algorithm Can an ANN of 2 neurons solve XOR?

weight and bias

The weights are multiplied by the perceptron’s signal to arrive at the weights used to calculate the bias; the value $y is added to the result. It is critical to find a minimum loss according to a set of parameters . Results show the training progress of both models (i.e., πt-neuron model and proposed model) for the 10-bit parity problem. The proposed model has achieved convergence while the πt-neuron model has not.

The proposed model has shown much smaller loss values than that of with πt-neuron model. Also, the proposed model has easily obtained the optimized value of the scaling factor in each case. Tessellation surfaces formed by the πt-neuron model and the proposed model have been compared in Figure 8 to compare the effectiveness of the models (considering two-dimensional input). Here, the larger scaling factor ‘bN’ accurately compensates for infinitesimally small gradient problems. Therefore, the larger scaling factor enforces a sharper transition to the sigmoid function and supports easier learning in case of higher dimensional parity problems.

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. As our XOR problem is a binary classification problem, we are using binary_crossentropy loss. Weight initialization is an important aspect of a neural network architecture. It abruptely falls towards a small value and over epochs it slowly decreases.

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It also showcases the mean and standard deviations of the predicted thresholds and bias values. 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. So let’s activate the neurons by knowing some famous activation functions.

The Xor Function: Why A Perceptron Can’t Learn It

It fails to map the output for XOR because the data points are in a non-linear arrangement, and hence we need a model which can learn these complexities. Adding a hidden layer will help the Perceptron to learn that non-linearity. This is why the concept of multi-layer Perceptron came in.

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. It generates a small range of input numbers from inputs of . It is possible to make a small change in output in these input spaces even if the change is large. To resolve this issue, there are several workarounds that are frequently based on algorithm or architecture . With such a low number of weights , sometimes random initialisation can create a combination that gets stuck easily.

The sample code from this post can be found here.

So you may need to try, check results and then re-start. I suggest you use a seeded xor neural network number generator for initialisation, and adjust the seed value if error values get stuck and do not improve. But it will increase the number of neurons more than two. XOR problem is illustrated by considering each input as one dimension and mapping the digital digit ‘0’ as negative axis and ‘1’ as the positive axis. Therefore, XOR data distribution is the areas formed by two of the axes ‘X1’ and ‘X2’, such that the negative area corresponds to class 1, and the positive area corresponds to class 2.

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It’s interesting to see that the neuron learned both the XOR function’s and its solution’s initialization parameters as a result of its initialization. There are more splits in a polynomial degree than in a non- polynomial degree. I began experimenting with polynomial neurons on the MNIST data set, but I will leave the findings to another article.


While creating these perceptrons, we will know why we need multi-layer neural networks. 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. However, these are much simpler, in both design and in function, and nowhere near as powerful as the real kind. In our X-OR problem, output is either 0 or 1 for each input sample. So, it is a two class or binary classification problem.


Following code gist shows the initialization of parameters for neural network. However, is it fair to assign different error values for the same amount of error? For example, the absolute difference between -1 and 0 & 1 and 0 is the same, however the above formula would sway things negatively for the outcome that predicted -1. Further, this error is divided by 2, to make it easier to differentiate, as we’ll see in the following steps. In the forward pass, we apply the wX + b relation multiple times, and applying a sigmoid function after each call.

Logical XOR

Backpropagation was discovered for the first time in the 1980s by Geoffrey Hinton. Convolutional Neural Network was founded by entrepreneur Yann LeCun as a collaboration between his network LeNet-5 and CNN. CNN processed the input signal as a feature of visual data by compressing it. In 2006, researchers Hinton and Bengio discovered that neural networks with multiple layers could be trained efficiently by weight initialization.

  • The XOR neural network is trained using the backpropagation algorithm.
  • Perceptrons are used to divide n-dimensional spaces into two regions known as True or False.
  • As part of backpropagation, we backfeed our errors from the final output neuron into the weights, which are then adjusted.
  • Activation used in our present model are “relu” for hidden layer and “sigmoid” for output layer.
  • This means that each data point can only belong to one class, and no two data points can be in the same class.

The of the second neuron should be the output of the XOR gate. However, the previous πt-neuron model has no such ability. This is possible in our model by providing the compensation to each input (as given in our proposed enhanced πt-neuron model by equation ). We have considered the input distribution similar to Figure 5 (i.e., the input varies between ) for each dimension. Results show that the effective scaling factor depends upon the dimension of input as well as the magnitude of the input.

The proposed model has accurately classified the considered dataset. Table 1 presents the list of variables used in this article with their meaning. Robotics, parity problems, and nonlinear time-series prediction are some of the significant problems suggested by the previous researchers where multiplicative neurons are applied.

Solving X-OR with multi layer perceptron in Keras

They are initialized to some random value or set to 0 and updated as the training progresses. The bias is analogous to a weight independent of any input node. Basically, it makes the model more flexible, since you can “move” the activation function around. MLP can solve the XOR problem efficiently by visualizing the data points in multi-dimensional spaces and then creating an n-variable equation to fit the output values. In this blog, we discuss the XOR problem and how it is solved using multi-layer perceptrons.

They have used the Vapnik-Chervonenkis dimension and the pseudo dimension to analyze the computational complexity of the multiplicative neuron models. The VC dimension is a theoretical tool that quantifies the computational complexity of neuron models. According to their investigation for a single product unit the VC dimension of a product unit with N-input variables is equal to N. It is observed by the results of Tables 5 and 6 that the πt-neuron model has a problem in learning highly dense XOR data distribution. However, the proposed neuron model has shown accurate classification results in each of these cases. Also, the loss function discerns heavy deviation as predicted and desired values of the πt-neuron model.

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. In our model, the activation function is a simple threshold function.

With the correct choice of functions and weight parameters, a Neural Network with one hidden layer is able to solve the XOR problem. 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. It works fine with Keras or TensorFlow using loss function ‘mean_squared_error’, sigmoid activation and Adam optimizer.

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 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. A single perceptron, therefore, cannot separate our XOR gate because it can only draw one straight line.

artificial neural network

Here, we cycle through the data indefinitely, keeping track of how many consecutive datapoints we correctly classified. If we manage to classify everything in one stretch, we terminate our algorithm. In the XOR problem, we are trying to train a model to mimic a 2D XOR function. I want to practice keras by code a xor, but the result is not right, the followed is my code, thanks for everybody to help me. My teature told me to make first XOR gate to make sure that the algorithm working . You can try to run like this, and see sometimes it works sometimes not, I have added your suggestion in code.

Dense is used to define layers of neural networks with parameters like the number of neurons, input_shape, and activation function. Only the hidden layer nodes produce Xo, and Yh represents the truth table’s actual input patterns. In this case, the expected_output and predicted_output must be fed back into each other until they converge.

I am testing this for different functions like AND, OR, it works fine for these. Shen, “Data-driven time series prediction based on multiplicative neuron model artificial neuron network,” Applied Soft Computing, vol. L1 loss obtained in these three experiments for the πt-neuron model, and the proposed model is provided in Table 3. This loss function is only used to visualize the comparison in the model. As mentioned earlier, we have used the binary cross-entropy loss function to train our model. Artificial Intelligence aims to mimic human intelligence using various mathematical and logical tools.

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. 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. The problem with a step function is that they are discontinuous. tested an implementation and the training of a simple neural network using pytorch. The implemented neural network evaluates XOR for two noisy inputs, A and B. The classical network consists of one input and one hidden layers. Further, the computational complexity of the proposed model is obtained from the investigation of Schmitt in . Schmitt has investigated the computational complexity of multiplicative neuron models.

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The first neuron acts as an OR gate and the second one as a NOT AND gate. Add both the neurons and if they pass the treshold it’s positive. You can just use linear decision neurons for this with adjusting the biases for the tresholds. The inputs of the NOT AND gate should be negative for the 0/1 inputs. This picture should make it more clear, the values on the connections are the weights, the values in the neurons are the biases, the decision functions act as 0/1 decisions . Egrioglu, “Threshold single multiplicative neuron artificial neural networks for nonlinear time series forecasting,” Journal of Applied Statistics, vol.


Although, after the suggested increment as well, the reported success ratio is ‘0.6’ only . It indicates the problem of training in the πt-neuron model for higher dimensional input. There are many other nonlinear data distributions resembling XOR. Both these problems are popular in the AI research domain and require a generalized single neuron model to solve them. We have seen that these problems require a model which can distinguish between positive and negative quantities. Interestingly, addition cannot easily separate positive and negative quantities, whereas multiplication has the basic property to distinguish between positive and negative quantities.