Implementing the XOR Gate using Backpropagation in Neural Networks by Siddhartha Dutta


After compiling the model, it’s time to fit the training data with an epoch value of 1000. After training the model, we will calculate the accuracy score and print the predicted output on the test data. We will use the Unit step activation function to keep our model simple and similar to traditional Perceptron.


Apart from the usual visualization and numerical libraries , we’ll use cycle from itertools . This is done since our algorithm cycles through our data indefinitely until it manages to correctly classify the entire training data without any mistakes in the middle. The XOR output plot — Image by Author using draw.ioOur algorithm —regardless of how it works — must correctly output the XOR value for each of the 4 points. We’ll be modelling this as a classification problem, so Class 1 would represent an XOR value of 1, while Class 0 would represent a value of 0. This process is repeated until the predicted_output converges to the expected_output. It is easier to repeat this process a certain number of times (iterations/epochs) rather than setting a threshold for how much convergence should be expected.

Define the output data matrix as numpy array:

You’ll notice that the loop never terminates, since a perceptron can only converge on linearly separable data. Linearly separable data basically means that you can separate data with a point in 1D, a line in 2D, a plane in 3D and so on. To visualize how our model performs, we create a mesh of datapoints, or a grid, and evaluate our model at each point in that grid. Finally, we colour each point based on how our model classifies it. So the Class 0 region would be filled with the colour assigned to points belonging to that class. If not, we reset our counter, update our weights and continue the algorithm.

  • I decided to model this network in Python, since it is the most popular language for Deep Learning because of the active development of packages like numpy, tensorflow, keras, etc.
  • In W1, the values of weight 1 to weight 9 (in Fig 6.) are defined and stored.
  • Whereas, to separate data points of XOR, we need two linear lines or can add a new dimension and then separate them using a plane.
  • In comparative terms to Picasso’s work, we closed the essential elements in a depiction that reduces language to the bull in line drawing.
  • For this ANN, the current learning rate (‘eta’) and the number of iterations (‘epoch’) are set at 0.1 and respectively.

This is our final equation when we go into the mathematics of xor neural network descent and calculate all the terms involved. To understand how we reached this final result, see this blog. A duplexed NVRAM is shown in this work to be as reliable as magnetic disk storage in this study.

A single neuron solution to the XOR problem

In RAID5 the following steps are required if the disks have an XOR capability. One must differentiate reduction errors from discrepancies between the FEM and the test model. The unadapted nPHRW, however, requires more time for convergence, where it fails to yield a correct result in the XOR5 version. We ascribe the superior behavior of these genotypes to the very compact genetic representation, paired with a good adaptation policy.

And with the support of python libraries like TensorFlow, Keras, and PyTorch, deciding these parameters becomes easier and can be done in a few lines of code. Stay with us and follow up on the next blogs for more content on neural networks. Using all the methods we discussed, we are going to put together the complete neural network model to learn the XOR truth table.

update the weights

We get our new weights by simply incrementing our original weights with the computed gradients multiplied by the learning rate. This is the most complicated of all the steps in designing a neural network. I am not going to go over all the details of the implementation but just give an intuition.


Next, we compute the number of input features , number of output features and set the number of hidden layer neurons. The beauty of this code is that you can reuse it for any input/output combinations as long as you shape the X and Y values correctly. The information of a neural network is stored in the interconnections between the neurons i.e. the weights. A neural network learns by updating its weights according to a learning algorithm that helps it converge to the expected output.

With the structure inspired by the biological neural network, the ANN is comprised of multiple layers — the input layer, hidden layer, and output layer — of nodes that send signals to each other. An activation function limits the output produced by neurons but not necessarily in the range or . This bound is to ensure that exploding and vanishing of gradients should not happen. The other function of the activation function is to activate the neurons so that model becomes capable of learning complex patterns in the dataset.


This leads to K(K − 1) interconnections if there are K nodes, with a wij weight on each. In this arrangement, the neurons transmit signals back and forth to each other in a closed-feedback loop, eventually settling in stable states. We consider a two-dimensional lattice of n × n qubits and define a set of products of Pauli operators acting on the n2 qubits. Then we partition the set into three subsets of operators; this process induces a partition of the Hilbert space, ⊗n2, into subspaces. Lastly, we discuss the construction of a subsystem code and the error correction procedure for the code.

The natural language for artificial intelligence

We will take the help of NumPy, a python library famous for its mathematical operations and multidimensional arrays. Then we will switch to Keras for building multi-layer Perceptron. Scan clock, which prevents the attacker from scanning in any intended values. Therefore, if the attacker keeps flushing the scan chain, an original or inverted scan in sequence shows up at the scan output after λ bits of zeros. Furthermore, as the protected obfuscation key has settled down after the whole chain is scanned, the Shadow chain does not impact the DFT launching or capturing process, such as when applying stuck-at or transition delay faults. The Shadow chain should be synchronously reset with the LFSR at any reset event.

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Also, towards the end of the session, we will use tensorflow deep-learning library to build a neural network, to illustrate the importance of building a neural network using a deep-learning framework. Today we’ll create a very simple neural network in Python, using Keras and Tensorflow to understand their behavior. We’ll implement an XOR logic gate and we’ll see the advantages of automated learning to traditional programming. In this project, I implemented a proof of concept of all my theoretical knowledge of neural network to code a simple neural network from scratch in Python without using any machine learning library. Remember the linear activation function we used on the output node of our perceptron model? You may have heard of the sigmoid and the tanh functions, which are some of the most popular non-linear activation functions.

Now, we will define a class MyPerceptron to include various functions which will help the model to train and test. The first function will be a constructor to initialize the parameters like learning rate, epochs, weight, and bias. The relationship defined by the XOR gate is adequate to the fact that the axiomatic and logical characteristics of language are inconceivable separately from each other. If the logical feature is ignored, there may be a chain, but the meaning is not constructed. In comparative terms to Picasso’s work, we closed the essential elements in a depiction that reduces language to the bull in line drawing. This notebook is created to coincide the 90th birth anniversary of pioneering psychologist and artificial intelligence researcher, Frank Rosenblatt, born July 11, 1928 – died July 11, 1971.

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This can be interpreted as a fractal nature of the language even if for the moment it is restricted only to a formal sequence of symbols. We will see that the fractal nature can also be seen in the higher level of interpretation of the natural language that is language as meaning creation. In other words, we have to separate the fractal nature as previously shown which characterizes a formal sequence of symbols from the higher level of language interpretation, which is also a fractal-like. Indeed, language is not a simple sequence of symbols but it is also a creative process of signification, that is, creation of meaning. We will show that also in the creation of meaning there is a recurrent process that can be interpreted as a fractal construction. To single out a more structured recurrence law, we need to remind some basic concepts on recursion in the language process.

Lastly, the logic table for the XOR logic gate is included as ‘inputdata’ and ‘outputdata’. As mentioned, a value of 1 was included with every input dataset to represent the bias. In their book, Perceptrons, Minsky and Papert suggested that “simple ANNs” were not computationally complex enough to solve the XOR logic problem . Spiking neural network is interesting both theoretically and practically because of its strong bio-inspiration nature and potentially outstanding energy efficiency. Unfortunately, its development has fallen far behind the conventional deep neural network , mainly because of difficult training and lack of widely accepted hardware experiment platforms.

Surprisingly, DeepMind developed a neural network that found a new multiplication algorithm that outperforms current, the best algorithm. In this article, we will discuss the research in more detail. Obviously, you can code the XOR with a if-else structure, but the idea was to show you how the network evolves with iterations in an-easy-to-see way.

Still, it is important to understand what is happening behind the scenes in a neural network. Coding a simple neural network from scratch acts as a Proof of Concept in this regard and further strengthens our understanding of neural networks. It just involves multiplication of weights with inputs, addition with biases and sigmoid function. Note that we are storing all the intermediate results in a cache. This is needed for the gradient computation in the back propagation step.

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The empty list ‘errorlist’ is created to store the error calculated by the forward pass function as the ANN iterates through the epoch. A simple for loop runs the input data through both the forward pass and backward pass functions as previously defined, allowing the weights to update through the network. Lastly, the list ‘errorlist’ is updated by finding the average absolute error for each forward propagation. This allows for the plotting of the errors over the training process. While there are many different activation functions, some functions are used more frequently in neural networks.

Identified a neural hierarchy corresponding to time scales for speech processing, placing these scales as something underlying the basic structure of language . Here, the loss function is calculated using the mean squared error . Backpropagation function reduces the prediction errors during each training step. Updating the weights of the network through backpropagation. Mathematically we need to compute the derivative of the activation function. When the boolean argument is set as true, the sigmoid function calculates the derivative of x.

  • Now, we will define a class MyPerceptron to include various functions which will help the model to train and test.
  • In this tutorial I will not discuss exactly how these ANNs work, but instead I will show how flexible these models can be by training an ANN that will act as a XOR logic gate.
  • Note that all functions are normalized in such a way that their slope at the origin is 1.
  • Decide the number of hidden layers and nodes present in them.

I hope that the mathematical explanation of neural network along with its coding in Python will help other readers understand the working of a neural network. Following code gist shows the initialization of parameters for neural network. In the forward pass, we apply the wX + b relation multiple times, and applying a sigmoid function after each call.

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