Advanced Optimization

[!NOTE] This module explores the core principles of Advanced Optimization, deriving solutions from first principles and hardware constraints to build world-class, production-ready expertise.

1. Module Overview

Welcome to Advanced Optimization. In this module, we bridge the gap between pure calculus and training modern Neural Networks.

Optimization is the engine of Machine Learning. It doesn’t matter how fancy your architecture is; if you can’t minimize the loss function, you have nothing.

What you will learn

  1. The Landscape: Understand Convexity, Saddle Points, and the treacherous terrain of high-dimensional loss functions.
  2. The Engine: How Momentum and Adaptive Learning Rates (Adam) speed up training by 10x.
  3. The Constraints: How to solve problems with rules (Lagrange Multipliers).
  4. The Magic: How Automatic Differentiation (AutoDiff) works under the hood of PyTorch.
  5. The Algorithm: Derive Backpropagation from scratch and solve the Vanishing Gradient problem.

2. Chapter List

  1. The Landscape of Learning: Convexity & Loss
    • Convex Sets, Jensen’s Inequality, and the 3D Terrain Explorer.
  2. Accelerating Descent: Momentum & Adam
    • Why SGD is slow, and how Physics (Momentum) fixes it.
  3. Playing by Rules: Lagrange Multipliers
    • Constrained Optimization and the Tangency condition.
  4. Automatic Differentiation: The Magic of PyTorch
    • Computational Graphs and Forward vs Reverse Mode.
  5. Case Study: Backpropagation from Scratch
    • Deriving the Chain Rule for Neural Networks.
  6. Module Review
    • Flashcards, Cheat Sheets, and Summary.

3. Prerequisites

  • Basic Calculus (Derivatives, Partial Derivatives).
  • Basic Linear Algebra (Vectors, Dot Products).
  • Python (Numpy).