Softmax

The softmax operator, defined by

$$ \sigma: \mathbb{R}^K \rightarrow (0,1)^K, $$

where $K > 1$, takes a tuple

$$ \mathbf{z} = (z_1, \ldots, z_K) \in \mathbb{R}^K $$

and computes each component of vector $\sigma(\mathbf{z}) \in (0,1)^K$ with

$$ \sigma(\mathbf{z})i = \frac{e^{z_i}}{\sum{j=1}^{K} e^{z_j}}. $$

We refer to $N = \sum e^{z_j}$ as the normalising factor. After applying softmax, each component will be in the interval (0,1), and the components will add up to 1, so that they can be interpreted as probabilities.

To avoid numerical instability, a variant called "safe softmax" replaces the original softmax. It subtracts the maximum value of the input vector from each element before applying the exponential. This simple trick doesn’t change the output, but it significantly improves numerical stability.

$$ \sigma(\mathbf{z})i = \frac{e^{z_i - z{max}}}{\sum_{j=1}^{K} e^{z_j - z_{max}}}. $$

The base used in softmax is irrelevant. $e$ is used for convenience, but other bases such as 2 and 3 can also be used, since softmax is translation-invariant (adding the same constant to every logit does nothing). That is, for any base $b \geq 2$, $\sigma_b(\mathbf{z})i = \frac{e^{(\ln b)\,z_i}}{\sum{j} e^{(\ln b)\,z_j}} = \sigma\left( (\ln b)\,\mathbf{z} \right)i$*. To maximise the domain, we pick $b = 2$. This is what we want to prove: $\sigma(\mathbf{z})i = \frac{2^{z_i - z_{max}}}{\sum (2^{z_j - z_{max}})}.$*

The normalising factor $N = \sum e^{z_j}$ needs to be known before any individual $\sigma(\mathbf{z})_i$ can be computed. This means that at least a two-step instruction is required.

Implementation

The sequence performs 14 virtual steps:

Step	Operation	Description
1	VirtualConst(0)	Initialize zero tensor
2	Gte	Compute ge0 = (z >= 0)
3	Sub	Compute neg_z = -z
4	Select	Compute abs_z = select(ge0, z, -z)
5	VirtualPow2	Compute `c = 2^{
6	VirtualConst(Q)	Constant quantization scalar
7	Div	Compute d_q_over_c = Q / c
8	Mul	Compute d_q_times_c = Q * c
9	Select	Select d = (z >= 0 ? Q * c : Q / c)
10	Sum	ReduceSum(d) to get total
11	Broadcast	Broadcast the sum
12	Mul	Compute f = Q * d
13	Div	Normalize g = f / e_sum
14	VirtualMove	Write final result to output tensor