# Parametric Learning and Monte Carlo Optimization

## TL;DR

This paper reveals the deep connection between Monte Carlo Optimization, parametric machine learning, and blackbox optimization, introducing new methods and demonstrating how PL techniques can enhance optimization performance.

## Contribution

It proves MCO is mathematically equivalent to a broad class of PL problems, enabling PL techniques to improve blackbox optimization methods.

## Key findings

- Cross-validation and bagging improve immediate sampling.
- Exploiting sample location information enhances MCO.
- Transforming BO into MCO broadens application of PL techniques.

## Abstract

This paper uncovers and explores the close relationship between Monte Carlo Optimization of a parametrized integral (MCO), Parametric machine-Learning (PL), and `blackbox' or `oracle'-based optimization (BO). We make four contributions. First, we prove that MCO is mathematically identical to a broad class of PL problems. This identity potentially provides a new application domain for all broadly applicable PL techniques: MCO. Second, we introduce immediate sampling, a new version of the Probability Collectives (PC) algorithm for blackbox optimization. Immediate sampling transforms the original BO problem into an MCO problem. Accordingly, by combining these first two contributions, we can apply all PL techniques to BO. In our third contribution we validate this way of improving BO by demonstrating that cross-validation and bagging improve immediate sampling. Finally, conventional MC and MCO procedures ignore the relationship between the sample point locations and the associated values of the integrand; only the values of the integrand at those locations are considered. We demonstrate that one can exploit the sample location information using PL techniques, for example by forming a fit of the sample locations to the associated values of the integrand. This provides an additional way to apply PL techniques to improve MCO.