Geometric Distributional Control: Learning Progress with Partial Structural Knowledge

Partial structural knowledge can constrain admissible decisions without specifying how to achieve task progress. We introduce Geometric Distributional Control (GDC), which formulates decision-making under such knowledge as learning and navigating conditional distributions of feasible action segments. The central principle is positive expected progress of a conditional action-segment distribution over supported horizons, rather than positive progress of every segment. This lets waiting or repositioning coexist with goal-directed behavior and extracts directional knowledge from imperfect trajectories without globally optimal demonstrations. Multi-horizon candidate selection lets the learned guidance respond to the temporal scale of the current context. GDC learns a progress-weighted distribution in a compact latent space and combines its score with available analytical structure to guide online geometric optimization. We establish conditions under which distributional approximation and candidate selection retain positive expected segment progress. A conditional objective-gap bound links representation coverage, value approximation, and online search accuracy to decision quality on a data-supported region. Experiments on traffic control and hierarchically coupled nonconvex merit optimization demonstrate effectiveness across changing constraints and problem scales.