The mathematical definition of the Softplus activation function is

with the derivative defined as

, which is actually the Sigmoid function. We have already discussed some efficient and stable implementations of the Sigmoid function here.

The Softplus 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:

**Softplus simplest implementation**

import numpy as np def Softplus(x): return np.log(1 + np.exp(-np.abs(x))) + np.maximum(x,0)

**oftplus gradient simplest implementation**

import numpy as np def Softplus_grad(x): return np.divide(1.,1.+np.exp(-x))

The above implementations are not stable enough, however, and can result in over/underflow (Reference: https://stackoverflow.com/questions/44230635/avoid-overflow-with-softplus-function-in-python)

The following is a stable implementation of the Softplus function

def Softplus(x): # Reference: https://stackoverflow.com/questions/44230635/avoid-overflow-with-softplus-function-in-python return np.log1p(np.exp(-np.abs(x))) + np.maximum(x, 0) # return np.log(1 + np.exp(-np.abs(x))) + np.maximum(x,0)

For the gradient of the Softplus function, we can make use of the stable and efficient implementation of the Sigmoid function, described here.

def Sigmoid(x): # Also known as logistic/soft step or even expit in scipy.special # Alternative 1 (Doesn't work with Numba as boolean masking (fancy indexing) is not supported for 2D arrays -https://stackoverflow.com/questions/57915632/numba-nopython-mode-cannot-accept-2-d-boolean-indexing ) # Hao Peng's answer from here https://stackoverflow.com/questions/51976461/optimal-way-of-defining-a-numerically-stable-sigmoid-function-for-a-list-in-pyth pos_mask = (x >= 0) # Boolean array inversion is faster than another comparison neg_mask = ~pos_mask z = np.zeros_like(x) z[pos_mask] = np.exp(-x[pos_mask]) z[neg_mask] = np.exp(x[neg_mask]) top = np.ones_like(x) top[neg_mask] = z[neg_mask] return top / (1. + z) def Softplus_grad(x): # This is simply the sigmoid function return Sigmoid(x)

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.

Furthermore, these implementations can still be 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:

**Softplus NUMBA implementation**

from numba import njit @njit(cache=True,fastmath=True) def Softplus(x): # Reference: https://stackoverflow.com/questions/44230635/avoid-overflow-with-softplus-function-in-python return np.log1p(np.exp(-np.abs(x))) + np.maximum(x, 0) # np.log(1 + np.exp(-np.abs(x))) + np.maximum(x,0)

**Softplus derivative (gradient) NUMBA implementation**

from numba import njit @njit(cache=True,fastmath=True, parallel=True) def Sigmoid(x): # Also known as logistic/soft step or even expit in scipy.special # Hao Peng's answer from here https://stackoverflow.com/questions/51976461/optimal-way-of-defining-a-numerically-stable-sigmoid-function-for-a-list-in-pyth # Works only for 2D arrays output = np.zeros((x.shape[0],x.shape[1]),dtype=x.dtype) for i in prange(x.shape[0]): for j in range(x.shape[1]): x_val = x[i,j] if x_val>=0: output[i,j] = 1. / ( 1. + np.exp(-x_val) ) else: e_x = np.exp(x_val) output[i,j] = e_x / ( 1. + e_x ) return output @njit(cache=True,fastmath=True) def Softplus_grad(x): # This is simply the sigmoid function # The following would be susceptible to over/underflow just like th Sigmoid function # return np.divide(1.,1.+np.exp(-x)) # Use this instead return Sigmoid(x)

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)

Moreover, 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 Softplus_cupy(x): # Reference: https://stackoverflow.com/questions/44230635/avoid-overflow-with-softplus-function-in-python return cp.log1p(cp.exp(-cp.abs(x))) + cp.maximum(x, 0) # np.log(1 + np.exp(-np.abs(x))) + np.maximum(x,0)

def Sigmoid_cupy(x): return cp.exp(-cp.logaddexp(0., -x)) def Softplus_grad_cupy(x): # This is simply the sigmoid function # The following would be susceptible to over/underflow just like th Sigmoid function # return cp.divide(1.,1.+cp.exp(-x)) # Use this instead return Sigmoid_cupy(x)

The above code is also used in the crysx_nn library.

To see how the crysx_nn implementations of Softplus 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.

Ph.D. researcher at Friedrich-Schiller University Jena, Germany. I’m a physicist specializing in computational material science. I write efficient codes for simulating light-matter interactions at atomic scales. I like to develop Physics, DFT, and Machine Learning related apps and software from time to time. Can code in most of the popular languages. I like to share my knowledge in Physics and applications using this Blog and a YouTube channel.