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backpropagation Creating neural net for xor function

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To train the neural network, we build the error function. The error function is calculated as the difference between the output vector from the neural network with certain weights and the training output vector for the given training inputs. 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. An iterative gradient descent finds the value of the coefficients for the parameters of the neural network to solve a specific problem.

  • On the contrary, the function drawn to the right of the ReLU function is linear.
  • The connections from the those units to the output would allow you to say ‘fire if the OR gate fires and the AND gate doesn’t’, which is the definition of the XOR gate.
  • Note that this is different from how you would train a neural network, where you wouldn’t try and correctly classify your entire training data.
  • Let’s bring everything together by creating an MLP class.

The function simply returns it’s input without applying any math, so it’s essentially the same as using no activation function at all. The first two params are training and target data, the third one is the number of epochs (learning iterations) and the last one tells keras how much info to print out during the training. If we imagine such a neural network in the form of matrix-vector operations, then we get this formula. Gradient descent is an iterative optimization algorithm for finding the minimum of a function.

Understanding Basics of Deep Learning by solving XOR problem

It uses known concepts to solve problems in neural networks, such as Gradient Descent, Feed Forward and Back Propagation. We are also using supervised learning approach to solve X-OR using neural network. The architecture of a network refers to its general structure — the number of hidden layers, the number of nodes in each layer and how these nodes are inter-connected. Remember that a perceptron must correctly classify the entire training data in one go. If we keep track of how many points it correctly classified consecutively, we get something like this.

This kind of architecture — shown in Figure 4 — is another feed-forward network known as a multilayer perceptron (MLP). And now let’s run all this code, which will train the neural network and calculate the error between the actual values of the XOR function and the received data after the neural network is running. The closer the resulting value is to 0 and 1, the more accurately the neural network solves the problem. Now let’s build the simplest neural network with three neurons to solve the XOR problem and train it using gradient descent. 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. One of the main problems historically with neural networks were that the gradients became too small too quickly as the network grew.

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. Real world problems require stochastic gradient descents which “jump about” as they descend giving them the ability to find the global minima given a long enough time. Two lines is all it would take to separate the True values from the False values in the XOR gate. A good resource is the Tensorflow Neural Net playground, where you can try out different network architectures and view the results. Let’s train our MLP with a learning rate of 0.2 over 5000 epochs.

Our 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. In any iteration — whether testing or training — these nodes are passed the input from our data. The perceptron basically works as a threshold function — non-negative outputs are put into one class while negative ones are put into the other class. The basic idea is to take the input, multiply it by the synaptic weight, and check if the output is correct.

Then, at the end, the pros (simple evaluation and simple slope) outweight the cons (dead neuron and non-differentiability at the origin). If you want to read another explanation on why a stack of linear layers is still linear, please access this Google’s Machine Learning Crash Course page. Sounds like we are making real improvements here, but a linear function of a linear function makes the whole thing still linear.

  • Such problems are said to be two class classification problem.
  • We’ll come back to look at what the number of neurons means in a moment.
  • Artificial neural networks (ANNs), or connectivist systems are computing systems inspired by biological neural networks that make up the brains of animals.
  • For example, some classification algorithms can also be used for regression, and some regression algorithms can be used for classification.
  • Created by the Google Brain team, TensorFlow presents calculations in the form of stateful dataflow graphs.

Using a random number generator, our starting weights are $.03$ and $0.2$. As we move downwards the line, the classification (a real number) increases. When we stops at the collapsed points, we have classification equalling 1. The last layer ‘draws’ the line over xor neural network representation-space points. All the previous images just shows the modifications occuring due to each mathematical operation (Matrix Multiplication followed by Vector Sum). Another way of think about it is to imagine the network trying to separate the points.

XOR-jax

However, these are much simpler, in both design and in function, and nowhere near as powerful as the real kind. The 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. Artificial Intelligence aims to mimic human intelligence using various mathematical and logical tools. These system were able to learn formal mathematical rules to solve problem and were deemed intelligent systems.

The AND logical function is a 2-variables function, AND(x1, x2), with binary inputs and output. Let’s see what happens when we use such learning algorithms. The images below show the evolution of the parameters values over training epochs. AND gate operation is a simple multiplication operation between the inputs.

The inputs remain the same with an additional bias input of 1. The table on the right below displays the output of the 4 inputs taken as the input. An interesting thing to notice here is that the total number of weights has increased to 9.

Perceptrons

Other approaches are unsupervised learning and reinforcement learning. To train our perceptron, we must ensure that we correctly classify all of our train data. Note that this is different from how you would train a neural network, where you wouldn’t try and correctly classify your entire training data. That would lead to something called overfitting in most cases. 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.

In this article, an application of YOLOv5, to detect items present in a retail store shelf is presented.

Apart from the usual visualization ( matplotliband seaborn) and numerical libraries (numpy), 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. So these new weights gave us a small adjustment, and our new output is $0.538$. We start with random synaptic weights, which almost always leads to incorrect outputs.

One of the most popular libraries is numpy which makes working with arrays a joy. Keras also uses numpy internally and expects numpy arrays as inputs. We import numpy and alias it as np which is pretty common thing to do when writing this kind of code. Jupyer notebook will help to enter code and run it in a comfortable environment.

OR logical function

In fact so small so quickly that the change in a deep parameter value causes such a small change in the output that it either gets lost in machine noise. Or, in the case of probabilistic models, lost in dataset noise. We now have a neural network (albeit a lousey one!) that can be used to make a prediction.