Conformal Predictive Programming for Chance Constrained Optimization

2024-02-12

arXiv.org

AI Summary

This paper addresses chance-constrained optimization (CCO) problems with stochastic constraints by proposing a novel framework termed Conformal Prediction Programming (CPP). CPP is the first to integrate conformal prediction into CCO, leveraging the quantile lemma to reformulate chance constraints as deterministic counterparts, thereby providing both prior and posterior guarantees on constraint satisfaction. The framework accommodates distributional shifts, class-conditional constraints, and joint chance constraints, and yields variants including Robust CPP and Mondrian CPP. Algorithmically, it encompasses three implementations: CPP-MIP (mixed-integer programming), CPP-Bilevel (bilevel optimization), and CPP-Discarding (sample-discard-based optimization). Extensive experiments across multiple case studies demonstrate that CPP significantly outperforms conventional scenario-based methods—achieving superior constraint satisfaction rates while maintaining high solution quality—without compromising theoretical rigor or computational scalability.

Technology Category

• Constraint Satisfaction and Optimization: Constraint Programming
• Reasoning under Uncertainty: Stochastic Optimization
• Machine Learning: Calibration & Uncertainty Quantification

Application Category

• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Economics, Online Markets and Human Computation: Cost models of using LLMs in production systems
• Systems and Infrastructure for Web, Mobile and WoT: Applied ML and AI for Web-based mobile applications

Abstract

We propose conformal predictive programming (CPP), a framework to solve chance constrained optimization problems, i.e., optimization problems with constraints that are functions of random variables. CPP utilizes samples from these random variables along with the quantile lemma - central to conformal prediction - to transform the chance constrained optimization problem into a deterministic problem with a quantile reformulation. CPP inherits a priori guarantees on constraint satisfaction from existing sample average approximation approaches for a class of chance constrained optimization problems, and it provides a posteriori guarantees that are of conditional and marginal nature otherwise. The strength of CPP is that it can easily support different variants of conformal prediction which have been (or will be) proposed within the conformal prediction community. To illustrate this, we present robust CPP to deal with distribution shifts in the random variables and Mondrian CPP to deal with class conditional chance constraints. To enable tractable solutions to the quantile reformulation, we present a mixed integer programming method (CPP-MIP) encoding, a bilevel optimization strategy (CPP-Bilevel), and a sampling-and-discarding optimization strategy (CPP-Discarding). We also extend CPP to deal with joint chance constrained optimization (JCCO). In a series of case studies, we show the validity of the aforementioned approaches, empirically compare CPP-MIP, CPP-Bilevel, as well as CPP-Discarding, and illustrate the advantage of CPP as compared to scenario approach.

Problem

• Solves chance constrained optimization using conformal prediction
• Transforms stochastic constraints into deterministic quantile reformulations
• Provides a priori and a posteriori constraint satisfaction guarantees

Innovation

• Uses conformal prediction for chance constraints
• Transforms problem into deterministic quantile reformulation
• Offers multiple solving strategies like MIP