### Table of contents

## Introduction

Backpropagation is an algorithm for training artificial neural networks, particularly feedforward networks with a multi-layered architecture (i.e., deep learning networks). It is a method of training the network by adjusting the weights and biases of the connections between the nodes (also known as "neurons") in the network in order to minimize the error between the predicted output and the true output.

## Loss function

A loss function is a measure of how well a machine learning model is able to predict the desired output given a set of input data. It is used to optimize the model's parameters (e.g., weights and biases) during the training process, by minimizing the error between the predicted output and the true output.

There are many different types of loss functions, depending on the type of problem being solved and the specific requirements of the model. One of the most common loss functions is mean squared error. Mean squared error (MSE) is a common loss function for regression problems, where the goal is to predict a continuous output value. MSE measures the average squared difference between the predicted output and the true output.

The choice of loss function depends on the specific problem being solved and the requirements of the model. The goal is to choose a loss function that accurately reflects the performance of the model and allows the model to be optimized effectively.

## How Backpropagation Works

The backpropagation algorithm consists of two phases: a forward pass and a backward pass.

In the forward pass, the input data is passed through the network, and the output is predicted based on the weights and biases of the connections between the nodes.

In the backward pass, the error between the predicted output and the true output is calculated using a loss function (e.g., mean squared error). This error is then backpropagated through the network, starting from the output layer and working backward through the hidden layers. At each layer, the gradients of the weights and biases with respect to the loss are calculated using the chain rule. These gradients are then used to update the weights and biases using an optimization algorithm (e.g., gradient descent).

This process is repeated for multiple iterations (i.e., epochs) until the error between the predicted output and the true output is minimized.

Backpropagation is a widely used algorithm for training artificial neural networks and has been successful in a variety of tasks, including image classification, natural language processing, and speech recognition. It is an important part of the training process for many deep learning models and has been a key enabler in the development of modern machine learning techniques.

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