Artificial Neural Network Fundamentals (Part 3): Exploring Neural Networks

Artificial Neural Network Fundamentals (Part 3): Exploring Neural Networks

Artificial Neural Networks (ANNs), inspired by the human brain, are a powerful tool for solving complex classification problems. Here's how they work, focusing on the backpropagation process.

The Backpropagation Process

The backpropagation process in an ANN optimizes the objective function (typically a loss function measuring output error) by iteratively adjusting the network's weights and biases to reduce prediction error. This process consists of four main steps:

1. Forward pass: Input data passes through the network layers, where each neuron computes a weighted sum of its inputs, adds a bias, and applies an activation function to produce an output. The final layer produces the network's prediction.
2. Error calculation: The network's output is compared to the true target values using a loss function such as Mean Squared Error (MSE) or cross-entropy. This quantifies the prediction error.
3. Backward pass: The error is propagated backward from the output layer to the input layer. Using the chain rule of calculus, the algorithm computes the gradient (partial derivative) of the loss function with respect to each weight and bias. This gradient indicates how each parameter influences the error.
4. Weights update: Using an optimization algorithm like gradient descent or stochastic gradient descent (SGD), weights and biases are updated by moving them in the direction that reduces error—typically subtracting a fraction of the gradient scaled by a learning rate. Advanced optimizers like Adam dynamically adjust learning rates for faster convergence.

This iterative process repeats over multiple epochs until the network's predictions closely match the targets, minimizing the objective function.

Detailed Explanation

Forward pass

During the forward pass, each neuron calculates and then applies an activation function (e.g., ReLU, softmax) to produce an output forwarded to the next layer.

Error/Loss calculation

The error/loss is computed as for regression or cross-entropy for classification.

Backward pass

In the backward pass, the backpropagation algorithm applies the chain rule to find gradients of the loss function w.r.t. weights by propagating error signals from output towards input layers, layer by layer.

Weight updates

These gradients guide the weight updates: , where is the learning rate.

Backpropagation thus enables the network to learn by systematically and efficiently adjusting parameters to optimize the objective function, reducing prediction error over time [1][3][5].

ANN Structure and Learning

An ANN is a collection of neurons connected in layers, producing an output when the neurons in the input layer are excited. The feed-forward process in an ANN computes the outputs for the individual neurons in the network based on the respective inputs and the weight of the neurons.

For a simple ANN, it consists of three layers: an input layer, a hidden layer, and an output layer. Each neuron in the hidden layer has a number of inputs , weights , and a bias term . Similarly, the output layer has one Neuron outputting the probability of the input point for being from the class.

A larger network with two hidden layers, each containing 5 Neurons, is constructed and applied to the training set of non-linearly separable data. The Neural Network constructed in the lesson uses only numpy in python and consists of three layers: the input layer, one hidden layer, and an output layer.

The learning in an ANN takes place by adjusting the weights such that the output resembles the true class label when these weights are plugged into the ANN. The weights are arranged in the form of matrices, one for each layer transition.

The number of neurons in the output layer depends on the number of classes in the dataset. For data that is not linearly separable, a network with multiple hidden layers is needed.

The weights are updated recursively, with the computed changes in the weights being multiplied by the activations at each previous layer, starting from the last layer.

Practical Application

The data is split into a training and a test split, which can be seen in figure 7. The ANN model is trained on the training-set by running an optimization algorithm (i.e., Gradient Descent) that finds the best values for the weights of the ANN. The result of such network is shown in figure 11.

The objective function includes a Regularization term to ensure that the model avoids overfitting. The lower values of the Regularization parameter may cause the model to fail to accommodate new variant examples in the test-set, while a large value might cause the model to not be fit enough for the data.

The ANN can be used to classify a set of data points belonging to two classes (0/1). The input takes two-dimensional data for simplicity.

References: [1] Goodfellow, I., Bengio, Y., & Courville, A. (2016). Deep Learning. MIT Press. [3] LeCun, Y., Bottou, L., Bengio, Y., & Haffner, P. (1998). Gradient-based learning applied to document recognition. Proceedings of the IEEE, 86(11), 2278-2324. [5] Rumelhart, D. E., Hinton, G. E., & Williams, R. J. (1986). Learning representations by back-propagating errors. Nature, 323(6088), 533-536.

Visualizing the connection between technology and artificial intelligence, the backpropagation process in an Artificial Neural Network (ANN) employs advanced optimization algorithms like gradient descent and Adam to iteratively adjust weights and biases according to calculated gradients of the loss function, thus minimizing prediction errors and optimizing the objective function over time [3][5]. Furthermore, the deep learning structure of an ANN, featuring multiple hidden layers, allows for complex classification tasks, such as classifying a set of data points belonging to two classes (0/1), by systematically learning from the training data [1].

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