The Complete Mathematics of Neural Networks and Deep Learning

The video introduces the topic of explaining the mathematics behind neural networks, specifically artificial neural networks and back propagation.

The lecture is not an introduction to neural networks, but rather focuses on the mathematics behind them. The prerequisites include basic linear algebra, multivariable calculus (specifically differential calculus), knowledge of Jacobians and gradients in multivariable functions, and a base understanding of machine learning concepts like cost function and gradient descent.

The lecture will cover the big picture of neural networks and back propagation, including a quick review of multi-variable calculus. It aims to explore the idea of viewing a single neuron as a function and how gradients are calculated in neural networks for optimization using gradient descent.

In machine learning, we use standard notation to represent key elements such as the size of the training set (m), the number of input variables (n), and the layers in a neural network. Each layer is denoted by lowercase 'l' and has specific weights and biases associated with it. The notation helps us define and understand these fundamental components.

Understanding Neural Networks as Functions Neural networks are complex functions made up of smaller functions, with weights and biases as parameters. The inputs to the function are a vector of variables for one training example, while the output can be probabilities or decisions. It's about transforming raw numbers into understandable outputs.

Viewing it as a Calculus Problem The focus is on minimizing the cost function through calculus operations and finding derivatives of the cost function with respect to every weight and bias. Back propagation aims to understand how each weight or bias impacts the final cost in order to optimize algorithms.

Exciting Developments in Deep Learning Deep learning has grown significantly in recent years despite being based on older math concepts like back propagation from 1986. This old math is now applied innovatively, leading to cutting-edge advancements that make learning about this field quite interesting.

Gradients are used to display the partial derivatives of a function that transforms a vector into a scalar. This is illustrated by taking an example function f(x, y) = x^2 + cos(y), and finding its partial derivatives with respect to x and y as 2x and -sin(y) respectively. The concept of transforming a vector into a scalar using such functions is explained through the visualization of input variables (x, y) being passed through f(x, y). The gradient of this function represents the vector containing all its partial derivatives.

Understanding Jacobians Jacobians take a vector to another vector, which can be of the same or different shape. This is demonstrated by considering a function that returns an r2 input and an r2 output. The process involves breaking it into two scalar functions (f1 and f2) and then calculating partial derivatives for each function with respect to the variables x and y.

Calculating the Jacobian The jacobian is calculated by assembling four partial derivatives in a 2x2 matrix format, where each line represents the gradient of each function with respect to x and y. The resulting jacobian for this specific example looks like: [ 2, 3y^2; -13, e^y ].

Jacobians and Partial Derivatives The Jacobian represents the partial derivatives of a vector function, displayed as a matrix. It's similar to the gradient but in matrix form, with rows representing functions and columns representing variables.

Jacobian Chain Rule The Jacobian chain rule provides an algorithmic approach to differentiating nested functions using intermediate function variables. By finding derivatives for each intermediate function and then substituting at the end, we can handle complex nested functions effectively.

Jacobian Chain Rule The Jacobian chain rule allows us to deal with vector-to-vector functions by computing the derivatives of intermediate and outer functions. By setting intermediate variables for the inside functions, we can turn them into a vector of intermediate functions. Then, we compute the Jacobians of both these vectors and use matrix multiplication to find the change in f when we change x.

Matrix Multiplication for Change Calculation To calculate the change in f when changing x using matrix multiplication, start with the outer function first followed by placing intermediate function at end. After simple matrix multiplication, substitute g1 and g2 values from previous steps to get final answer: 2x*cos(x^2+y), cos(x^2+y), 0, 3y^2/y.

When using the chain rule with Jacobians, it's important to consider what happens when not all functions are chain-rule-able. In such cases, an intermediate function may not exist for certain functions, and in those instances, a placeholder value of 1 can be used to ensure safe multiplication of Jacobians. This is crucial to keep in mind during computations involving functions that do not have intermediate functions.

Understanding Single Neurons in Neural Networks In a neural network, a single neuron takes inputs and computes the weighted sum of these inputs. This is represented mathematically as the summation of each input multiplied by its corresponding weight plus a bias term. The result is then passed through an activation function such as sigmoid or relu to produce a scalar output. This process can also be described using dot products, where the inner product of input vector x and weight vector w yields the same result.

Representation of Neurons in Two Steps A single neuron's computation can be divided into two steps: first, it calculates the weighted sum (x transpose w) plus bias (b), denoted as z; secondly, it applies an activation function to z to obtain the final output value 'a'. Essentially, this means that a single neuron acts as a function that takes in a vector and outputs a scalar value. In subsequent chapters we will explore layers of neurons and their representation using matrices.

Notation for Neural Network Weights The notation for the weights of a neural network can be tricky, especially when dealing with multiple nodes. The standard notation is wljk, where l represents the layer weight is going into, j is the number of the neuron in that layer, and k denotes the node in the previous layer. This notation helps identify specific weights within a network.

Biases in Neural Networks Biases are generally scalar values assigned to each node in a layer. They are denoted by bl and may vary across different nodes within a layer (denoted as bj). Understanding bias notation is important for working with neural networks effectively.

Representation of Neurons in a Neural Network In this chapter, we explore the representation of computations for an entire layer of a neural network. We start by understanding how multiple neurons are connected to inputs through weighted sums and produce scalar outputs. These scalars form the new vector of inputs for the next layer, which is represented as 'a' (activations). We then introduce the concept of representing weights for an entire layer using a weight matrix ('W'). Each entry in this weight matrix represents connections between neurons from one layer to another.

Understanding Weight Matrix Notation This chapter delves into understanding how each entry in the weight matrix corresponds to connections between neurons from one layer to another. The rows represent destination neurons while columns represent source neurons within a specific layer. By examining individual entries like w_jk, where j is the destination neuron and k is the source neuron, we can understand their positions within layers and visualize their connectivity within neural networks.

Neural Network Weights Computation In this chapter, we compute the weights matrix for a simple neural network by hand. The weights are assigned real values to better understand their impact on the network's output.

Weight Matrix Construction The weight matrix is constructed based on the inputs and nodes of the neural network. It involves locating elements in specific positions using row and column indices.

Function Notation Explanation We explain how function notation is used to represent input-to-layer calculations in a single neuron, including activation functions and weighted sums.

Feed Forward Operation Overview 'Feed forward' refers to passing an input through a neural network to get an output. This process involves multiplying weights by inputs, summing them up with biases, applying activation functions, and obtaining scalar outputs that become vectors.

Back Propagation Introduction Back propagation focuses on lowering the cost of final output or minimizing error by adjusting weights and biases throughout the algorithm.

Cost Function Mean Squared Error 'Mean squared error' is introduced as one of the most popular cost functions used in evaluating model performance during training.

Understanding Neural Network Derivatives The video discusses the process of rewriting a dot product as a new function and adding more layers to the chain rule. It explains how to find derivatives with respect to weights and biases, using narrative storytelling.

Derivative Calculation for Weights The derivative calculation for weights involves finding intermediate functions, simplifying expressions, applying piecewise functions (such as max), and understanding how errors affect weight changes in neural networks.

Derivative Calculation for Biases Similar to the derivative calculation for weights, this section covers finding derivatives with respect to biases by segmenting solutions into intermediate functions. It also explores graphical intuition behind gradient descent.

Understanding 2D Plane Explaining the concept of a 2D plane created by w1 and w2 to graph components and vectors.

Vector Components on Graph Illustrating how vector components are sourced from a given vector in the plane, showing amounts in different directions.

Impact of Input Value on Derivative Discussing how input value affects derivative or weight changes, emphasizing larger inputs causing significant cost change.

'Dead' Nodes in Neural Network Training 'Dead' nodes with small inputs don't learn much due to minimal impact on overall cost function during training process.

Error Magnification by Inputs Explaining error magnification when x1 is large compared to x2, leading to more significant impact on weights and costs.

Directional Pointing Towards Steepest Cost Descent Describing arrow direction pointing towards steepest descent for lowest cost using gradient descent formula.

Understanding Back Propagation Explaining the process of back propagation and its role in calculating errors for each layer of a neural network. The video subtitles are condensed to explain how the error is calculated using equations 1 and 2, leading to an understanding of how back propagation works.

Derivatives with Respect to Weights and Biases Discussing the derivatives of weights and biases with respect to cost, focusing on equation 3 which calculates bias derivatives based on error terms. Equation 4 is explored as it finds weight derivatives by considering activation values from previous layers.