# Efficient implementation of Softmax activation function and its Derivative (jacobian) in Python The mathematical definition of the Softmax activation function is $S\left(z_{i}\right)=\frac{e^{z_{i}}}{\sum_{j=1}^{K} e^{z_{j}}}$

with the derivative defined as $\frac{\partial S\left(z_{i}\right)}{\partial z_{j}}= \begin{cases}S\left(z_{i}\right) \times\left(1-S\left(z_{i}\right)\right) & \text { if } i=j \\ -S\left(z_{i}\right) \times S\left(z_{j}\right) & \text { if } i \neq j\end{cases}$

The Softmax function and its derivative for a batch of inputs (a 2D array with nRows=nSamples and nColumns=nNodes) can be implemented in the following manner:
Softmax simplest implementation

import numpy as np
def Softmax(x):
'''
Performs the softmax activation on a given set of inputs
Input: x (N,k) ndarray (N: no. of samples, k: no. of nodes)
Returns:
Note: Works for 2D arrays only(rows for samples, columns for nodes/outputs)
'''
max_x = np.amax(x, 1).reshape(x.shape,1) # Get the row-wise maximum
e_x = np.exp(x - max_x ) # For stability
return e_x / e_x.sum(axis=1, keepdims=True)


Softmax gradient (technically jacobian) simplest implementation

import numpy as np
def Softmax_grad(x): # Best implementation (VERY FAST)
'''Returns the jacobian of the Softmax function for the given set of inputs.
Inputs:
x: should be a 2d array where the rows correspond to the samples
and the columns correspond to the nodes.
Returns: jacobian
'''
s = Softmax(x)
a = np.eye(s.shape[-1])
temp1 = np.zeros((s.shape, s.shape, s.shape),dtype=np.float32)
temp2 = np.zeros((s.shape, s.shape, s.shape),dtype=np.float32)
temp1 = np.einsum('ij,jk->ijk',s,a)
temp2 = np.einsum('ij,ik->ijk',s,s)
return temp1-temp2


Please note, and I can’t stress this enough, the above and the following implementations are only tested and fine-tuned for a batch of inputs, i.e., the expected input for the functions is a 2d array with rows representation different samples, and columns representing different nodes.

However, these implementations can be further accelerated (sped-up) by using Numba (https://numba.pydata.org/). Numba is a Just-in-time (JIT) compiler that

translates a subset of Python and NumPy code into fast machine code.

To use numba, install it as:

pip install numba


Also, make sure that your numpy is compatible with Numba or not, although usually pip takes care of that. You can get the info here: https://pypi.org/project/numba/

Accelerating the above functions using Numba is quite simple. Just modify them in the following manner:

Softmax NUMBA implementation

from numba import njit
@njit(cache=True,fastmath=True)  # Best implementation (VERY FAST)
def Softmax(x):
'''
Performs the softmax activation on a given set of inputs
Input: x (N,k) ndarray (N: no. of samples, k: no. of nodes)
Returns:
Note: Works for 2D arrays only(rows for samples, columns for nodes/outputs)
'''
max_x = np.zeros((x.shape,1),dtype=x.dtype)
for i in range(x.shape):
max_x[i,0] = np.max(x[i,:])
e_x = np.exp(x - max_x)
return e_x / e_x.sum(axis=1).reshape((-1, 1)) # Alternative of keepdims=True for Numba compatibility


Softmax derivative (jacobian) NUMBA implementation

from numba import njit
@njit(cache=True,fastmath=True)
def Softmax_grad(x): # Best implementation (VERY FAST)
'''Returns the jacobian of the Softmax function for the given set of inputs.
Inputs:
x: should be a 2d array where the rows correspond to the samples
and the columns correspond to the nodes.
Returns: jacobian
'''
s = Softmax(x)
a = np.eye(s.shape[-1])
temp1 = np.zeros((s.shape, s.shape, s.shape),dtype=np.float32)
temp2 = np.zeros((s.shape, s.shape, s.shape),dtype=np.float32)
# Einsum is unsupported with Numba (nopython mode)
# temp1 = np.einsum('ij,jk->ijk',s,a)
# temp2 = np.einsum('ij,ik->ijk',s,s)
for i in range(s.shape):
for j in range(s.shape):
for k in range(s.shape):
temp1[i,j,k] = s[i,j]*a[j,k]
temp2[i,j,k] = s[i,j]*s[i,k]

return temp1-temp2


This is quite fast and competitive with Tensorflow and PyTorch (https://github.com/manassharma07/crysx_nn/blob/main/benchmarks_tests/Performance_Activation_Functions_CPU.ipynb).

It is in fact also used in the CrysX-Neural Network library (crysx_nn)

Furthermore, the above implementations can be further accelerated using Cupy (CUDA), if using single precision (float32) is not a problem.

CuPy is an open-source array library for GPU-accelerated computing with Python. CuPy utilizes CUDA Toolkit libraries to make full use of the GPU architecture.

The Cupy implementations look as follows:

import cupy as cp
def Softmax_cupy(x):
'''
Performs the softmax activation on a given set of inputs
Input: x (N,k) ndarray (N: no. of samples, k: no. of nodes)
Returns:
Note: Works for 2D arrays only(rows for samples, columns for nodes/outputs)
'''
max_x = cp.amax(x, 1).reshape(x.shape,1)
e_x = cp.exp(x - max_x) # For stability as it is prone to overflow and underflow
#     return e_x / e_x.sum(axis=1, keepdims=True) # Alternative 1
return e_x / e_x.sum(axis=1).reshape((-1, 1)) # Alternative 2

def Softmax_grad_cupy(x): # Best implementation (VERY FAST)
'''Returns the jacobian of the Softmax function for the given set of inputs.
Inputs:
x: should be a 2d array where the rows correspond to the samples
and the columns correspond to the nodes.
Returns: jacobian
'''
s = Softmax_cupy(x)
a = cp.eye(s.shape[-1])
temp1 = cp.zeros((s.shape, s.shape, s.shape),dtype=cp.float32)
temp2 = cp.zeros((s.shape, s.shape, s.shape),dtype=cp.float32)
temp1 = cp.einsum('ij,jk->ijk',s,a)
temp2 = cp.einsum('ij,ik->ijk',s,s)
return temp1-temp2


The above code is also used in the crysx_nn library.
To see how the crysx_nn implementations of Softmax compare with TensorFlow and PyTorch, click here.

I hope you found this information useful.

If you did, then don’t forget to check out my other posts on Machine Learning and efficient implementations of activation/loss functions in Python. [wpedon id="7041" align="center"]