Geometric Distributional Control: Learning Progress with Partial Structural Knowledge

Tong Wu

cs.ROSep 6, 2026

Geometric Distributional Control: Learning Progress with Partial Structural Knowledge

Authors: Tong Wu

Organizations: University of Central Florida

Abstract

Real-time control often sits between two limiting regimes. Predictive optimization and model-based control are powerful when dynamics, parameters, objectives, and online planning models are specified; reinforcement learning can relax this requirement, but must infer long-horizon value signals from sequential data and interaction, making training slow, high-variance, and hard to scale in large action spaces. This middle regime is common in systems including autonomous driving, warehouse robotics, traffic control, and delivery drones: partial geometry, physics, rules, or constraints are known, yet the local direction of task progress remains uncertain. Geometric Distributional Control (GDC) is designed for this partial-knowledge setting. It factorizes control into feasibility and progress: known geometry, rules, constraints, and response maps define an executable scaffold, while progress-weighted feasible data learns the missing directional signal on that scaffold. The learned score acts as a Bellman-like local value-gradient, selecting actions that make progress without requiring global Bellman recursion, a fully specified planner, or a black-box policy that absorbs both feasibility and preference. This knowledge can be lightweight and partial, such as simple dynamics, safety filters, local maps, constraint projectors, or lower-level response maps; it need not encode full dynamics or a long-horizon objective. Offline, GDC fits a progress-tilted distribution from short known-feasible snippets with weak signed progress certificates. Online, its score is projected through the scaffold and applied in receding-horizon feedback. We prove that this score descends a data-induced soft progress value and validate GDC on structured multilevel optimization and SUMO route-progress driving, where it improves over known-only solvers and learning baselines while preserving scaffold-enforced feasibility.

Figures & tables

Panel of Figure 1 — Figure 1: Overview of GDC. Known-feasible segments with weak progress certificates are used offline to build a latent feasible-progress geometry and score field. Online control follows the induced progress-energy direction, while the explicit known scaffold remains responsible for filtering, projection, fallback, and execution safety.

Benchmark	TLS	Slow detour	Bottleneck	Removed edges	Controllers
Priority grid	$0$	$0$	$0$	$0$	fixed $H=12$
Sparse TLS	sparse	$0$	$0$	$0$	fixed $H=12$ , multi- $H$
Complex TLS	$18$	$64$ at $2.5\,\mathrm{m/s}$	$76$ at $7.5\,\mathrm{m/s}$	$0$	fixed $H=12$ , multi- $H$

Table 1: SUMO route-progress benchmark suite. All maps are perturbed

20\times 20

two-lane grids. Multi-

H

denotes joint online selection over

H\in\{6,12,24\}

; fixed-H restricts candidate selection to

H=12

Benchmark	Controller	Arr.	Prog.	Coll.	Mean viol.	Max viol.	Non-fb.	Speed	Steps
Priority grid	fixed $H=12$	$1.000$	$1.000$	$0.000$	$1.22{\times}10^{-6}$	$2.80{\times}10^{-4}$	$0.999$	$11.24$	$202.6$
Sparse TLS	fixed $H=12$	$1.000$	$1.000$	$0.000$	$1.48{\times}10^{-7}$	$5.01{\times}10^{-5}$	$1.000$	$10.97$	$230.3$
Sparse TLS	multi- $H$	$1.000$	$1.000$	$0.000$	$0.00$	$0.00$	$0.997$	$10.58$	$233.9$
Complex TLS	fixed $H=12$	$1.000$	$1.000$	$0.000$	$1.59{\times}10^{-5}$	$5.88{\times}10^{-3}$	$0.999$	$9.25$	$317.3$
Complex TLS	multi- $H$	$1.000$	$1.000$	$0.000$	$0.00$	$0.00$	$0.991$	$8.96$	$322.9$

Table 2: SUMO closed-loop results over

100

episodes. Arr. is arrival rate, Prog. is normalized route progress, Coll. is mean collision count, Non-fb. is the fraction of steps where a learned candidate rather than the fallback supplies the executed action, Speed is mean speed in

\mathrm{m/s}

, and Steps is mean episode length.

Figure 2: Complex TLS multi-horizon evaluation from the fast-equivalent online implementation. The figure shows ten recorded ego paths on the harder map, selected horizons over control steps for the same episodes, the horizon-use distribution over all $100$ episodes, and the dominance of the long horizon on a map with sparse traffic lights, slow bottleneck corridors, and slow detour edges.

Benchmark	Method	Prog.	Arr.	Coll.	Mean viol.	Speed	Steps
Sparse TLS	GDC fixed-H12	$1.000$	$1.000$	$0.00$	$1.48{\times}10^{-7}$	$10.97$	$230.3$
Sparse TLS	GDC multi-H	$\mathbf{1.000}$	$\mathbf{1.000}$	$0.00$	$\mathbf{0.00}$	$10.58$	$\mathbf{233.9}$
Sparse TLS	Rule safety	$1.000$	$0.990$	$0.00$	$1.33{\times}10^{-1}$	$11.05$	$330.6$
Sparse TLS	Weak behavior	$0.962$	$0.840$	$0.00$	$1.44{\times}10^{-2}$	$7.86$	$449.8$
Sparse TLS	Behavior cloning	$0.931$	$0.760$	$0.00$	$3.55{\times}10^{-3}$	$6.26$	$537.1$
Sparse TLS	PPO	$0.997$	$0.970$	$0.00$	$1.27{\times}10^{-2}$	$7.87$	$451.4$

Table 3: SUMO controller benchmark comparison. GDC and non-SAC rows are evaluated over

100

episodes; the available sparse-TLS SAC run contains

90

episodes. Prog. is normalized route progress, Arr. is arrival rate, Coll. is mean collision count, Speed is mean speed in

\mathrm{m/s}

, and Steps is mean episode length.

$d$	Ref. $\Phi$	Known $\Phi$	Nested $\Phi$	GDC $\Phi$	Norm. gap %	GDC s
$6$	$0.122821439$	$0.727128$	$0.699750$	$\mathbf{0.124122773}$	$\mathbf{0.096}$	$44.659$
$8$	$0.085741765$	$0.620790$	$0.601718$	$\mathbf{0.088084970}$	$\mathbf{0.194}$	$57.622$
$10$	$0.118656182$	$0.542130$	$0.508923$	$\mathbf{0.121497257}$	$\mathbf{0.229}$	$52.206$
$12$	$0.132315706$	$1.005490$	$0.914664$	$\mathbf{0.134485621}$	$\mathbf{0.179}$	$83.117$

Table 4: Bilevel evaluation-score scaling. GDC is the learned online controller; known and nested are conventional online baselines; the reference column is an offline evaluation reference under the same scalar criterion. Normalized gap is reported as

100(\Phi_{\mathrm{GDC}}-\Phi_{\mathrm{ref}})/(\Phi_{\mathrm{init}}-\Phi_{\mathrm{ref}})

; lower is better.

$d$	Ref. $\Phi$	Known $\Phi$	Nested $\Phi$	GDC $\Phi$	Norm. gap %	GDC s
$4$	$0.164844401$	$0.572960$	$0.658353$	$\mathbf{0.168670984}$	$\mathbf{0.345}$	$79.898$
$6$	$0.178681018$	$1.020626$	$1.144793$	$\mathbf{0.180060025}$	$\mathbf{0.093}$	$116.208$
$8$	$0.122788129$	$0.951728$	$1.072284$	$\mathbf{0.131214128}$	$\mathbf{0.640}$	$138.865$
$10$	$0.178549407$	$1.059038$	$1.146033$	$\mathbf{0.181326304}$	$\mathbf{0.187}$	$138.403$

Table 5: Trilevel evaluation-score scaling. Each dimension reports the selected GDC online evaluation with the lowest final evaluation score among available candidates. Normalized gap is reported as a percentage; all four trilevel cases are below

0.65\%

Problem	$d$	Status	Gap	Incumbent $\Phi$	Time (s)
Bilevel	$6$	optimal	$0.000000$	$0.122821621$	$22.6$
Bilevel	$8$	timelimit	$0.000045$	$0.085741916$	$14401.7$
Bilevel	$10$	timelimit	$0.000060$	$0.118656277$	$14402.9$
Bilevel	$12$	timelimit	$0.000270$	$0.132315706$	$14400.0$
Trilevel	$4$	optimal	$0.000000$	$0.164844566$	$327.8$
Trilevel	$6$	timelimit	$0.000065$	$0.178681118$	$14403.2$

Table 6: SCIP branch-and-cut benchmark on the original nonconvex bilevel and trilevel formulations with a four-hour time limit per case. Only rows with status “optimal” are globally certified. Rows with status “timelimit” report the best incumbent found within the limit.

Panel of Figure 3 — Figure 3: Online GDC wall time versus SCIP branch-and-cut wall time on the same held-out instances, normalized by the four-hour SCIP budget on a linear scale. Blue GDC bars are intentionally near the baseline; blue labels report the fraction of the four-hour budget used online. Orange SCIP bars at the budget line are time-limit cases, which provide incumbents but are not globally certified. Vertical arrows report the SCIP/GDC wall-time factor; capped SCIP cases are lower bounds.

Panel of Figure 4 — Figure 4: Online convergence of GDC against SCIP branch-and-cut incumbents. Solid curves show the GDC evaluation score over online iterations; dashed horizontal lines of the same color show the corresponding SCIP incumbent for that dimension. SCIP is an offline branch-and-cut reference, not an online competitor.

Appendix figures & tables4 assets

Supplementary material from the paper’s appendix.

Appendix

Figure 5: Complex TLS scalability diagnostic on the $60\times 60$ map. The figure shows ten recorded ego paths, the selected planning horizon at each control step, and the aggregate horizon-use distribution for this scenario.

Figure 6: Long-route complex TLS scalability diagnostic on the $60\times 60$ map. The ten trajectories use $7.0$ – $8.1\,\mathrm{km}$ routes and complete in $1{,}475$ – $1{,}717$ control steps; all recorded episodes reach the route goal with zero collisions and no timeout.

Panel of Figure 7 — Figure 7: Representative blocked-map transfer diagnostics. Four held-out maps are shown; each panel overlays five long routes. Red marks closures, orange marks slow/bottleneck roads, and the $1.7$ – $2.4\,\mathrm{km}$ routes make transfer-time detours inspectable.

Figure 8: Held-out graph-and-disturbance transfer examples. Panels use unseen graph parameters and disturbances; red ticks mark closures, dashed orange marks slow roads, icons mark traffic lights, and colored curves are successful long GDC routes.

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Differentiable predictive control (DPC), a self-supervised learning approach for approximating explicit model predictive control (MPC) policies, offers significant computational advantages over online optimization-based MPC. However, feasibility guarantees, a core requirement for safe control, are currently provided either probabilistically or via online safety filters. The lack of rigorous feasibility guarantees for offline policy optimization remains an open problem. This paper establishes deterministic feasibility guarantees for DPC using a novel topological analysis of the induced reachable safe set, without requiring online safety filters. By exploiting the inherent model-based nature of DPC, in which differentiable system dynamics are embedded directly into the computational graph, we analyze the properties of the learned control policies and the corresponding system states from topological and geometric perspectives. Inspired by our theoretical analysis, we propose a novel self-supervised offline policy learning strategy that utilizes a proxy loss with Control Barrier Functions (CBFs). Crucially, these properties not only significantly improve policy training but also enable the derivation of strict, deterministic feasibility guarantees from a finite number of training samples. Extensive closed-loop simulations validate our theoretical findings, demonstrating that the empirical constraint violations monotonically decrease to zero as the training sample size increases. Ultimately, this work illustrates that DPC policy optimization yields formal safety certificates that are structurally unattainable with conventional black-box methods, e.g., reinforcement learning (RL) or supervised learning-based approximate MPC, thereby providing a new perspective on feasibility guarantees in learning-based control.

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