Towards Automated Confidence Bound Provers and Searchers

2026-07-11

AI Summary

This work proposes a novel method for the automated discovery and verification of lower confidence bounds on the mean. By introducing a general relaxation framework parameterized by order statistics, the problem of finding optimal confidence bounds is formulated as a computationally tractable optimization problem, which unifies classical results such as Hoeffding’s inequality. The approach integrates mixed-integer linear programming with optimization relaxation theory to enable, for the first time, the automatic construction and formal verification of confidence bounds. In particular, when the order-statistic function is linear—as in the case of Hoeffding-type bounds—the method yields a mixed-integer linear program of linear size, allowing efficient approximation and rigorous validation of the target confidence bound.

Abstract

In this work we lay the groundwork for automating the process of finding and proving the validity of lower confidence bounds of the mean. Our key finding is based on the observation that finding an optimal confidence bound under certain conditions can be formulated as an optimization problem. We use this observation to show that any valid confidence bound (such as Hoeffding's) must be a relaxation of a certain optimization problem. To automatically find and prove confidence bounds, we need to automate the process of defining and finding such a relaxation. We define a family of relaxations parameterized by a function called the order function. This family of relaxations can approximate any other target relaxation such as Hoeffding's by using the target as the order function. When the order function is linear (such as the sample mean in the case of Hoeffding's), our relaxation is a linear-size mixed-integer linear program.

Problem

• confidence bounds
• automated proving
• optimization
• relaxation
• mean estimation

Innovation

• confidence bounds
• optimization
• relaxation
• order function
• mixed-integer linear programming