#006 PyTorch Solving the famous XOR problem using Linear classifiers with PyTorch - IMP Concursos

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24 de abril de 2024

#006 PyTorch Solving the famous XOR problem using Linear classifiers with PyTorch

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 https://forexhero.info/ train it using gradient descent. The problem itself was described in detail, along with the fact that the inputs for XOR are not linearly separable into their correct classification categories.

Targets and Error function

The choice appears good for solving this problem and can also reach to a solution easily. The XOR-Problem is a classification problem, where you only have four datapoints with two features. The training set and the test set are exactlythe same in this problem. So the interesting question is only if the model isable to find a decision boundary which classifies all four points correctly. Artificial neural networks (ANNs), or connectivist systems are computing systems inspired by biological neural networks that make up the brains of animals.

MGMT 4190/6560 Introduction to Machine Learning Applications @Rensselaer

On the surface, XOR appears to be a very simple problem, however, Minksy and Papert (1969) showed that this was a big problem for neural network architectures of the 1960s, known as perceptrons. In the case of XOR problem, unsupervised learning techniques like clustering or dimensionality reduction can be used to find patterns in data without any labeled examples. For example, we can use k-means clustering algorithm to group input data into two clusters based on their features. RNNs suffer from the vanishing gradient problem which occurs when the gradient becomes too small to update weights and biases during backpropagation. This makes it difficult for them to learn long-term dependencies in data.

  1. These hidden layers allow the network to learn non-linear relationships between the inputs and outputs.
  2. If the input is the same(0,0 or 1,1), then the output will be 0.
  3. 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.
  4. As we move downwards the line, the classification (a real number) increases.

Need for linear separability in neural networks

The neurons in the layer above provide inputs to every neuron in these layers. Weights are a set of parameters learned during training, multiplied by the inputs before being added together. 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. As a result, we will have the necessary values of weights and biases in the neural network and output values on the neurons will be the same as the training vector. To solve the XOR problem using neural networks, one can use either Multi-Layer Perceptrons or a neural network that consists of an input layer, a hidden layer, and an output layer. As the neural network processes data through forward propagation, the weights of each layer are adjusted accordingly and the XOR logic is executed.

We can therefore expect the trained network to be 100% accurate in its predictions and there is no need to be concerned with issues such as bias and variance in the resulting model. 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. For the XOR problem, we can use a network with two input neurons, two hidden neurons, and one output neuron. The AND logical function is a 2-variables function, AND(x1, x2), with binary inputs and output. Now that we have defined everything we need, we’ll create a training function.

To solve the XOR problem with LSTMs, we need to use a network with one input neuron, two hidden layers each with four LSTM neurons, and one output neuron. During training, we adjust weights and biases based on the error between predicted output and actual output until we achieve a satisfactory level of accuracy. In the case of multi-layer feedforward networks, backpropagation helps in solving the XOR problem by adjusting weights and biases at each layer based on the error between predicted output and actual output. This allows the network to learn complex patterns in data by using multiple layers of neurons. Neural networks have revolutionized artificial intelligence and machine learning.

We need to look for a more general model, which would allow for non-linear decision boundaries, like a curve, as is the case above. Out of all the 2 input logic gates, the XOR and XNOR gates are the only ones that are not linearly-separable. The algorithm only terminates when correct_counter hits 4 — which is the size of the training set — so this will go on indefinitely. To visualize how our model performs, we create a mesh of datapoints, or a grid, and evaluate our model at each point in that grid. Finally, we colour each point based on how our model classifies it. So the Class 0 region would be filled with the colour assigned to points belonging to that class.

In these fields, it’s critical to understand the decision-making process. 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 some practical cases e.g. when collecting product reviews online for various parameters and if the parameters are optional fields we may get some missing input values. In such case, we can use various approaches like setting the missing value to most occurring value of the parameter or set it to mean of the values. One interesting approach could be to use neural network in reverse to fill missing parameter values. For, X-OR values of initial weights and biases are as follows[set randomly by Keras implementation during my trial, your system may assign different random values].

For learning to happen, we need to train our model with sample input/output pairs, such learning is called supervised learning. Supervised learning approach has given amazing result in deep learning when applied to diverse tasks like face recognition, object identification, NLP tasks. Most of the practically applied deep learning models in tasks such as robotics, automotive etc are based on supervised learning approach only. Other approaches are unsupervised learning and reinforcement learning. Minsky and Papert used this simplification of Perceptron to prove that it is incapable of learning very simple functions. Learning by perceptron in a 2-D space is shown in image 2.

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. Let’s bring everything together by creating an MLP class.

Notice the left-hand side image which is based on the Cartesian coordinates. 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. xor neural network This exercise brings to light the importance of representing a problem correctly. If we represent the problem at hand in a more suitable way, many difficult scenarios become easy to solve as we saw in the case of the XOR problem.

These networks are especially well-suited for sequential data processing, in which the context and sequence of the data points play a critical role. However, disappearing or exploding gradients pose a problem for RNNs when dealing with extended sequences, making training stable and efficient RNNs difficult without gating mechanisms. That’s why they are often called “black boxes.” These models have an intricate structure. They have possibly millions of parameters and nonlinear transformations. This structure makes it hard to understand how they work. It’s especially true in industries like healthcare and law.


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