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Geometric Distributional Control: Learning Progress with Partial Structural Knowledge

Tong Wu

arXiv:2609.06331v1cs.ROcs.SYeess.SY

TL;DR

In partially structured control, known constraints can define feasible actions without supplying a complete value signal to rank them. GDC learns a progress direction from feasible snippets and applies it through an explicit scaffold; evaluations report gains over known-only and learning baselines while retaining scaffold-enforced feasibility.

  • Problem

    Many control settings have actionable partial structure but lack a complete model or Bellman value for ranking feasible actions by downstream consequences.

    Source: §1.1 · pp. 1–2

  • Method

    GDC learns a progress score from feasible action segments and uses it with an explicit scaffold that enforces known constraints during control.

    Source: §2.2 · p. 5, §3.1 · p. 8

  • Results

    On complex TLS routes, GDC multi-H reaches 100% arrival versus 91% for the rule controller, 51% for behavior cloning, 81% for PPO, and 79% for SAC; in multilevel optimization it reaches near-reference Φ values where the known-only solver stalls.

    Source: Numerical results · p. 19, §6 · p. 23

  • Takeaways & Limitations

    The results support using feasible-progress data to supply local direction while explicit known constraints continue to define deployment-time feasibility.

    Source: §6 · p. 23

    Implicit feasibility is a statistical property of feasible demonstrations with progress certificates, not a proof of feasibility for all implicit physics.

    Source: §2.2 · p. 5, §3.2 · p. 11

Abstract

from arXiv · show

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.

1 Introduction

GDC addresses control problems where known structure constrains feasible actions but does not provide a complete value function for ranking them. It learns a local progress direction from feasible data and composes that direction with an explicit scaffold.

1.1 Background and motivation

  • When dynamics, objectives, or cross-level couplings are incomplete, partial structure can rule out infeasible actions without ranking the remaining feasible directions.The missing information is value-like guidance, not simply another way to enforce feasibility.Source: §1.1 · pp. 1–2
  • GDC learns a local progress-value gradient from short, known-feasible segments with weak signed progress certificates, rather than recovering a global Bellman function or complete expert policy.Segments need not be globally optimal or improve monotonically; their progress-tilted distribution guides motion in data-supported contexts.Source: §1.1 · p. 2

1.2 Related work

  • Unlike full predictive control or dataset-supported value learning, GDC targets a progress direction on a known-feasible scaffold, without requiring a complete model, Bellman value, or expert rollouts.The learned score ranks feasible local motions while explicit constraints remain part of execution.Source: §1.2 · p. 2
  • GDC uses a learned conditional density’s score as a control direction, not as a policy to execute, and combines it with known-constraint potentials and feasibility layers.This differs from diffusion-policy interfaces that sample actions and from approaches that treat constraints only as costs or post-hoc corrections.Source: §1.2 · pp. 2–3

1.3 Contributions

  • Its contributions include a data-supported feasible-progress geometry, progress-tilted paths that allow non-monotone snippets, and score-guided control that keeps explicit physics responsible for deployment feasibility.Diffusion-style paths are one implementation; other differentiable path or density models yielding a score can also be used.Source: §1.3 · p. 3
  • Under a maximum-entropy demonstration model, GDC’s score approximates the negative gradient of a data-induced soft progress value; the methodology is tested on multilevel optimization and SUMO driving.The paper also derives stability, robustness, drift, action-error, and physics–data interpolation results.Source: §1.3 · pp. 3–4

2 Problem Formulation

The formulation separates structure enforced analytically from residual feasibility and progress represented statistically. It uses feasible short segments and signed progress certificates to learn a direction, while retaining explicit known constraints in control.

2.1 Notation

  • The notation distinguishes observed context, goals, short action segments, signed progress certificates, known constraints, and a progress-tilted segment distribution.Latent segments use an encoder and horizon-conditioned decoder; the decoder’s pullback metric defines Riemannian gradients.Source: §2.1 · p. 4

2.2 Problem setting and data

  • GDC assumes some structure, such as constraints, actuator limits, geometry, conservation laws, or low-order dynamics, can be enforced reliably online.Residual dynamics and feasibility patterns may be complex, coupled, or expensive to model and optimize.Source: §2.2 · p. 5
  • The explicit/implicit split is bookkeeping, not necessarily an identifiable physical decomposition: explicit parts are enforced analytically, while residual structure is represented statistically.The hard execution guarantee concerns known constraints; implicit feasibility is supported by data rather than analytically certified beyond that support.Source: §2.2 · pp. 5–6
  • Training uses offline known-feasible trajectory segments with weak signed certificates; snippets may wait, back up, or reposition and need not reach the goal or always make progress.The tilt biases the conditional distribution toward positive expected progress in supported contexts, without requiring dense rewards or complete success labels.Source: §2.2 · p. 5
  • GDC selects the shortest candidate horizon whose progress tilt is statistically reliable and positive, using feasible windows from the relevant context neighborhood.The confidence assessment uses weighted progress statistics and effective sample size; horizons with no feasible windows are inadmissible.Source: §2.2 · pp. 5–6
  • If no reliable positive-progress horizon is available, GDC makes no positive-progress guarantee and falls back to known-feasible control or increases the known-physics weight.Feasible data may still describe executable segments even when progress guidance is disabled or downweighted.Source: §2.2 · p. 6
  • The controller seeks safe, task-consistent, directionally correct feedback using the feasible-progress geometry and distributional information.The ideal objective is expressed using the full safety set and a task cost, while GDC enforces known constraints explicitly.Source: §2.2 · p. 6

3 Geometric Distributional Control Method

GDC learns a progress-tilted map over known-feasible action segments and uses its score to guide online control. The method combines learned progress structure with known physics, task costs, and safety constraints, then decodes and projects actions for execution.

3.1 Overview and offline construction

  • The progress tilt increases probability of positive-certificate snippets without deleting feasible waiting, yielding, or repositioning behavior, and applies only in data-supported regions after the horizon test.The tilt is soft, not a hard filter or a guarantee of feasibility beyond the empirical support.Source: §3.1 · p. 7
  • GDC first embeds known-feasible action segments, then learns a progress-tilted path over that scaffold; its density score supplies the online progress direction.The map learns feasible action-segment coordinates, while the separate path learns where progress-biased behavior lies and which direction to move.Source: §3.1 · pp. 8–9

3.2 Geometry and progress score

  • Under the decoder-pullback metric, GDC’s Riemannian gradient flow is invariant to diffeomorphic reparameterizations of the latent manifold.The metric is induced by the action-segment decoder and ensures positive definiteness.Source: §3.2 · p. 10
  • Known constraints anchor the path to a feasible scaffold, while its learned distribution approximates missing physics, preference structure, and cross-level coupling.The path does not need to rediscover known physics or explore the full trajectory space.Source: §3.1 · p. 9, §3.2 · p. 10
  • A diffusion-style implementation predicts clean latent behavior and derives a score for control, but diffusion is optional and implicit feasibility remains statistical rather than guaranteed.Known constraints are still enforced explicitly through conditioning, barriers, anchor penalties, and candidate filtering.Source: §3.2 · pp. 10–11

3.3 Online progress-energy navigation

  • At deployment, GDC freezes its learned objects for online control.The deployed objects include the action-segment posterior, decoder, horizon selector, and progress-tilted latent density.Source: §3.3 · p. 11
  • Unlike GPC, GDC derives tangential direction from a progress-tilted distribution learned from feasible-progress segments, using it as a combinable energy term rather than executing it as a policy.The online energy combines this data term with scaffold restoration, known physics and constraints, and runtime task costs.Source: §3.3 · pp. 11–12
  • GDC localizes candidates to the current known-feasible latent set, follows a capped fixed-depth geometric descent with scaffold correction, then decodes, selects, and projects an action or falls back.The velocity cap and retraction are numerical guards, not optimizers; the final projection is an execution guard.Source: §3.3 · pp. 11–13
  • Online cost is O(KNr(Cθ + Cknown) + Cproj); GDC uses frozen models and does not retrain online or solve a full nonlinear planning problem.The repeated operation evaluates the frozen score and energy, integrates the bounded latent field, decodes candidates, and projects the action.Source: §3.3 · p. 13

4 Formal Assumptions and Main Guarantees

GDC’s guarantees are conditional on data-supported feasible progress and an explicit execution scaffold. Under those conditions, its learned score supplies a local progress direction, while feasibility and approximation guarantees depend on additional regularity and consistency assumptions.

Assumption 4.1

  • The data assumption requires feasible action segments within the explicit scaffold and positive expected signed progress when the horizon selector returns a horizon.The guarantees are stated for contexts in the modeled operating envelope.Source: §4 · p. 13

Theorem 4.3

  • Under a data-supported feasible-progress scaffold and a finite, positive, differentiable continuation partition function, the learned score approximates descent on a data-induced soft progress value.The score approximation error bounds the difference from the soft-progress-value gradient; the guarantee applies where the horizon selector does not abstain.Source: §4 · pp. 13–14

Proposition 4.4

  • When the projection layer is available, every executed action is in the current known-feasible set; filtering, projection, and a known-feasible fallback preserve this guarantee.The execution assumptions include a regular latent map and a nonempty closed feasible set whenever projection is invoked.Source: §4 · pp. 13–14

Corollary 4.5

  • With local strong geodesic convexity, bounded score error, and the stated execution safeguards, the asymptotic action-error bound combines integration drift, score approximation, and decoder error without growing with rollout horizon.The same bound holds pointwise once a receding-horizon refinement reaches the stated robustness tube.Source: §4 · p. 14

Theorem 4.6

  • If the learned potential is value-consistent with the oracle objective and refinement is sufficiently accurate, GDC’s executed first action approaches the constrained optimum as the value, refinement, decoder, and projection errors vanish.The guarantee is restricted to the selected data-supported region and requires the theorem’s convexity and approximation conditions.Source: §4 · p. 15

Corollary 4.7

  • Training-anchor coverage controls the optimality and action-error rates through the fill distance; quasi-uniform coverage gives hn = O(n^-1/d), while iid coverage with density bounded below gives hn = Op((log n/n)^(1/d)).The rate result also assumes Lipschitz potentials and bounded anchor error.Source: §4 · p. 15

Remark 4.8

  • An affine mixing parameter interpolates between known-physics control and data-driven progress completion while retaining the known scaffold’s support restrictions, filters, and final projection.When data support is sparse or the horizon selector abstains, taking λ toward 1 removes the data term and recovers the known-feasible controller.Source: §4 · p. 16

Scope

  • These guarantees do not establish global optimality or implicit feasibility outside the data-supported scaffold; approximate optimality requires the stronger value-consistency condition.Without that condition, the stated operational guarantee is limited to known-physics feasibility and a local direction where positive expected progress is certified.Source: §4 · pp. 13, 16

5 Experimental Validation

The experiments test GDC across SUMO route progress and nonconvex multilevel optimization, where known constraints provide a scaffold and learning supplies progress directions. Both domains evaluate whether that separation improves results while retaining explicit feasibility constraints.

Task and known scaffold

  • SUMO supplies traffic and actuator constraints, signals, and route information; the missing decision is which feasible action advances route progress, with a conservative feasible fallback available.The fallback tracks target speed, brakes for nearby leaders or restrictive signals, and avoids discretionary lane changes.Source: Task and known scaffold · p. 16, §5 · p. 16

Progress data and online decision rule

  • Diverse weak driving traces—not expert labels—receive signed route-progress certificates; the learned distribution ranks safe segments, executes only the first action, and replans after the next observation.Training uses 100,000 weak rollouts, and a fallback action is used when no learned candidate improves sufficiently.Source: Progress data and online decision rule · p. 17, Numerical results · p. 17

Numerical results

  • On complex TLS, multi-H removes fixed H = 12’s 5.88 × 10^-3 maximum violation, trading 0.29 m/s in speed and 5.6 mean episode steps; longer horizons dominate ordinary progress.Shorter horizons appear near local traffic or constraint-sensitive events.Source: Figure 2 · p. 18, Numerical results · pp. 18–19
  • On complex TLS, GDC multi-H reaches 100% arrival versus 91% for the rule controller, 51% for behavior cloning, 81% for PPO, and 79% for SAC.It is the only method with 100% arrival on both sparse-TLS and complex-TLS maps, with zero collisions and zero mean known-constraint violation in the multi-horizon setting.Source: Numerical results · p. 19, Table 3 · p. 19

Problem family and known structural knowledge

  • The multilevel tests use nonconvex bilevel and trilevel problems with analytic objectives, gradients, projections, and local updates, but no globally certified response or optimum.The scaffold maintains explicit box and step constraints while GDC learns joint motion across levels from solver traces.Source: Problem family and known structural… · p. 20, Numerical results · p. 21

Progress model and online solver

  • GDC learns from H = 8 solver-trace segments with signed progress-minus-effort certificates, then scores projected joint motions against known-only updates and falls back on failure.Only the first increment is applied; feasibility or progress failure triggers the known-only fallback.Source: Progress model and online solver · p. 20

Protocol and metrics

  • Held-out parameter tests use 50,000 traces and fixed-weight evaluation criteria; offline multistart references assess scores but are not online runtime competitors.SCIP is a separate offline branch-and-cut benchmark, with time-limit cases reported as incumbents rather than globally certified solutions.Source: Protocol and metrics · p. 20, Numerical results · p. 21

Numerical results

  • GDC’s normalized gaps are below 0.23% for bilevel and 0.65% for trilevel problems, reaching scores close to offline references while known and nested online baselines occupy worse basins.The largest trilevel gap is 0.64% at d = 8.Source: Numerical results · p. 21, Table 4 · p. 21, Table 5 · p. 21

6 Discussion and Conclusion

GDC targets partial-physics settings where complete model-based optimization or behavioral cloning is not already sufficient. It keeps known constraints explicit while using feasible-progress data to learn useful directions, with results spanning multilevel optimization and SUMO driving.

6 Discussion and Conclusion

  • In multilevel optimization, the known-only solver preserves feasibility but can stall in poor nonconvex basins, while GDC reaches near-reference Φ values.The paper also reports that in SUMO, GDC improves arrival relative to rule, imitation, and RL baselines while maintaining collision-free route completion.Source: §6 · p. 23
  • GDC is intended for partial-physics settings, not those where complete model-based optimization or pure behavioral cloning is already sufficient.Source: §6 · p. 23
  • Across the two settings, known constraints determine what may be executed, while learned progress determines which feasible direction is useful.Source: §6 · p. 23

A Additional SUMO Diagnostics

Additional SUMO diagnostics examine scalability on a larger complex-TLS map. The two figures cover separate scenarios, and Table 7 links diagnostic figures to numerical conclusions.

A Additional SUMO Diagnostics

  • Figures 5 and 6 show separate scenarios on the 60 × 60 complex-TLS map: recorded scalability routes and newly recorded long-route episodes, respectively.Table 7 links each SUMO diagnostic figure to the numerical conclusion it supports.Source: A Additional SUMO Diagnostics · p. 23
  • Figure 5 presents recorded ego paths, selected planning horizons, and aggregate horizon use for the scalability routes.Source: A Additional SUMO Diagnostics · p. 23
  • Figure 6 presents newly recorded long-route episodes; traffic-light constraints use signal markers, while star markers are reserved for non-TLS roadblocks absent from this TLS-only map.Source: A Additional SUMO Diagnostics · p. 23

A.1 SUMO Transfer Diagnostics

The transfer diagnostics cover held-out SUMO maps with either fixed topology and changed road disturbances or changed graph parameters and disturbances. Reported outcomes include successful route completion and, for blocked-map transfer, zero known violations.

A.1 SUMO Transfer Diagnostics

  • Blocked-map transfer changes held-out closures and slow or bottleneck edges on a fixed base topology, whereas graph-and-disturbance transfer changes graph parameters and disturbances together.In both settings, the controller receives current known map constraints; transfer tests the learned direction field on a new scaffold, not blind map inference.Source: §A.1 · p. 23
  • On the 100-map blocked-map benchmark, GDC achieves 1.000 arrival and progress, zero known violations, and nonfallback learned control on 99.7% of decisions.The base perturbed topology is fixed while held-out closures and slow roads vary.Source: §A.1 · p. 23
  • Across 20 shown graph-and-disturbance transfer traces, routes reach their goals with zero collisions despite changed graphs and local disturbances.The shown maps span 18 × 19 to 22 × 22 grids, 8–26 closure pairs, and 33–48 bottleneck pairs.Source: §A.1 · p. 23
  • The diagnostics show long blocked-map detours and held-out graph routes; Figure 6’s separate scalability episodes use 7.0–8.1 km routes and finish in 1,475–1,717 control steps with zero collisions and no timeout.Figure 7 overlays five routes on each of four held-out blocked maps; Figure 8 depicts routes on unseen graph parameters and disturbances.Source: Figure 6 · p. 24, Figure 7 · p. 25, Figure 8 · p. 26

B Additional Theoretical Analysis

The additional theoretical analysis examines fixed objects learned offline and used online. It analyzes the empirical manifold, progress-tilted density, score field, and induced latent control flow, rather than adding offline training steps.

B Additional Theoretical Analysis

B.1 Notation and standing assumptions for the proofs

The appendix establishes notation and regularity assumptions for the learned latent path and execution scaffold, then defines data energy and its score on the empirical manifold.

B.1 Notation and standing assumptions for the proofs

  • Proofs condition on observation, goal, known constraints, and optionally a selected horizon, using Riemannian volume and metric-aware gradients and norms.The manifold is connected and finite-dimensional with a uniformly elliptic metric on the data-supported region; boundary conditions are absent or reflecting/no-flux.Source: §B.1 · p. 27
  • Conditional latent path densities and their ideal targets are strictly positive and C1 on the data-supported region, with bounded-below C1 potentials.Potential functions are C2 on geodesically convex neighborhoods when results require Hessians or strong convexity.Source: §B.1 · p. 27
  • The decoder is Lipschitz with bounded first-action reconstruction error, and projection is invoked only for a closed, nonempty feasible set with a feasible fallback.Score errors use the dual Riemannian norm, and progress certificates are bounded on selected finite horizons.Source: §B.1 · pp. 27–28

Data energy function

  • Low data energy identifies latent behaviors consistent with feasible demonstrations and their progress certificates under the observation, goal, and constraint context.The data energy is defined over the empirical manifold from the conditional latent path distribution.Source: Data energy function · p. 28

Score function on the empirical manifold

  • The learned-distribution score is defined on the empirical manifold, and the clean-behavior predictor induces the score at a noise level.The displayed definition relates the score to the gradient of the data energy.Source: Score function on the empirical manifold · p. 28

Implicit progress value theorem

The theorem links a learned latent score to a data-induced soft progress value, using progress-weighted feasible trajectories for directional information and known dynamics and constraints to define the scaffold. Under data-support, fitting-error, and modeling assumptions, it characterizes progress under the data policy and provides stability guarantees for the associated Riemannian gradient flow.

Implicit progress value theorem

  • The score equals the gradient of a data-induced soft progress value when the learned density matches the maximum-entropy feasible-progress model under the theorem’s regularity conditions.The empirical aggregate posterior’s agreement with that model is an assumption, not a consequence of the offline objective.Source: Implicit progress value theorem · p. 29
  • Progress-weighted feasible segments bias selected-horizon trajectories toward expected certificate improvement, while data-supported waiting, backtracking, or repositioning can remain possible.The resulting implicit value reflects likely local progress under the demonstrations, not optimal final success; the connection is strongest with diverse, relevant coverage.Source: Implicit progress value theorem · p. 30

Structural role separation in the control law

  • The score selects among known-feasible options and captures learned directional information, while the known-constraint potential removes explicitly modeled violations; their distinct roles preserve direction against high-bandwidth feasibility signals.Role separation preserves the directional information created by progress-certificate training; it does not create that information.Source: Structural role separation in the… · p. 30

Total control potential on the empirical manifold; Riemannian gradient flow on the empirical manifold

  • GDC combines known physics, the data-induced implicit-information approximation, and task cost in a total potential, then uses score-guided Riemannian gradient flow on the behavior manifold.Unlike diffusion-policy sampling in action space, the structured flow uses the path score for direction and geometry and constraints for feasibility.Source: Total control potential on the… · p. 31, Riemannian gradient flow on the… · p. 31

Singular perturbation structure for known constraints; Discrete-time implementation

Manifold Langevin SDE; Stochastic score-guided control on the empirical manifold; Euler–Maruyama discretization

Stationary distribution; Implications for control

  • Under the stated finiteness condition, the manifold Langevin dynamics have a Gibbs stationary distribution balancing data consistency, known-constraint satisfaction, and task optimality.The distribution is unique when the data-supported region is connected and diffusion is uniformly elliptic and nonexplosive there.Source: Stationary distribution · p. 33, Implications for control · p. 33

The Fokker–Planck equation on the manifold; Free energy and convergence

Trajectory path model on the empirical horizon manifold

  • For coupled inter-step constraints, GDC plans on an empirical trajectory manifold where known dynamics and constraints form the scaffold and feasible trajectory data supplies implicit coupling.The manifold is built by encoding and decoding feasible horizon action sequences; residual couplings not captured by known dynamics are learned from feasible trajectory data.Source: Receding-horizon trajectory control · p. 34, Trajectory path model on the empirical… · p. 34

Receding-horizon trajectory control

  • The trajectory-level extension applies when inter-step constraints couple multiple time steps, using known dynamics and constraints alongside progress-certified feasible trajectories.Source: Receding-horizon trajectory control · p. 34

Trajectory energy and score

  • Trajectory energy separates known constraints from task objectives, and its score-guided dynamics use the Riemannian metric on the trajectory manifold.Source: Trajectory energy and score · p. 35

Receding horizon execution

  • At each receding-horizon step, the controller selects a trajectory minimizing its trajectory potential, shifts the horizon, and repeats while enforcing known trajectory-level constraints.The path model’s distributional knowledge approximates implicit trajectory information.Source: Receding horizon execution · p. 35

Progress consistency across horizon

Suppose also that the learned trajectory path satisfies

  • Progress certificates encode directional information in short-horizon trajectory distributions, so GDC need not plan far enough to see the goal as model-predictive control does.The distribution can favor higher-progress turns while retaining waiting or yielding behavior supported by data; it does not guarantee every segment reaches the goal.Source: Suppose also that the learned… · p. 36

Lyapunov stability of Riemannian gradient flow

  • A strict local minimum is Lyapunov stable, its sublevel sets are forward invariant, and strong geodesic convexity yields exponential convergence at rate m.These guarantees assume a continuously differentiable, bounded-below potential and an interior strict local minimum in the data-supported region.Source: Lyapunov stability of Riemannian… · p. 36

Robustness to score estimation error

The robustness result bounds the effect of score estimation error under strong geodesic convexity, while a guarded update rule prevents discretization drift from accumulating over time.

Robustness to score estimation error

  • When score error is bounded by ε and the total potential is m-strongly geodesically convex, approximate dynamics converge to a neighborhood of radius ε/m around the minimizer.The distance-squared derivative is negative outside that radius.Source: Robustness to score estimation error · pp. 36–37
  • The robustness guarantee applies on a geodesically convex, data-supported neighborhood containing the trajectory and minimizer.The score error is assumed uniformly bounded in the dual Riemannian norm.Source: Robustness to score estimation error · pp. 36–37

Non-accumulation of discretization drift

  • Accepted updates use capped metric length and a retraction/backtracking guard that bounds distance from the local known-feasible latent scaffold.The bound is uniform in time and vanishes as the step size tends to zero, so deviation does not accumulate over the rollout horizon.Source: Non-accumulation of discretization drift · p. 37

End-to-end action error bound

The section bounds receding-horizon action error, gives conditional guarantees for approaching a constrained optimum, and describes how known-physics and data potentials can be interpolated.

End-to-end action error bound

  • Action error decomposes into geometric integration drift, score approximation error, and decoder error, and does not accumulate with the number of future rollout steps.For accepted updates within the stated robustness tube, the same bound holds pointwise; the final projection is non-expansive for convex known-constraint sets.Source: End-to-end action error bound · pp. 37–38

Then the oracle suboptimality gap and latent distance satisfy

  • Quasi-uniform deterministic coverage gives h_n = O(n^-1/d), while iid anchors with density bounded below give h_n = O_p((log n/n)^(1/d)); the value gap is O(r_n+s_n+q_K) under the stated error rates.The rate combines value-consistency, coverage, and online-refinement errors.Source: Then the oracle suboptimality gap and… · p. 40

Approximate optimality under value consistency

Interpolated control potential under partial physics

  • The interpolated control potential combines λ-weighted known-physics potential, (1−λ)-weighted data potential, and task potential.The definition places both potentials on the scaffold.Source: Interpolated control potential under… · p. 40

Interpolation theorem

The controller interpolates explicit known-physics and data-driven progress potentials, with a continuous minimizer path under uniqueness; its evolution is also characterized as a Wasserstein gradient flow.

Interpolation theorem

  • At λ = 1 the controller uses known-physics and task potentials, at λ = 0 it uses data-driven progress and task potentials, and intermediate values yield a hybrid controller.The hybrid retains known-feasible support, anchor penalties, and the scaffold's filters and projection layer.Source: Interpolation theorem · p. 41

Data-physics tradeoff path

  • On a compact region with continuous potential, minimizers exist for every λ; if each minimizer is unique, they trace a continuous data-physics tradeoff path.The result concerns minimizers of the interpolated potential.Source: Data-physics tradeoff path · p. 41

B.9 Online control used in the theory

  • The theory's online setting freezes learned models, checks known constraints after decoding, and executes only the first action of the selected segment.The interpolation analysis retains the known-feasible scaffold during online control.Source: §B.9 · p. 41, Interpolation theorem · p. 41

GDC as a Wasserstein gradient flow on the empirical manifold

  • On the empirical manifold, GDC's evolution is a Wasserstein gradient flow that minimizes expected control potential while maximizing behavioral entropy.The potential includes known physics, data-induced implicit information, and task cost.Source: GDC as Wasserstein gradient flow on… · pp. 41–42
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