Exact Real Search: Formalised Optimisation and Regression in Constructive Univalent Mathematics

TL;DR

This paper develops a formal framework within constructive univalent mathematics for exact real arithmetic, enabling verified search, optimisation, and regression algorithms on real number types using Agda.

Contribution

It introduces a novel constructive, type-theoretic approach to formalising and verifying optimisation and regression algorithms on exact real number types.

Findings

Formalised convergence properties of optimisation and regression algorithms.

Implemented verified algorithms on ternary signed-digit encodings of reals.

Extended the theory of searchable types within a constructive setting.

Abstract

The real numbers are important in both mathematics and computation theory. Computationally, real numbers can be represented in several ways; most commonly using inexact floating-point data-types, but also using exact arbitrary-precision data-types which satisfy the expected mathematical properties of the reals. This thesis is concerned with formalising properties of certain types for exact real arithmetic, as well as utilising them computationally for the purposes of search, optimisation and regression. We develop, in a constructive and univalent type-theoretic foundation of mathematics, a formalised framework for performing search, optimisation and regression on a wide class of types. This framework utilises Mart\'in Escard\'o's prior work on searchable types, along with a convenient version of ultrametric spaces -- which we call closeness spaces -- in order to consistently search certain infinite types using the functional programming language and proof assistant Agda. We formally define and prove the convergence properties of type-theoretic variants of global optimisation and parametric regression, problems related to search from the literature of analysis. As we work in a constructive setting, these convergence theorems yield computational algorithms for correct optimisation and regression on the types of our framework. Importantly, we can instantiate our framework on data-types from the literature of exact real arithmetic, allowing us to perform our variants of search, optimisation and regression on ternary signed-digit encodings of the real numbers, as well as a simplified version of Hans-J. Boehm's functional encodings of real numbers. Furthermore, we contribute to the extensive work on ternary signed-digits by formally verifying the definition of certain exact real arithmetic operations using the Escard\'o-Simpson interval object specification of compact intervals.