The Theory and Practice of MAP Inference over Non-Convex Constraints

2026-02-09

AI Summary

This work addresses the challenge of efficiently computing maximum a posteriori (MAP) estimates in safety-critical probabilistic machine learning systems under non-convex algebraic constraints and non-log-concave posteriors, where existing methods struggle to balance accuracy and scalability. The paper identifies, for the first time, a tractable subclass of MAP inference problems that admit efficient exact solutions under non-convex constraints and introduces a general framework that overcomes the traditional reliance on convexity or log-concavity. By partitioning the non-convex feasible region into convex subregions and alternating between scalable message passing and numerical constrained optimization, the proposed approach achieves both precision and scalability. Experiments demonstrate significant improvements over constraint-ignoring baselines on synthetic and real-world benchmarks, and the method scales to complex density scenarios beyond the reach of current exact solvers.

Technology Category

• Machine Learning: Probabilistic Circuits and Graphical Models
• Reasoning under Uncertainty: Probabilistic Programming
• Search and Optimization: Non-convex Optimization

Application Category

• Security and Privacy: Large-scale security measurements
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Applied ML and AI for Web-based mobile applications

Abstract

In many safety-critical settings, probabilistic ML systems have to make predictions subject to algebraic constraints, e.g., predicting the most likely trajectory that does not cross obstacles. These real-world constraints are rarely convex, nor the densities considered are (log-)concave. This makes computing this constrained maximum a posteriori (MAP) prediction efficiently and reliably extremely challenging. In this paper, we first investigate under which conditions we can perform constrained MAP inference over continuous variables exactly and efficiently and devise a scalable message-passing algorithm for this tractable fragment. Then, we devise a general constrained MAP strategy that interleaves partitioning the domain into convex feasible regions with numerical constrained optimization. We evaluate both methods on synthetic and real-world benchmarks, showing our approaches outperform constraint-agnostic baselines, and scale to complex densities intractable for SoTA exact solvers.

Problem

• constrained MAP inference
• non-convex constraints
• probabilistic ML
• safety-critical prediction
• algebraic constraints

Innovation

• constrained MAP inference
• non-convex constraints
• message-passing algorithm
• domain partitioning
• probabilistic ML