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
Challenges: Local Minima and Saddle Points
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- Deep Learning, Ian Goodfellow, Yoshua Bengio, and Aaron Courville, 2016 (MIT Press) - A foundational textbook that provides comprehensive coverage of optimization challenges, including local minima and saddle points, specifically within the context of deep learning.
- The Loss Surfaces of Multilayer Networks, Anna Choromanska, Mikael Henaff, Michael Mathieu, Gerard Ben Arous, Yann LeCun, 2015 Proceedings of Machine Learning Research, Vol. 38 (Proceedings of Machine Learning Research) - This paper discusses the properties of loss surfaces in deep neural networks, suggesting that many local minima may be equivalent in terms of performance, influencing the perception of local minima as less problematic for very deep models.