The mathematical definition of the Softmax activation function is

with the derivative defined as

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[0],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[0], s.shape[1], s.shape[1]),dtype=np.float32) temp2 = np.zeros((s.shape[0], s.shape[1], s.shape[1]),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[0],1),dtype=x.dtype) for i in range(x.shape[0]): 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[0], s.shape[1], s.shape[1]),dtype=np.float32) temp2 = np.zeros((s.shape[0], s.shape[1], s.shape[1]),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[0]): for j in range(s.shape[1]): for k in range(s.shape[1]): 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[0],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[0], s.shape[1], s.shape[1]),dtype=cp.float32) temp2 = cp.zeros((s.shape[0], s.shape[1], s.shape[1]),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.