6. Verification and Certificates🔗

This gives a precise verification question:

\forall x\in B,\qquad P(\operatorname{denote}(g,\theta,x)).

This part develops several ways to answer it: interval and affine bounds, compiler-correctness proofs, autograd theorems, numerical error bounds, optimizer laws, and replayed certificates. We will also feed bad artifacts to the checkers and see exactly where they fail.

1. 6.1. Neural Network Verification
2. 6.2. Proof Systems Beyond Bounds
3. 6.3. Autograd Proofs
4. 6.4. Runtime Approximation
5. 6.5. Learning Theory
6. 6.6. Optimization Theory
7. 6.7. Self-Supervised Objectives
8. 6.8. Approximation Theory
9. 6.9. Structural Model Proofs
10. 6.10. Probability and Local Gradient Proofs
11. 6.11. Scientific ML Verification
12. 6.12. Matrix Factorizations: Cholesky and QR
13. 6.13. Verification Certificates
14. 6.14. Two-Stage Verification Workflows