Phase 1: Calculus & Optimization · ~120 minutes · Python

Numerical Stability

Floating point is a leaky abstraction. It will bite you during training, and you will not see it coming.

What you will learn

Implement numerically stable softmax and log-sum-exp using the max-subtraction trick
Identify overflow, underflow, and catastrophic cancellation in floating-point computations
Verify analytical gradients against numerical gradients using centered finite differences
Explain why bfloat16 is preferred over float16 for training and how loss scaling prevents gradient underflow

Introduction

Type: Build
Language: Python
Prerequisites: Earlier lessons in this course
Time: ~120 minutes

Objective

Learning objectives

Implement numerically stable softmax and log-sum-exp using the max-subtraction trick
Identify overflow, underflow, and catastrophic cancellation in floating-point computations
Verify analytical gradients against numerical gradients using centered finite differences
Explain why bfloat16 is preferred over float16 for training and how loss scaling prevents gradient underflow

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You've read the first 2 sections. The rest of this lesson covers The Problem, The Concept, Build, Check Yourself, Key Terms & Next — plus a hands-on lab, quiz, and project artifact.

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Key terms in this lesson

Softmax