Perceptron Algorithm and Binary Classification 1

Perceptron Perceptron The perceptron is an algorithm for learning a binary classifier called a threshold function: a function that maps its input x(a real-valued vector) to an output value f(x)(a single binary value): Here is Heaviside step function (where an input of>0 outputs 1; otherwise 0 is the output ), is a vector of real-valued weights, is the dot product where is the number of inputs to the perceptron, and is the bias. The appropriate weights are applied to the inputs, and the resulting weighted sum passed to a function that produces the output o. • In the context of neural networks, a perceptron is an artificial neuron using the Heaviside step function as the activation function. • The perceptron algorithm is also termed the single-layer perceptron, to distinguish it from a multilayer perceptron, which is a misnomer for a more complicated neural network. • As a linear classifier, the singlelayer perceptron is the simplest In a feedforward network, feedforward neural network. information always moves in one direction; it never goes • A diagram showing a perceptron updating its linear boundary as more training examples are added Gradient Descent • People are stuck in the mountains and are trying to get down (i.e., trying to find the global minimum). There is heavy fog such that visibility is extremely low. Therefore, the path down the mountain is not visible • They can use the method of gradient descent, which involves looking at the steepness of the hill at their current position, then proceeding in the direction with the steepest descent (i.e., downhill). • Using this method, they would eventually find their way down the mountain or possibly get stuck in some hole (i.e., local minimum or saddle point), like a mountain lake. • However, assume also that the steepness of the hill is not immediately obvious with simple observation, but rather it requires a sophisticated instrument to measure, which the persons happen to have at the moment. • It takes quite some time to measure the steepness of the hill with the instrument, thus they should minimize their use of the instrument if they wanted to get down the mountain before sunset. • The difficulty then is choosing the frequency at which they should measure the steepness of the hill so not to go off track. • In this analogy, the persons represent the algorithm, and the path taken down the mountain represents the sequence of parameter settings that the algorithm will explore. • The steepness of the hill represents the slope of the function at that point. The instrument used to measure steepness is differentiation. The direction they choose to travel in aligns with the gradient of the function at that point. The amount of time they travel before taking another measurement is the step size. Gradient Descent • Step 1. Training Machine Learning Models Neural networks are trained using Gradient Descent (or its variants) in combination with backpropagation. Backpropagation computes the gradients of the loss function with respect to each parameter (weights and biases) in the network by applying the chain rule. The process involves: Forward Propagation: Computes the output for a given input by passing data through the layers. Backward Propagation: Uses the chain rule to calculate gradients of the loss with respect to each parameter (weights and biases) across all layers. Gradients are then used by Gradient Descent to update the parameters layer-by-layer, moving toward minimizing the loss function. • Step 2. Minimizing the Cost Function The algorithm minimizes a cost function, which quantifies the error or loss of the model’s predictions compared to the true labels