve scalability.

A second kind of remedies is based on approximating the softmax (during training) with modified loss functions that avoid the calculation of the full normalization factor. These include methods that restrict the normalization sum to a sample of outcomes (e.g. Importance Sampling, Target Sampling).

Numerical algorithms

The standard softmax is numerically unstable because of large exponentiations. The safe softmax is the largest score involved. Subtracting by it guarantees that the exponentiations result in at most 1, see translation invariance in the mathematical properties.

The attention mechanism in transformersThe standard softmax method involves several loops over the inputs, which would be bottlenecked by memory bandwidth.

It can be efficiently computed on a GPU cluster using the FlashAttention algorithm.

Mathematical properties

Geometrically the softmax function maps the Euclidean space to the boundary of the (K-1)-simplex-dimensional space), due to the linear constraint that all output sum to 1 meaning it lies on a hyperplane.

: equal scores yield equal probabilities.

and is positive.

The standard logistic function is the special case for a 1-dimensional axis in 2-dimensional space, say the x-axis in the

Gradients

The softmax function is also the gradient of the LogSumExp function,

where the Kronecker delta.

History

The softmax function was used in statistical mechanics as the Boltzmann distribution in the foundational paper , formalized and popularized in the influential textbook .

The use of the softmax in decision theory is credited to R. Duncan Luce, who used the axiom of independence of irrelevant alternatives in rational choice theory to deduce the softmax in Luce's choice axiom for relative preferences.

In machine learning, the term "softmax" is credited to John S. Bridle in two 1989 conference papers, : and :

Example

With an input of , the softmax is approximately . The output has most of its weight where the "4" was in the original input. This is what the function is normally used for: to highlight the largest values and suppress values which are significantly below the maximum value. But note: a change of temperature changes the output. When the temperature is multiplied by 10, the inputs are effectively and the softmax is approximately . This shows that high temperatures de-emphasize the maximum value.

Computation of this example using Python code:

Alternatives

The softmax function generates probability predictions densely distributed over its support. Other functions like sparsemax or α-entmax can be used when sparse probability predictions are desired. Also the Gumbel-softmax reparametrization trick can be used when sampling from a discrete-discrete distribution needs to be mimicked in a differentiable manner.