ArtificialTheorems: Autoformalization of Theoretical Foundations of AI/ML

This repo is a library of Lean formalizations of theoretical foundations of AI and ML. We explicitly allow and encourage AI-generated / AI-assisted proofs, with the following quality assurance:

• The formal theorem statements (in directory ArtificialTheoremsSpec/) are vetted by human experts.
• The proofs (in ArtificialTheorems/) are checked using secure verifiers (Comparator, SafeVerify) to ensure that they prove exactly the statements in ArtificalTheoremsSpec/

Verification

To verify that the proofs match their specifications, run:

./scripts/verify.sh

This script:

1. Builds both ArtificialTheorems and ArtificialTheoremsSpec
2. Runs lean4checker on all implementation modules to validate the olean files
3. Runs safe_verify on each spec/impl pair to ensure the implementations match their specifications exactly

All checks must pass for the proofs to be considered valid.

Wish List

Contributions are appreciated! Both formal theorem statements vetted by human experts, and autoformalizations of proofs. I am particularly interested in these areas:

• Universal representation theorems for deep neural nets
• Generalization theory
  • A recent Lean formalization of generalization error bound by Radmacher complexity: https://github.com/auto-res/lean-rademacher
• Implicit regularization (how training via SGD can achieve generalization)
• RL theory
  • A recent Lean formalization of convergence of Q-Learning: https://github.com/ShangtongZhang/rl-theory-in-lean
• Bayesian learning; and perhaps building on top of that, Singular Learning Theory.