Softmax 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.

Formally, instead of considering the arg max as a function with categorical output ${\displaystyle 1,\dots ,n}$ (corresponding to the index), consider the arg max function with one-hot representation of the output (assuming there is a unique maximum arg): $${\displaystyle \mathrm{a}\mathrm{r}\mathrm{g}\mathrm{m}\mathrm{a}\mathrm{x}(z_1,\dots ,z_n)=(y_1,\dots ,y_n)=(0,\dots ,0,1,0,\dots ,0),}$$ where the output coordinate ${\displaystyle y_i=1}$ if and only if ${\displaystyle i}$ is the arg max of ${\displaystyle (z_1,\dots ,z_n)}$, meaning ${\displaystyle z_i}$ is the unique maximum value of ${\displaystyle (z_1,\dots ,z_n)}$. For example, in this encoding ${\displaystyle \mathrm{a}\mathrm{r}\mathrm{g}\mathrm{m}\mathrm{a}\mathrm{x}(1,5,10)=(0,0,1),}$ since the third argument is the maximum.

This can be generalized to multiple arg max values (multiple equal ${\displaystyle z_i}$ being the maximum) by dividing the 1 between all max args; formally ⁠${\displaystyle \frac{1}{k}}$⁠ where ⁠${\displaystyle k}$⁠ is the number of arguments assuming the maximum. For example, ${\displaystyle \mathrm{a}\mathrm{r}\mathrm{g}\mathrm{m}\mathrm{a}\mathrm{x}(1,5,5)=(0,\frac{1}{2},\frac{1}{2}),}$ since the second and third argument are both the maximum. In case all arguments are equal, this is simply ${\displaystyle \mathrm{a}\mathrm{r}\mathrm{g}\mathrm{m}\mathrm{a}\mathrm{x}(z,\dots ,z)=(\frac{1}{n},\dots ,\frac{1}{n}).}$ Points z 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 ⁠${\displaystyle \beta \to \mathrm{\infty }}$⁠, 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 z as ⁠${\displaystyle \beta \to \mathrm{\infty }}$⁠, ${\displaystyle {\sigma }_{\beta }(\mathbf{z})\to \mathrm{a}\mathrm{r}\mathrm{g}\mathrm{m}\mathrm{a}\mathrm{x}(\mathbf{z}).}$ However, softargmax does not converge uniformly to arg max, meaning intuitively that different points converge at different rates, and may converge arbitrarily slowly. In fact, softargmax is continuous, but arg max is not continuous at the singular set where two coordinates are equal, while the uniform limit of continuous functions is continuous. The reason it fails to converge uniformly is that for inputs where two coordinates are almost equal (and one is the maximum), the arg max is the index of one or the other, so a small change in input yields a large change in output. For example, ${\displaystyle {\sigma }_{\beta }(1,1.0001)\to (0,1),}$ but ${\displaystyle {\sigma }_{\beta }(1,0.9999)\to (1,0),}$ and ${\displaystyle {\sigma }_{\beta }(1,1)=\frac{1}{2}}$ for all inputs: the closer the points are to the singular set ${\displaystyle (x,x)}$, the slower they converge. However, softargmax does converge compactly on the non-singular set.

Conversely, as ⁠${\displaystyle \beta \to -\mathrm{\infty }}$⁠, 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 β, if one input ⁠${\displaystyle z_i}$⁠ is much larger than the others relative to the temperature, ${\displaystyle T=1/\beta }$, the output is approximately the arg max. For example, a difference of 10 is large relative to a temperature of 1: $${\displaystyle \sigma (0,10):={\sigma }_1(0,10)=\left(\frac{1}{1+e^{10}},\frac{e^{10}}{1+e^{10}}\right)\approx (0.00005,0.99995)}$$ However, if the difference is small relative to the temperature, the value is not close to the arg max. For example, a difference of 10 is small relative to a temperature of 100: $${\displaystyle {\sigma }_{1/100}(0,10)=\left(\frac{1}{1+e^{1/10}},\frac{e^{1/10}}{1+e^{1/10}}\right)\approx (0.475,0.525).}$$ As ⁠${\displaystyle \beta \to \mathrm{\infty }}$⁠, temperature goes to zero, ${\displaystyle T=1/\beta \to 0}$, 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): the index set ${\displaystyle 1,\dots ,k}$ are the microstates of the system; the inputs ${\displaystyle z_i}$ are the energies of that state; the denominator is known as the partition function, often denoted by Z; and the factor β 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 K distinct linear functions, and the predicted probability for the jth class given a sample tuple x and a weighting vector w is:

$${\displaystyle P(y=j\mid \mathbf{x})=\frac{e^{{\mathbf{x}}^{\mathsf{T}}{\mathbf{w}}_j}}{\sum _{k=1}^K{e}^{{\mathbf{x}}^{\mathsf{T}}{\mathbf{w}}_k}}}$$

This can be seen as the composition of K linear functions ${\displaystyle \mathbf{x}\mapsto {\mathbf{x}}^{\mathsf{T}}{\mathbf{w}}_1,\dots ,\mathbf{x}\mapsto {\mathbf{x}}^{\mathsf{T}}{\mathbf{w}}_K}$ and the softmax function (where ${\displaystyle {\mathbf{x}}^{\mathsf{T}}\mathbf{w}}$ denotes the inner product of ${\displaystyle \mathbf{x}}$ and ${\displaystyle \mathbf{w}}$). The operation is equivalent to applying a linear operator defined by ${\displaystyle \mathbf{w}}$ to tuples ${\displaystyle \mathbf{x}}$, thus transforming the original, probably highly-dimensional, input to vectors in a K-dimensional space ${\displaystyle {\mathbb{R}}^K}$.

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.

Since the function maps a tuple and a specific index ${\displaystyle i}$ to a real value, the derivative needs to take the index into account:

$${\displaystyle \frac{\mathrm{\partial }}{\mathrm{\partial }{q}_k}\sigma (\mathbf{\text{q}},i)=\sigma (\mathbf{\text{q}},i)({\delta }_{ik}-\sigma (\mathbf{\text{q}},k)).}$$

This expression is symmetrical in the indexes ${\displaystyle i,k}$ and thus may also be expressed as

$${\displaystyle \frac{\mathrm{\partial }}{\mathrm{\partial }{q}_k}\sigma (\mathbf{\text{q}},i)=\sigma (\mathbf{\text{q}},k)({\delta }_{ik}-\sigma (\mathbf{\text{q}},i)).}$$

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.

If the function is scaled with the parameter ${\displaystyle \beta }$, then these expressions must be multiplied by ${\displaystyle \beta }$.

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: $${\displaystyle P_t(a)=\frac{\exp (q_t(a)/\tau )}{\sum _{i=1}^n\exp (q_t(i)/\tau )}\text{,}}$$

where the action value ${\displaystyle q_t(a)}$ corresponds to the expected reward of following action a and ${\displaystyle \tau }$ is called a temperature parameter (in allusion to statistical mechanics). For high temperatures (${\displaystyle \tau \to \mathrm{\infty }}$), all actions have nearly the same probability and the lower the temperature, the more expected rewards affect the probability. For a low temperature (${\displaystyle \tau \to 0^{+}}$), the probability of the action with the highest expected reward tends to 1.

Computational complexity and remedies

In neural network applications, the number K 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. This can make the calculations for the softmax layer (i.e. the matrix multiplications to determine the ${\displaystyle z_i}$, 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 from ${\displaystyle O(K)}$ to ${\displaystyle O({\log }_{2}K)}$. 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 method calculates instead$${\displaystyle \sigma (\mathbf{z}{)}_i=\frac{e^{\beta (z_i-m)}}{\sum _{j=1}^K{e}^{\beta (z_j-m)}}}$$where ${\displaystyle m=\underset{i}{max}{z}_i}$ 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 transformers takes three arguments: a "query vector" ${\displaystyle q}$, a list of "key vectors" ${\displaystyle k_1,\dots ,k_N}$, and a list of "value vectors" ${\displaystyle v_1,\dots ,v_N}$, and outputs a softmax-weighted sum over value vectors:$${\displaystyle o=\sum _{i=1}^N\frac{e^{q^T{k}_i-m}}{\sum _{j=1}^N{e}^{q^T{k}_j-m}}{v}_i}$$The 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 ${\displaystyle {\mathbb{R}}^K}$ to the boundary of the standard ${\displaystyle (K-1)}$-simplex, cutting the dimension by one (the range is a ${\displaystyle (K-1)}$-dimensional simplex in ${\displaystyle K}$-dimensional space), due to the linear constraint that all output sum to 1 meaning it lies on a hyperplane.

Along the main diagonal ${\displaystyle (x,x,\dots ,x),}$ softmax is just the uniform distribution on outputs, ${\displaystyle (\frac{1}{n},\dots ,\frac{1}{n})}$: equal scores yield equal probabilities.

More generally, softmax is invariant under translation by the same value in each coordinate: adding ${\displaystyle \mathbf{c}=(c,\dots ,c)}$ to the inputs ${\displaystyle \mathbf{z}}$ yields ${\displaystyle \sigma (\mathbf{z}+\mathbf{c})=\sigma (\mathbf{z})}$, because it multiplies each exponent by the same factor, ${\displaystyle e^c}$ (because ${\displaystyle e^{z_i+c}=e^{z_i}\cdot e^c}$), so the ratios do not change: $${\displaystyle \sigma (\mathbf{z}+\mathbf{c}{)}_j=\frac{e^{z_j+c}}{\sum _{k=1}^K{e}^{z_k+c}}=\frac{e^{z_j}\cdot e^c}{\sum _{k=1}^K{e}^{z_k}\cdot e^c}=\sigma (\mathbf{z}{)}_j.}$$

Geometrically, softmax is constant along diagonals: this is the dimension that is eliminated, and corresponds to the softmax output being independent of a translation in the input scores (a choice of 0 score). One can normalize input scores by assuming that the sum is zero (subtract the average: ${\displaystyle \mathbf{c}}$ where $c=\frac{1}{n}\sum z_i$), and then the softmax takes the hyperplane of points that sum to zero, $\sum z_i=0$, to the open simplex of positive values that sum to 1$\sum \sigma (\mathbf{z}{)}_i=1$, analogously to how the exponent takes 0 to 1, ${\displaystyle e^0=1}$ and is positive.

By contrast, softmax is not invariant under scaling. For instance, ${\displaystyle \sigma ((0,1))=(\frac{1}{1+e},\frac{e}{1+e})}$ but ${\displaystyle \sigma ((0,2))=(\frac{1}{1+e^2},\frac{e^2}{1+e^2}).}$

The standard logistic function is the special case for a 1-dimensional axis in 2-dimensional space, say the x-axis in the (x, y) plane. One variable is fixed at 0 (say ${\displaystyle z_2=0}$), so ${\displaystyle e^0=1}$, and the other variable can vary, denote it ${\displaystyle z_1=x}$, so the standard logistic function and its complement are $${\displaystyle \frac{e^{z_1}}{e^{z_1}+e^{z_2}}=\frac{e^x}{e^x+1},\frac{e^{z_2}}{e^{z_1}+e^{z_2}}=\frac{1}{e^x+1}.}$$ The 1-dimensional input could alternatively be expressed as the line ${\displaystyle (\frac{1}{2}x,-\frac{1}{2}x)}$, with outputs $${\displaystyle \frac{e^{x/2}}{e^{x/2}+e^{-x/2}}=\frac{e^x}{e^x+1},\frac{e^{-x/2}}{e^{x/2}+e^{-x/2}}=\frac{1}{e^x+1}.}$$

Gradients

The softmax function is also the gradient of the LogSumExp function, ⁠${\displaystyle \mathrm{LSE}(z_1,\dots ,z_K)=\log \left(\exp (z_1)+\cdots +\exp (z_K)\right)}$⁠: $${\displaystyle \frac{\mathrm{\partial }}{\mathrm{\partial }{z}_i}\mathrm{LSE}(\mathbf{z})=\frac{\exp z_i}{\sum _{k=1}^K\exp z_k}=\sigma (\mathbf{z}{)}_i,}$$ for ⁠${\displaystyle i=1,\dots ,K}$⁠, and ⁠${\displaystyle \mathbf{z}=(z_1,\dots ,z_K)\in {\mathbb{R}}^K}$⁠.

The gradient of softmax is $${\displaystyle \frac{\mathrm{\partial }}{\mathrm{\partial }{z}_j}\sigma (\mathbf{z}{)}_i=\frac{\mathrm{\partial }}{\mathrm{\partial }{z}_j}\frac{\exp z_i}{\sum _{k=1}^K\exp z_k}={\sigma }_i({\delta }_{ij}-{\sigma }_j),}$$ where the Kronecker delta ${\displaystyle {\delta }_{ij}}$ equals ${\displaystyle 1}$ if ${\displaystyle i=j}$ and is otherwise equal to ${\displaystyle 0}$.

History

The softmax function was used in statistical mechanics as the Boltzmann distribution in the foundational paper Boltzmann (1868), formalized and popularized in the influential textbook Gibbs (1902).

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, Bridle (1990a): and Bridle (1990b):

We are concerned with feed-forward non-linear networks (multi-layer perceptrons, or MLPs) with multiple outputs. We wish to treat the outputs of the network as probabilities of alternatives (e.g. pattern classes), conditioned on the inputs. We look for appropriate output non-linearities and for appropriate criteria for adaptation of the parameters of the network (e.g. weights). We explain two modifications: probability scoring, which is an alternative to squared error minimisation, and a normalised exponential (softmax) multi-input generalisation of the logistic non-linearity.

For any input, the outputs must all be positive and they must sum to unity. ...

Given a set of unconstrained values, ⁠${\displaystyle V_j(x)}$⁠, we can ensure both conditions by using a Normalised Exponential transformation: $${\displaystyle Q_j(x)=e^{V_j(x)}/\sum _k{e}^{V_k(x)}}$$ This transformation can be considered a multi-input generalisation of the logistic, operating on the whole output layer. It preserves the rank order of its input values, and is a differentiable generalisation of the 'winner-take-all' operation of picking the maximum value. For this reason we like to refer to it as softmax.

Example

With an input of (1, 2, 3, 4, 1, 2, 3), the softmax is approximately (0.024, 0.064, 0.175, 0.475, 0.024, 0.064, 0.175). 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 (0.1, 0.2, 0.3, 0.4, 0.1, 0.2, 0.3) and the softmax is approximately (0.125, 0.138, 0.153, 0.169, 0.125, 0.138, 0.153). This shows that high temperatures de-emphasize the maximum value.

Computation of this example using Python code:

<span id="mwAdw"></span><span class="gp" id="mwAd0">>>> </span><span class="kn" id="mwAd4">import</span><span class="w" id="mwAd8"> </span><span class="nn" id="mwAeA">numpy</span><span class="w" id="mwAeE"> </span><span class="k" id="mwAeI">as</span><span class="w" id="mwAeM"> </span><span class="nn" id="mwAeQ">np</span> <span class="gp" id="mwAeU">>>> </span><span class="n" id="mwAeY">z</span> <span class="o" id="mwAec">=</span> <span class="n" id="mwAeg">np</span><span class="o" id="mwAek">.</span><span class="n" id="mwAeo">array</span><span class="p" id="mwAes">([</span><span class="mf" id="mwAew">1.0</span><span class="p" id="mwAe0">,</span> <span class="mf" id="mwAe4">2.0</span><span class="p" id="mwAe8">,</span> <span class="mf" id="mwAfA">3.0</span><span class="p" id="mwAfE">,</span> <span class="mf" id="mwAfI">4.0</span><span class="p" id="mwAfM">,</span> <span class="mf" id="mwAfQ">1.0</span><span class="p" id="mwAfU">,</span> <span class="mf" id="mwAfY">2.0</span><span class="p" id="mwAfc">,</span> <span class="mf" id="mwAfg">3.0</span><span class="p" id="mwAfk">])</span> <span class="gp" id="mwAfo">>>> </span><span class="n" id="mwAfs">beta</span> <span class="o" id="mwAfw">=</span> <span class="mf" id="mwAf0">1.0</span> <span class="gp" id="mwAf4">>>> </span><span class="n" id="mwAf8">np</span><span class="o" id="mwAgA">.</span><span class="n" id="mwAgE">exp</span><span class="p" id="mwAgI">(</span><span class="n" id="mwAgM">beta</span> <span class="o" id="mwAgQ">*</span> <span class="n" id="mwAgU">z</span><span class="p" id="mwAgY">)</span> <span class="o" id="mwAgc">/</span> <span class="n" id="mwAgg">np</span><span class="o" id="mwAgk">.</span><span class="n" id="mwAgo">sum</span><span class="p" id="mwAgs">(</span><span class="n" id="mwAgw">np</span><span class="o" id="mwAg0">.</span><span class="n" id="mwAg4">exp</span><span class="p" id="mwAg8">(</span><span class="n" id="mwAhA">beta</span> <span class="o" id="mwAhE">*</span> <span class="n" id="mwAhI">z</span><span class="p" id="mwAhM">))</span> <span class="go" id="mwAhQ">array([0.02364054, 0.06426166, 0.1746813, 0.474833, 0.02364054,</span> <span class="go" id="mwAhU"> 0.06426166, 0.1746813])</span>

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.