🤖 AI Summary
This paper establishes necessary and sufficient conditions for almost-sure convergence (i.e., consistency) of stochastic iterative global optimization algorithms on Lipschitz-continuous functions. The core insight is that consistency holds if and only if the algorithm asymptotically samples the entire search space with positive probability—termed “full-space sampling.” To formalize this, the authors introduce a generic framework for stochastic iterative optimization, modeling iterations via Markov kernels and constructing a rigorous sequence of probability measures using the Ionescu-Tulcea theorem. Crucially, they provide the first machine-checked proof of this equivalence in the theorem prover Lean, leveraging the Mathlib library. The result rigorously characterizes the equivalence between consistency and full-space sampling, thereby furnishing a unified theoretical foundation and an extensible formal modeling paradigm for the design, analysis, and verification of stochastic optimization algorithms.
📝 Abstract
We formalize a proof that any stochastic and iterative global optimization algorithm is consistent over Lipschitz continuous functions if and only if it samples the whole search space. To achieve this, we use the L$exists$$forall$N theorem prover and the Mathlib library. The major challenge of this formalization, apart from the technical aspects of the proof itself, is to converge to a definition of a stochastic and iterative global optimization algorithm that is both general enough to encompass all algorithms of this type and specific enough to be used in a formal proof. We define such an algorithm as a pair of an initial probability measure and a sequence of Markov kernels that describe the distribution of the next point sampled by the algorithm given the previous points and their evaluations. We then construct a probability measure on finite and infinite sequences of iterations of the algorithm using the Ionescu-Tulcea theorem.