Automated Synthesis of Problem-Specific Concentration Inequalities

The paper by Affeldt et al. (2025) on formalizing concentration inequalities in the Rocq proof assistant is foundational. They're building the rigorous infrastructure. But what if we could put a powerful search engine on top of that infrastructure? Think about the Efron-Stein PAC-Bayesian inequalities from Kuzborskij & Szepesvari (2019). Their beauty is their adaptability; the bound depends on the data-dependent variance estimate. But there are dozens of such inequalities (martingale-based, matrix-based, etc.), and choosing the best one for a specific problem is an art form. This research aims to automate that art. The goal is to develop a system where a user inputs the structure of their random variable (e.g., "a function of dependent variables with bounded variance") and the system, using the formal library from Affeldt et al., explores different proof strategies. It could try martingale methods, matrix concentration, or Efron-Stein approaches, automatically generating the tightest possible bound and then formally verifying its correctness in Rocq. This goes beyond simply formalizing existing inequalities; it's about synthesizing novel, problem-specific ones. It challenges the very workflow of using concentration inequalities, transforming it from a manual selection process to an automated, verifiable synthesis. The impact would be enormous, making advanced probabilistic guarantees more accessible, reliable, and optimally applied in fields where correctness is paramount, like aerospace and medical AI.

References:

1. Formalizing Concentration Inequalities in Rocq: Infrastructure and Automation. Reynald Affeldt, Alessandro Bruni, Cyril Cohen, Pierre Roux, Takafumi Saikawa (2025). International Conference on Interactive Theorem Proving.
2. Efron-Stein PAC-Bayesian Inequalities. Ilja Kuzborskij, Csaba Szepesvari (2019). arXiv.org.