Calculus Essentials for Machine Learning
Chapter 1: Introduction: Why Calculus Matters in Machine Learning
Functions and Models in ML
The Concept of Optimization in ML
Measuring Change: The Role of Derivatives
Calculus as a Tool for Understanding Algorithms
Quiz for Chapter 1
Chapter 2: Single-Variable Calculus: Derivatives and Optimization
Understanding Limits: The Foundation
Defining the Derivative
Common Differentiation Rules
Higher-Order Derivatives
Finding Minima and Maxima using Derivatives
Application: Simple Cost Function Optimization
Hands-on Practical: Calculating Derivatives with Python
Quiz for Chapter 2
Chapter 3: Multivariable Calculus: Gradients and Direction
Functions of Multiple Variables
Partial Derivatives
The Gradient Vector
Directional Derivatives
The Hessian Matrix
Multivariable Optimization Concepts
Hands-on Practical: Computing Gradients with NumPy
Quiz for Chapter 3
Chapter 4: Gradient Descent Algorithms
The Intuition Behind Gradient Descent
The Gradient Descent Algorithm Steps
The Learning Rate Parameter
Batch Gradient Descent
Stochastic Gradient Descent (SGD)
Mini-batch Gradient Descent
Challenges: Local Minima and Saddle Points
Hands-on Practical: Implementing Simple Gradient Descent
Quiz for Chapter 4
Chapter 5: The Chain Rule and Backpropagation
Revisiting the Chain Rule for Single Variables
The Chain Rule for Multivariable Functions
Introduction to Neural Networks as Composite Functions
Backpropagation: Applying the Chain Rule
Calculating Gradients for Weights and Biases
Computational Graphs
Hands-on Practical: Manual Backpropagation Example
Quiz for Chapter 5
The Hessian Matrix
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- Calculus: Early Transcendentals, James Stewart, 2015 (Cengage Learning) - Provides a detailed introduction to multivariable calculus, covering the definition and properties of the Hessian matrix, Clairaut's Theorem, and the second derivative test for extrema.
- Convex Optimization, Stephen Boyd and Lieven Vandenberghe, 2004 (Cambridge University Press) DOI: 10.1017/CBO9780511804441 - A standard text that clarifies the relationship between the Hessian matrix and function convexity, fundamental for understanding optimization landscapes.
- Numerical Optimization, Jorge Nocedal and Stephen J. Wright, 2006 (Springer) DOI: 10.1007/978-0-387-40065-5 - Examines the application of the Hessian matrix in second-order optimization methods, such as Newton's method, and discusses the associated computational considerations.