# Neural Representation of AND, OR, NOT, XOR and XNOR Logic Gates Perceptron Algorithm by Obumneme Stanley Dukor

The input is arranged as a matrix where rows represent examples and column represent features. So, if we have say m examples and n features then we will have an m x n matrix as input. This notebook is an excellent example of choosing Relu instead of sigmoid to avoid the vanishing gradient problem. Later we will require the derivative of this function so we can add in a factor of 0.5 which simplifies the derivative. 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.

- The ⊕ (“o-plus”) symbol you see in the legend is conventionally used to represent the XOR boolean operator.
- 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.
- A drawback of the gradient descent method is the need to calculate partial derivatives for each of the input values.

In the above figure, we can see that above the linear separable line the red triangle is overlapping with the pink dot and linear separability of data points is not possible using the XOR logic. So now let us understand how to solve the XOR problem with neural networks. If we change weights on the next step of gradient descent methods, we will minimize the difference between output on the neurons and training set of the vector.

## Building and training XOR neural network

Talking about the weights of the overall network, from the above and part 1 content we have deduced the weights for the system to act as an AND gate and as a NOR gate. We will be using those weights for the implementation of the XOR gate. For layer 1, 3 of the total 6 weights would be the same as that of the NOR gate and the remaining 3 would be the same as that of the AND gate. Therefore, the weights for the input to the NOR gate would be [1,-2,-2], and the input to the AND gate would be [-3,2,2]. Now, the weights from layer 2 to the final layer would be the same as that of the NOR gate which would be [1,-2,-2]. Before starting with part 2 of implementing logic gates using Neural networks, you would want to go through part1 first.

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. 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. Neural networks are a type of program that are based on, very loosely, a human neuron.

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- For a particular choice of the parameters w and b, the output ŷ only depends on the input vector x.
- This plot code is a bit more complex than the previous code samples but gives an extremely helpful insight into the workings of the neural network decision process for XOR.
- A L-Layers XOR Neural Network using only Python and Numpy that learns to predict the XOR logic gates.
- Backpropagation is an algorithm for update the weights and biases of a model based on their gradients with respect to the error function, starting from the output layer all the way to the first layer.

At least, that’s essentially what we want the neural net to learn over time. The value [0, 0] means 0, [0, 1] means 1 and so on and so forth. Let’s look at a simple example of using gradient descent to solve an equation with a quadratic function. XOR is an exclusive or (exclusive disjunction) logical operation that outputs true only when inputs differ.

## The architecture

If this was a real problem, we would save the weights and bias as these define the model. The problem with a step function is that they are discontinuous. This creates problems with the practicality of the mathematics (talk to any derivatives trader about the problems in hedging barrier options at the money). Thus we tend to use a smooth functions, the sigmoid, which is infinitely differentiable, allowing us to easily do calculus with our model. A neural network is essentially a series of hyperplanes (a plane in N dimensions) that group / separate regions in the target hyperplane. In the forward pass, we apply the wX + b relation multiple times, and applying a sigmoid function after each call.

## How to Choose Loss Functions When Training Deep Learning Neural Networks – Machine Learning Mastery

That is why I would like to “start” with a different example. Therefore, the network gets stuck when trying to perform linear regression on a non-linear problem. Connect and share knowledge within a single location that is structured and easy to search. Stack Exchange network consists of 183 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

## Training algorithm

A converged result should have hyperplanes that separate the True and False values. We can plot the hyperplane separation of the decision boundaries. The sigmoid is a smooth function so there is no discontinuous boundary, rather we plot the transition from True into False. It is very important in large networks to address exploding parameters as they are a sign of a bug and can easily be missed to give spurious results. It is also sensible to make sure that the parameters and gradients are cnoverging to sensible values.

There are no fixed rules on the number of hidden layers or the number of nodes in each layer of a network. The best performing models are obtained through trial and error. You’ll notice that the training 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.

## Perceptron: Whats, whys, and hows

A total of 6 weights from the input layer to the 2nd layer and a total of 3 weights from the 2nd layer to the output layer. It is the setting of the weight variables that gives the network’s author control over the process of converting input values to an output value. It is the weights that determine where the classification line, the line that separates data points into classification groups, is drawn. If all data points on one side of a classification line are assigned the class of 0, all others are classified as 1.

The next post in this series will feature a implementation of the MLP architecture described here, including all of the components necessary to train the network to act as an XOR logic gate. In order for the neural network to be able to make the right adjustments to the weights we need to be able to tell how good our model xor neural network is performing. Or to be more specific, with neural nets we always want to calculate a number that tells us how bad our model performs and then try to get that number lower. All the inner arrays in target_data contain just a single item though. Each inner array of training_data relates to its counterpart in target_data.

In the image above we see the evolution of the elements of \(W\). Notice also how the first layer kernel values changes, but at the end they go back to approximately one. I believe they do so because the gradient descent is going around a hill (a n-dimensional hill, actually), over the loss function. It doesn’t matter how many linear layers we stack, they’ll always be matrix in the end. In this representation, the first subscript of the weight means “what hidden layer neuron output I’m related to? The second subscript of the weight means “what input will multiply this weight?

The outputs generated by the XOR logic are not linearly separable in the hyperplane. So In this article let us see what is the XOR logic and how to integrate the XOR logic using neural networks. For the system to generalize over input space and to make it capable of predicting accurately for new use cases, we require to train the model with available inputs. During training, we predict the output of model for different inputs and compare the predicted output with actual output in our training set.