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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