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

Leander Kurscheidt, Gabriele Masina, Roberto Sebastiani, Antonio Vergari

Keywords: probabilistic machine learning, SMT(LRA)-constraints, constrained optimization

TL;DR: We study exact and approximate MAP inference under non-convex algebraic constraints and show that exploiting constraint structure yields more reliable and efficient predictions than constraint-agnostic methods.

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 in an efficient and reliable way 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 structure aware approach outperforms constraint-agnostic baselines.

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Submission Number: 98