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.

翻译：在许多安全关键场景中，概率机器学习系统必须在满足代数约束的条件下进行预测，例如预测不穿越障碍物的最可能轨迹。这些现实约束很少是凸的，所考虑的密度函数也通常不具备（对数）凹性。这使得高效可靠地计算这种带约束的最大后验预测变得极具挑战性。本文首先探究在何种条件下能够对连续变量执行精确且高效的带约束最大后验推断，并针对该可处理片段设计了一种可扩展的消息传递算法。随后，我们提出了一种通用的带约束最大后验策略，该策略将定义域分割为凸可行区域与数值约束优化过程交替进行。我们在合成与真实基准测试中评估了两种方法，结果表明我们的方法优于无视约束的基线方案，并能扩展到当前最先进精确求解器无法处理的复杂密度函数。