## 30 Aug Implementing Logic Gates using Neural Networks Part 2 by Vedant Kumar

Contents:

Truth Table for XORThe goal of the neural network is to classify the input patterns according to the above truth table. If the input patterns are plotted according to their outputs, it is seen that these points are not linearly separable. Hence the neural network has to be modeled to separate these input patterns using decision planes. 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.

We use the xor neural network of the logistic sigmoid function. But in other cases, the output could be a probability, a number greater than 1, or anything else. Normalizing in this way uses something called an activation function, of which there are many. The basic idea is to take the input, multiply it by the synaptic weight, and check if the output is correct. If it is not, adjust the weight, multiply it by the input again, check the output and repeat, until we have reached an ideal synaptic weight.

So you may need to try, check results and then re-start. I suggest you use a seeded random number generator for initialisation, and adjust the seed value if error values get stuck and do not improve. It can take a surprisingly large number of epochs to train the minimal network using batched or online gradient descent. Most usual mistake is to set it too high, so the network will oscillate or diverge instead of learn. Otherwise you risk that input signal to a neuron might be large from the start in which case learning for that neuron is slow. You might also want to decrease learning rate and increase number of iterations.

There are several workarounds for this problem which largely fall into architecture (e.g. ReLu) or algorithmic adjustments (e.g. greedy layer training). We should check the convergence for any neural network across the paramters. A single perceptron, therefore, cannot separate our XOR gate because it can only draw one straight line. The problem with a step function is that they are discontinuous.

## THE MATH BEHIND GRADIENT DESCENT

This will, therefore, be classified as 1 after passing through the sigmoid function. The loss function we used in our MLP model is the Mean Squared loss function. Though this is a very popular loss function, it makes some assumptions on the data and isn’t always convex when it comes to a classification problem. It was used here to make it easier to understand how a perceptron works, but for classification tasks, there are better alternatives, like binary cross-entropy loss. Let’s go with a single hidden layer with two nodes in it. We’ll be using the sigmoid function in each of our hidden layer nodes and of course, our output node.

XOR is a classification problem and one for which the expected outputs are known in advance. It is therefore appropriate to use a supervised learning approach. The XOR gate consists of an OR gate, NAND gate and an AND gate. 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. We now have a neural network (albeit a lousey one!) that can be used to make a prediction.

- So if you want to find out more, have a look at this excellent article by Simeon Kostadinov.
- So among the various logical operations, XOR logical operation is one such problem wherein linear separability of data points is not possible using single neurons or perceptrons.
- Escaping all the complexities, data professionals use python libraries and frameworks to implement models.
- However, it doesn’t ever touch 0 or 1, which is important to remember.
- Adding input nodes — Image by Author using draw.ioFinally, we need an AND gate, which we’ll train just we have been.
- These are some basic steps one must follow to train a neural network.

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. As I said, there are many different kinds of activation functions – tanh, relu, binary step – all of which have their own respective uses and qualities. For this example, we’ll be using what’s called the logistic sigmoid function.

## Parameter updates

Stay with us and follow up on the next blogs for more content on neural networks. The designing process will remain the same with one change. We will choose one extra hidden layer apart from the input and output layers.

We know that a datapoint’s evaluation is expressed by the relation wX + b . We define a threshold (θ) which classifies our data. Generally, this threshold is set to 0 for a perceptron. This is often simplified and written as a dot- product of the weight and input vectors plus the bias. Error/Loss vs Weights GraphOur goal is to find the weight vector corresponding to the point where the error is minimum i.e. the minima of the error gradient.

## More than only one neuron , the return (let’s use a non-linearity)

I am trying to learn how to use scikit-learn’s MLPClassifier. For a very simple example, I thought I’d try just to get it to learn how to compute the XOR function, since I have done that one by hand as an exercise before. The linear separable data points appear to be as shown below. The XOR gate can be usually termed as a combination of NOT and AND gates and this type of logic finds its vast application in cryptography and fault tolerance. The basic principle of matrix multiplication says if the shape of X is and W is , then only they can be multiplied, and the shape of XW will be .

Define the https://forexhero.info/ to update the parameters. These are some basic steps one must follow to train a neural network. The first step is to import all the modules and define training and testing data as we did for single-layer Perceptron. First, we need to understand that the output of an AND gate is 1 only if both inputs are 1. Coding a neural network from scratch strengthened my understanding of what goes on behind the scenes in a neural network.

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. This completes a single forward pass, where our predicted_output needs to be compared with the expected_output. Based on this comparison, the weights for both the hidden layers and the output layers are changed using backpropagation. Backpropagation is done using the Gradient Descent algorithm.

## NeuralNetwork

No straight line can entirely separate all the True values from the False values.A neural network is essentially a series of hyperplanes that group / separate regions in the target hyperplane. Yes, you will have to pay attention to the progression of the error rate. In larger problem instances, you would typically pay attention to the development of the error function on your test set. This is done by measuring the accuracy of the network after a period of training. XNOR GateNow that we are done with the necessary basic logic gates, we can combine them to give an XNOR gate. Complete introduction to deep learning with various architechtures.

If we compile the whole code of a single-layer perceptron, it will exceed 100 lines. To reduce the efforts and increase the efficiency of code, we will take the help of Keras, an open-source python library built on top of TensorFlow. As we can see, the Perceptron predicted the correct output for logical OR. Similarly, we can train our Perceptron to predict for AND and XOR operators.

## How Neural Networks Solve the XOR Problem

These weights will need to be adjusted, a process I prefer to call “learning”. On the contrary, the function drawn to the right of the ReLU function is linear. Applying multiple linear activation functions will still make the network linear. I was trying to implement an XOR gate with tensorflow. I succeeded in implementing that, but i don’t fully understand why it works. So both with one hot true and without one hot true outputs.

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Further, this error is divided by 2, to make it easier to differentiate, as we’ll see in the following steps. Hence, it signifies that the Artificial Neural Network for the XOR logic gate is correctly implemented. Is there a magic sequence of parameters to allow the model to infer correctly from the data it hasn’t seen before?

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. We get our new weights by simply incrementing our original weights with the computed gradients multiplied by the learning rate.

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So keeping this in mind, the weight matrix W will be . From the diagram, the OR gate is 0 only if both inputs are 0. Using a random number generator, our starting weights are $.03$ and $0.2$. It is also sensible to make sure that the parameters and gradients are cnoverging to sensible values. Furthermore, we would expect the gradients to all approach zero. In larger networks the error can jump around quite erractically so often smoothing (e.g. EWMA) is used to see the decline.

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Following code gist shows the initialization of parameters for neural network. This tutorial is very heavy on the math and theory, but it’s very important that you understand it before we move on to the coding, so that you have the fundamentals down. In the next tutorial, we’ll put it into action by making our XOR neural network in Python. This meant that neural networks couldn’t be used for a lot of the problems that required complex network architecture. Where y_output is now our estimation of the function from the neural network. (from Kevin Swingler via Lucas Araújo)The trick is to realise that we can just logically stack two perceptrons.

Data Science Stack Exchange is a question and answer site for Data science professionals, Machine Learning specialists, and those interested in learning more about the field. Initialize the value of weight and bias for each layer. The number of nodes in the input layer equals the number of features. To design a hidden layer, we need to define the key constituents again first. We are using a more simple optimization technique here.