The mathematical definition of the ReLU activation function is

and its derivative is defined as

The ReLU 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:

**ReLU simplest implementation**

import numpy as np def ReLU(x): return np.maximum(0.,x)

**ReLU derivative simplest implementation**

import numpy as np def ReLU_grad(x): return np.greater(x, 0.).astype(np.float32)

I should also note here that the above implementations use broadcasting. So in case, you find it to be slow (I haven’t tested it enough to say something decisive), you can try the following alternative:

import numpy as np def ReLU(x): # If Broadcasting seems expensive compared to TF and PyTorch a = np.zeros(x.shape,dtype=np.float32) return np.maximum(a,x)

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:

**ReLU NUMBA implementation**

from numba import njit @njit(cache=True,fastmath=True) def ReLU(x): # Broadcasting seems expensive compared to TF and PyTorch # return np.maximum(0.,x) a = np.zeros(x.shape,dtype=np.float32) return np.maximum(a,x)

**ReLU derivative NUMBA implementation**

from numba import njit @njit(cache=True,fastmath=True) def ReLU_grad(x): return np.greater(x, 0.).astype(np.float32)

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 ReLU_cupy(x): # Broadcasting seems expensive compared to TF and PyTorch # return np.maximum(0.,x) a = cp.zeros(x.shape,dtype=cp.float32) return cp.maximum(a,x)

def ReLU_grad_cupy(x): return cp.greater(x, 0.).astype(cp.float32)

The above code is also used in the crysx_nn library.

To see how the crysx_nn implementations of ReLU 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.

I’m a physicist specializing in computational material science with a PhD in Physics from Friedrich-Schiller University Jena, Germany. 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.