Understanding Basics of Deep Learning by solving XOR problem by Lalit Kumar Analytics Vidhya

It can be used also for non-separable data sets, where the aim is to find a perceptron with a small number of misclassifications. Below is an example of a learning algorithm for a single-layer perceptron with a single output unit. For a single-layer perceptron with multiple output units, since the weights of one output unit are completely separate from all the others’, the same algorithm can be run for each output unit. 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. One of the main problems historically with neural networks were that the gradients became too small too quickly as the network grew.

Single-layer Perceptron

The final weights can be interpreted as the learned representation of the XOR function, allowing the network to classify the inputs correctly. One neuron with two inputs can form a decisive surface in the form of an arbitrary line. 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. This means that no matter what values are assigned to weights and thresholds, a single-layer neural network is unable to reproduce the relationship between input and output required to represent the XOR function. In the hidden layer, the network effectively transforms the input space into a new space where the XOR problem becomes linearly separable.

• However, these are much simpler, in both design and in function, and nowhere near as powerful as the real kind.
• Each connection (similar to a synapse) between artificial neurons can transmit a signal from one to the other.
• The model is trained using the Adam optimizer and binary cross-entropy loss function.
• A single-layer perceptron can solve problems that are linearly separable by learning a linear decision boundary.
• Now let’s build the simplest neural network with three neurons to solve the XOR problem and train it using gradient descent.

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. There are several workarounds for this problem which largely fall into architecture (e.g. ReLu) or algorithmic adjustments (e.g. greedy layer training). Two lines is all it would take to separate the True values from the False values in the XOR gate. 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. Neural networks are now widespread and are used in practical tasks such as speech recognition, automatic text translation, image processing, analysis of complex processes and so on. Used to store information about the time a sync with the lms_analytics cookie took place for users in the Designated Countries.

It is often incorrectly believed that they also conjectured that a similar result would hold for a multi-layer perceptron network. However, this is not true, as both Minsky and Papert already knew that multi-layer perceptrons were capable of producing an XOR function. (See the page on Perceptrons (book) for more information.) Nevertheless, the often-miscited Minsky and Papert text caused a significant decline in interest and funding of neural network research. Now that we’ve looked at real neural networks, we can start discussing artificial neural networks. Like the biological kind, an artificial neural network has inputs, a processing area that transmits information, and outputs.

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With these weights and biases, the network produces the correct XOR output for each input pair (A, B). At the core of a neural network are “neurons,” which work together to solve problems. Each friend processes part of the puzzle, and their combined insights help solve it. If the training set is linearly separable, then the perceptron is guaranteed to converge after making finitely many mistakes.36 The theorem is proved by Rosenblatt et al.

They allow finding the minimum of error (or cost) function with a large number of weights and biases in a reasonable number of iterations. A drawback of the gradient descent method is the need to calculate partial derivatives for each of the input values. Very often when training neural networks, we can get to the local minimum of the function without finding an adjacent minimum with the best values. Also, gradient descent can be very slow and makes too many iterations if we are close to the local minimum. The outputs generated by the XOR logic are not linearly separable in the hyperplane.

Convergence of one perceptron on a linearly separable dataset

The XOR problem serves as a fundamental example in neural network training, demonstrating the necessity of non-linear architectures. Before we dive deeper into the XOR problem, let’s briefly understand how neural networks work. Neural networks are composed of interconnected nodes, called neurons, which are organized into layers. The input layer receives the input data passed through the hidden layers. Each neuron in the network performs a weighted sum of its inputs, applies an activation function to the sum, and passes the result to the next layer. For multilayer perceptrons, where a hidden layer exists, more sophisticated algorithms such as backpropagation must be used.

A single-layer perceptron cannot solve the XOR problem due to its inability to represent non-linear decision boundaries. To address this, multilayer perceptrons (MLPs) are employed, which consist of multiple layers of neurons that can learn complex patterns. A linear neural network, which can be represented as a single-layer perceptron, struggles to classify these inputs correctly. The inability to find a linear decision boundary highlights the need for deeper architectures or non-linear activation functions.

Following code gist shows the initialization of parameters for neural network. In my next post, I will show how you can write a simple python program that uses the Perceptron Algorithm to automatically update the weights of these Logic gates. To speed things up with the beauty of computer science – when we run this iteration 10,000 times, it gives us an output of about $.9999$. This is very close to our expected value of 1, and demonstrates that the network has learned what the correct output should be. A converged result should have hyperplanes that separate the True and False values.

A single perceptron, therefore, cannot separate our XOR gate because it can only draw one straight line. While taking the Udacity Pytorch Course by Facebook, I found it difficult understanding how the Perceptron works with Logic gates (AND, OR, NOT, and so on). I decided to check online resources, but as of the time of https://traderoom.info/neural-network-for-xor/ writing this, there was really no explanation on how to go about it. So after personal readings, I finally understood how to go about it, which is the reason for this medium post. This approach helps in mitigating the vanishing gradient problem, allowing for better training of deeper networks. The basic idea is to take the input, multiply it by the synaptic weight, and check if the output is correct.

Example Code Snippet

It works by propagating the error backwards through the network and updating the weights using gradient descent. Some machine learning algorithms like neural networks are already a black box, we enter input in them and expect magic to happen. Still, it is important to understand what is happening behind the scenes in a neural network.

We want to find the minimum loss given a set of parameters (the weights and biases). Recalling some AS level maths, we can find the minima of a function by minimising the gradient (each minima has zero gradient). Also luckily for us, this problem has no local minima so we don’t need to do any funny business to guarantee convergence. Created by the Google Brain team, TensorFlow presents calculations in the form of stateful dataflow graphs. The library allows you to implement calculations on a wide range of hardware, from consumer devices running Android to large heterogeneous systems with multiple GPUs.

XOR Problem Solver Using Neural Networks

A simple feedforward neural network built using TensorFlow to solve the classic XOR problem. This project demonstrates how neural networks can be applied to solve non-linear classification problems using Python. In addition to MLPs and the backpropagation algorithm, the choice of activation functions also plays a crucial role in solving the XOR problem.