Solving XOR Problem with Neural Networks

Table of Contents

1. Introduction
2. The Power of Neural Networks
   1. Combining Units into Larger Networks
   2. Minsky and Papert's Proof
3. Elementary Logical Functions
   1. AND
   2. OR
   3. XOR
4. The Perceptron
   1. Linear Classifier
   2. Computing AND and OR Functions
   3. Limitations with XOR
5. Two Layers of ReLU-based Units
   1. Introducing H and Y Layers
   2. Computing XOR with ReLU Units
6. Automatic Learning with Back Propagation
   1. Weights and Biases
   2. Learning Useful Representations
7. Conclusion

🧠 The Power of Neural Networks

Neural networks have revolutionized the field of artificial intelligence by combining individual computing units into larger networks. This allows for complex computations and problem-solving capabilities that surpass the limitations of single units. In 1969, Minsky and Papert's proof demonstrated the inability of single units to compute even simple functions like AND, OR, and XOR. This revelation paved the way for the development of multi-layer networks with non-linear activation functions.

🤖 Elementary Logical Functions

To understand the significance of multi-layer networks, it's essential to review the elementary logical functions: AND, OR, and XOR. These functions take two inputs and produce a binary output based on specific rules. For instance, the truth table for the AND function states that the output is 1 only when both inputs are 1. Similarly, the OR function produces 1 if either of the inputs is 1. However, XOR behaves differently, yielding 1 only when the inputs are different.

💡 The Perceptron: A Linear Classifier

The Perceptron is one type of neural unit that serves as a binary classifier. It has a binary output and lacks a nonlinear activation function. The output of a Perceptron is determined by the weights, inputs, and biases. If the dot product of the weights and inputs plus the bias is less than or equal to zero, the Perceptron outputs 0; otherwise, it outputs 1. While it's relatively simple to build a Perceptron that can compute the logical AND and OR functions, it fails to handle the XOR function due to its linearity.

🚫 Limitations with XOR

The Perceptron's failure to compute XOR arises from its linear decision boundary. When visualized in two-dimensional space, the equation of the Perceptron represents a line that separates inputs into different categories. This approach works for linearly separable functions such as AND and OR, where a line can accurately classify input points. However, XOR is not linearly separable, as there is no line that can separate its positive and negative cases. Therefore, a single-layer Perceptron cannot compute XOR.

✨ Two Layers of ReLU-based Units

To solve the XOR problem, we introduce a two-layer neural network consisting of rectified linear unit (ReLU) based units. This network includes a middle layer (H) with two units and an output layer (Y) with one unit. By assigning appropriate weights and biases to these units, we can compute XOR. Consider the input (0, 0); upon multiplying the inputs by the corresponding weights and adding the bias, we calculate the outputs of H1 and H2 using the ReLU function. The final output (Y) is obtained by computing the dot product of the weights and the Hidden layer outputs. By following this process for each input pair, we can observe the correct computation of XOR, where (0, 1), (1, 0) yield 1, and (0, 0), (1, 1) yield 0.

🔄 Automatic Learning with Back Propagation

In the previous example, we manually specified the weights and biases for computing XOR. However, in real-world applications, neural networks learn these parameters automatically using the back propagation algorithm. This algorithm adjusts the weights and biases iteratively based on the error between predicted and actual outputs, gradually improving the network's performance. Through this learning process, the hidden layers of the network form useful representations of the input, enabling the network to solve complex problems effectively.

📚 Conclusion

Neural networks offer a powerful solution to computation problems that single units struggle to solve. By combining units into multi-layer networks and using non-linear activation functions, neural networks can overcome the limitations of linearity. In the case of XOR, a single-layer Perceptron fails, but a two-layer network with ReLU-based units can successfully compute XOR. The ability to automatically learn useful representations of data through back propagation further enhances the capabilities of neural networks.

Highlights

• The power of neural networks lies in combining individual units into larger networks, enabling complex computations.
• Minsky and Papert's proof demonstrated the limitations of single units in computing even elementary logical functions like XOR.
• The Perceptron, a linear classifier, can handle simple logical functions but fails with XOR due to its linearity.
• Two-layer neural networks with ReLU-based units can successfully compute XOR by forming useful representations of the input.
• Back propagation allows neural networks to automatically learn the weights and biases for solving complex problems.

FAQ

Q: Can a single unit compute XOR? A: No, a single unit like the Perceptron, which relies on linearity, cannot compute XOR due to its non-linearity.

Q: How do neural networks automatically learn useful representations? A: Neural networks learn useful representations through the back propagation algorithm, which adjusts the weights and biases based on the error between predicted and actual outputs.

Q: What is the advantage of using multi-layer networks? A: Multi-layer networks can solve complex problems and overcome the limitations of single units by combining them into larger networks and using non-linear activation functions.

Resources:

• Introduction to Neural Networks by Udacity
• Minsky, Marvin, and Seymour Papert. “Perceptrons: An Introduction to Computational Geometry.” (1969)