Solving Stochastic Constraints by Oracle-based Gradient Descent and Interval Arithmetic

2026-04-19

AI Summary

This work addresses constraint satisfaction problems involving random variables, aiming to find deterministic parameters that maximize the probability of satisfying all constraints under uncertainty. To this end, the authors propose a novel approach that integrates oracle-based stochastic gradient descent with interval arithmetic: the former efficiently explores high-quality parameter configurations, while the latter provides rigorous and successively tighter lower bounds on the satisfaction probability. This is the first method to synergistically combine these two techniques for solving stochastic constraints, offering both high-probability convergence guarantees and computational reliability. Experimental results on stochastic satisfiability modulo theories (SSMT) and stochastic trajectory planning tasks demonstrate that the proposed method efficiently generates sequences of reliable lower bounds that closely approximate the true optimal value.

Technology Category

Reasoning under Uncertainty: Stochastic OptimizationConstraint Satisfaction and Optimization: SatisfiabilitySearch and Optimization: Sampling/Simulation-based Search

Application Category

Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphsSearch and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for rankingWeb Mining and Content Analysis: Robustness and generalizability of Web mining methods

Abstract

Stochastic constraints, which incorporate both deterministic parameters and random variables, extend classical deterministic constraints by explicitly accounting for uncertainty. These constraints are increasingly prevalent in data science, artificial intelligence, and bioinformatics; however, solving them requires addressing quantitative satisfaction problems that remain a significant challenge in computer science. In this paper, we propose a novel framework for deciding deterministic parameters that maximize the satisfaction probability. Our approach features a unique synergy between stochastic optimization and symbolic techniques: at the high level, it employs \emph{oracle-based stochastic gradient descent} to identify high-quality parameter candidates, while at the low level, it utilizes \emph{interval arithmetic} to compute rigorously certified lower bounds. This framework produces a sequence of sound and increasingly tight lower bounds for the true maximum satisfaction probability, supported by a high-probability convergence guarantee. We demonstrate the effectiveness and efficiency of our approach through its application to Stochastic Satisfiability Modulo Theories (SSMT) problems and a stochastic trajectory planning task.

Problem

• stochastic constraints
• satisfaction probability
• uncertainty
• parameter optimization
• quantitative satisfaction

Innovation

• oracle-based stochastic gradient descent
• interval arithmetic
• stochastic constraints
• satisfaction probability
• SSMT