Settling improves inference for outputs with complex valid sets
Settling: Equilibrium Inference for Non-Convex Validity Sets
Machine Learning
Summary
Many machine learning models give one best guess even when multiple different answers could be correct, which can cause problems when the correct answers aren't in one simple group. The authors call this problem "conditional mean collapse" and create a new method called Settling that starts with a guess and then adjusts it to find a stable, valid answer. They tested this idea with tricky geometric examples and found Settling worked almost perfectly, while older methods struggled or gave rough results. This approach could help make smarter decisions in cases where valid answers form complicated shapes or sets.
What this means in practice
- •For robotics developers: Generate stable, valid motion plans even when feasible paths are disconnected or complex, improving robot navigation near obstacles.
- •For autonomous vehicle engineers: Improve decision-making systems by reliably refining candidate outputs toward valid driving scenarios in challenging environments.
Tested on simulated data.
Authors
Lyes Saad Saoud
Abstract
Many learning systems return a single point estimate even when admissible outputs form disconnected or non-convex sets. Under squared loss, an ambiguous conditional distribution can therefore have a Bayes-optimal conditional mean that is invalid. We formalize this failure as conditional mean collapse and introduce Settling, an equilibrium-based inference operator that separates proposal generation, consistency evaluation, and test-time equilibrium selection. The operator treats a mean-seeking proposal as an initialization and refines it toward a locally stable configuration; conditional on initialization, refinement is deterministic. We establish exact-gradient descent, local convergence, and an inexact-gradient robustness condition relevant to learned consistency critics. In a reproducible 100-context geometric diagnostic, the mean-seeking baseline succeeds in 0/100 contexts, stochastic denoising in 100/100, and Settling in 99/100 while producing substantially lower trajectory roughness. A 1,200-run sensitivity study yields 97-100% success across obstacle-jitter ranges up to 0.20 and 94-100% across one-time initialization perturbations from 0.05 to 0.50. Cross-domain panels remain mechanism illustrations; learned high-dimensional validation remains an open empirical test.