Project 24: Numerical Stability and Conditioning Stress Lab

Build a failure harness that reveals where finite precision, scaling, and conditioning break ML pipelines.

Quick Reference

Attribute Value
Difficulty Level 4: Expert (The Systems Architect)
Time Estimate 2 weeks
Main Programming Language Python
Alternative Programming Languages C++, Julia, Rust
Coolness Level Level 4: Hardcore Tech Flex
Business Potential 3. The “Service & Support” Model (B2B Utility)
Knowledge Area Numerical Methods / Stability Engineering
Main Book “Numerical Linear Algebra” by Trefethen and Bau

1. Learning Objectives

  1. Measure conditioning and sensitivity in model components.
  2. Reproduce instability signatures reliably (overflow, NaN, divergence).
  3. Apply stabilization patterns (scaling, log-sum-exp, clipping, regularization).
  4. Build convergence criteria resilient to noise.

2. All Theory Needed (Per-Concept Breakdown)

Concept A: Floating-Point Limits

Fundamentals Finite precision creates rounding, underflow, overflow, and cancellation.

Deep Dive into the concept ML pipelines often fail silently when values leave representable ranges or when tiny updates vanish. Stable implementations manage value scale explicitly.

Concept B: Conditioning

Fundamentals Condition numbers quantify output sensitivity to input perturbations.

Deep Dive into the concept Ill-conditioned objectives magnify data and arithmetic noise. This can make identical algorithms appear “randomly unreliable” across datasets.

Concept C: Stable Optimization Primitives

Fundamentals Stable primitives reduce catastrophic arithmetic.

Deep Dive into the concept Log-sum-exp, feature standardization, regularization, and gradient clipping are not hacks; they are numerical prerequisites for trustworthy optimization.

3. Build Blueprint

  1. Build baseline stable training loop.
  2. Add perturbation sweeps (precision, learning rate, scaling, regularization).
  3. Log condition numbers and gradient/update norms.
  4. Generate mitigation recommendations automatically.

4. Real-World Outcome (Target)

$ python stability_lab.py --task logistic_regression --sweep precision,lr,lambda

Run float64 lr=0.01 lambda=1e-3: converged
Run float32 lr=0.2  lambda=0: diverged (loss=inf at epoch 14)
Condition number(X^T X): 2.4e7
Recommendation: scale features + lower lr + add L2

5. Core Design Notes from Main Guide

Core Question

“Is the model wrong, or is the computation unstable?”

Common Pitfalls

  • Ignoring condition numbers
  • Stopping criteria based on noisy single-step deltas
  • Comparing unstable and stable runs without matched seeds/configs

Definition of Done

  • Reproduces at least three numerical failure signatures
  • Logs stability metrics and mitigation outcomes
  • Includes stable softmax/log-loss implementation
  • Documents reproducible troubleshooting playbook

6. Extensions

  1. Add mixed-precision policy evaluation.
  2. Add auto-detection of instability phases during training.
  3. Add per-layer stability attribution in deep nets.