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

In order for the network to implement the XOR function specified in the table above, you need to position the line so that the four points are divided into two sets. Trying to draw such a straight line, we are convinced that this is impossible. Backpropagation is a way to update the weights and biases of a model starting from the output layer all the way to the beginning. The main principle behind it is that each parameter changes in proportion to how much it affects the network’s output.

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If this was a real problem, we would save the weights and bias as these define the model. The gif outputs are plots of the XOR(A,B)evaluated for a mesh grid of A and B during the training. A multi-layer perceptron implementation using python and numpy for the https://forexhero.info/ XOR problem. After running model.fit(), Tensorflow will feed the input data 5000 times and try to fit the model. The goal of our network is to train a network to receive two boolean inputs and return True only when one input is True and the other is False.

How to build a neural network on Tensorflow for XOR

In practice you can insert this column anywhere you like, but we typically place it either as (1) the first entry in the feature vector or (2) the last entry in the feature vector. The next step is to create the LogisticRegression() class. To be able to use it as a PyTorch model, we will pass torch. Then, we will define the init() function by passing the parameter self.

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We then applied the same backpropagation + Python implementation to a subset of the MNIST dataset to demonstrate that the algorithm can be used to work with image data as well. The biggest advantage of using the above solution for the XOR problem is that now, xor neural network we don’t have non-differentiable step-functions anymore. Instead, we have differential equations sigmoid activation functions which means we can start with a random guess for each of the parameters in our model and later, optimize according to our dataset.

1. This function allows us to fit the output in a way that makes more sense.
2. The overall components of an MLP like input and output nodes, activation function and weights and biases are the same as those we just discussed in a perceptron.
3. We’ll use Stochastic Gradient Descent as optimisation method and Binary Crossentropy as the loss function.
4. Observe how the green points are below the plane and the red points are above the plane.

Weights and Biases

The first phase of the backward pass is to compute our error, or simply the difference between our predicted label and the ground-truth label (Line 91). Since the final entry in the activations list A contains the output of the network, we can access the output prediction via A[-1]. The value y is the target output for the input data point x. We’ll start by reviewing each of these phases at a high level. From there, we’ll implement the backpropagation algorithm using Python.

The closer the resulting value is to 0 and 1, the more accurately the neural network solves the problem. TensorFlow is an open-source machine learning library designed by Google to meet its need for systems capable of building and training neural networks and has an Apache 2.0 license. Hence, it signifies that the Artificial Neural Network for the XOR logic gate is correctly implemented. Let’s go with a single hidden layer with two nodes in it.

There is no intuitive way to distinguish or separate the green and the blue points from each other such that they can be classified into respective classes. So, by shifting our focus from a 2-dimensional visualization to a 3-dimensional one, we are able to classify the points generated by the XOR operator far more easily. This is clearly a wrong hypothesis and doesn’t hold valid simply because the intersection point of the green and the red line segment cannot lie in both the half-spaces. This means that the intersection point of the two lines cannot be classified in both Class 0 as well as Class 1. The image on the left cannot be considered to be a convex set as some of the points on the line joining two points from \(S\) lie outside of the set \(S\).

An iterative gradient descent finds the value of the coefficients for the parameters of the neural network to solve a specific problem. To bring everything together, we create a simple Perceptron class with the functions we just discussed. We have some instance variables like the training data, the target, the number of input nodes and the learning rate.

This value p will be passed through every layer in the network, propagating until we reach the final output prediction. First of all, you should think about how your targets look like. Forclassification problems, one usually takes as many output neurons as one hasclasses. Then the softmax function is applied.1 The softmax function makes sure that the output of every single neuron is in \([0, 1]\) and the sum of all outputs is exactly \(1\).

Then for the X_train and y_tarin, we will take the first 150 numbers, and then for the X_test and y_test, we will take the last 150 numbers. Finally, we will return X_train, X_test, y_train, and y_test. Until the 2000s, choosing the transformation was done manually for problems such as vision, speech, etc. using histogram of gradients to study regions of interest.

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. As discussed, it’s applied to the output of each hidden layer node and the output node. Its differentiable, so it allows us to comfortably perform backpropagation to improve our model. The overall components of an MLP like input and output nodes, activation function and weights and biases are the same as those we just discussed in a perceptron. These parameters are what we update when we talk about “training” a model.

A good resource is the Tensorflow Neural Net playground, where you can try out different network architectures and view the results. There’s a lot to cover when talking about backpropagation. So if you want to find out more, have a look at this excellent article by Simeon Kostadinov. What we now have is a model that mimics the XOR function. The ⊕ (“o-plus”) symbol you see in the legend is conventionally used to represent the XOR boolean operator. A converged result should have hyperplanes that separate the True and False values.

Backpropagation can be considered the cornerstone of modern neural networks and deep learning. A large number of methods are used to train neural networks, and gradient descent is one of the main and important training methods. It consists of finding the gradient, or the fastest descent along the surface of the function and choosing the next solution point.