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
Stochastic Gradient Descent (SGD)
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- A Stochastic Approximation Method, Herbert Robbins and Sutton Monro, 1951 The Annals of Mathematical Statistics, Vol. 22 (Institute of Mathematical Statistics) DOI: 10.1214/aoms/1177729586 - This seminal paper introduced the stochastic approximation method, which forms the theoretical basis for Stochastic Gradient Descent, laying the groundwork for iterative optimization with noisy estimates.
- Deep Learning, Ian Goodfellow, Yoshua Bengio, and Aaron Courville, 2016 (MIT Press) - Chapter 8, "Optimization for Training Deep Models," provides an extensive discussion of Stochastic Gradient Descent, covering its mechanisms, advantages, and practical considerations in machine learning.
- Large-scale machine learning with stochastic gradient descent, Léon Bottou, 2010 Proceedings of COMPSTAT'2010 (Physica-Verlag, Heidelberg) DOI: 10.1007/978-3-7908-2604-2_14 - This work provides valuable insights into the practical aspects and effectiveness of Stochastic Gradient Descent for handling large datasets, highlighting its computational efficiency and convergence properties.