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Control-Geometry Straightening for Sampling-Based Latent Planning

Ziang Fu, Ning Ning

arXiv:2609.35603v2cs.LGstat.ML

TL;DR

Planning with latent world models remains difficult when predictive accuracy does not make the planning objective easy to optimize. CGS matches action similarities to corresponding latent-transition similarities to shape control geometry. Across multiple environments and planners, it improves planning and sample efficiency, with reported success-rate gains up to 20 and 12.6 percentage points over LeWM and LeWM+TS, respectively.

  • Problem

    How representation geometry can make good action sequences easier to find with limited sampled candidates and refinement steps remains an open planning question.

    Source: §1 · p. 1

  • Method

    CGS learns control geometry from local transitions by matching pairwise cosine similarities among actions to those among corresponding latent differences.

    Source: §1 · p. 2, §3.2 · p. 4

  • Results

    CGS improves planning performance and sample efficiency across multiple planners, with success-rate gains up to 20 and 12.6 percentage points over LeWM and LeWM+TS, respectively.

    Source: §1 · p. 2

  • Takeaways & Limitations

    Local geometric regularization can shape the full-horizon optimization landscape without modifying the planner at test time.

    Source: §7 · p. 9

    The theoretical analysis assumes linear latent dynamics and, in its main case, d_a = d with invertible B.

    Source: §4.1 · p. 5

Abstract

from arXiv · show

Joint-embedding predictive architectures enable planning with latent world models, but accurate transition prediction alone does not ensure that the planning objective is easy to optimize. We introduce Control-Geometry Straightening (CGS), a single auxiliary loss that learns planner-friendly representations by directly straightening control geometry for sampling-efficient planning. CGS matches pairwise cosine similarities among actions to those among corresponding latent differences only using local transitions from pixel-action pairs. The loss can be applied across world-model architectures using end-to-end learned or pretrained representations. Under linear-dynamics, our theoretical analysis connects this objective to temporal straightening and more balanced terminal-cost curvature across the full planning horizon, yielding finite-budget guarantees for MPPI, local contraction results for CEM, and convergence bounds for gradient descent. Across four control environments and multiple planners, CGS improves planning with fewer sampled candidates and refinement steps, achieving success-rate gains up to 20 and 12.6 percentage points over LeWorldModel (LeWM) and its temporal-straightening variant (LeWM+TS), respectively, with sampling-based planners using 128 candidates per update. Probes, comparisons with DINO-WM architecture, and planner-side ablations clarify how latent motion organization, state dependence, and dynamical context shape planning behavior. Straightening control geometry thus makes good action sequences easier to find under limited planning budgets.

1 INTRODUCTION

Predictive accuracy alone does not ensure that latent representations make planning easy, and existing temporal straightening does not explicitly structure how actions shape transitions. CGS directly matches action geometry to latent transition geometry and is reported to improve planning across planners and budgets.

1 INTRODUCTION

  • Planning time is a bottleneck because each refinement requires many multi-step model rollouts, motivating representations that help find good sequences with limited candidates and refinements.Source: §1 · p. 1
  • Temporal Straightening regularizes latent-trajectory curvature, but does not explicitly structure how actions shape latent transitions.LeWM also exhibits implicit temporal straightening, while adding explicit TS has environment-dependent effects.Source: §1 · p. 1
  • CGS matches pairwise cosine similarities among actions and corresponding latent differences, learning planner-friendly representations from local transitions.The loss applies across end-to-end learned and pretrained world-model representations.Source: §1 · p. 2
  • CGS is reported to achieve success-rate gains up to 20 and 12.6 percentage points over LeWM and LeWM+TS, respectively.The paper also reports improved performance with fewer sampled candidates and refinement steps.Source: §1 · p. 2

2 RELATED WORK

Prior latent-world-model and JEPA work addresses predictive representation learning and planning in several ways, while how representation geometry supports action-sequence optimization remains underdetermined. CGS addresses action geometry, a distinct focus from temporal straightening and sampling-based planner proposal or update designs.

2 RELATED WORK

  • Predictive-learning mechanisms address representation collapse, but how representation geometry should help planners find good action sequences remains underdetermined.Source: §2 · p. 2
  • Temporal Straightening aligns consecutive latent displacements to reduce trajectory curvature, without structuring how actions shape latent transitions.Source: §2 · p. 3
  • Sampling-based MPC methods differ in proposal updates: CEM fits proposals to low-cost elites, while MPPI weights candidates by rollout cost.Related approaches also adapt covariance, reuse samples, or use controller and learned-prior proposals.Source: §2 · p. 3

3 CONTROL-GEOMETRY STRAIGHTENING

CGS learns control geometry by matching action and latent-difference similarities while retaining the LeWM predictive-learning objective. At planning time, a frozen model predicts candidate rollouts for terminal-goal matching.

3.1 LATENT WORLD MODEL AND PLANNING SETUP

  • The latent world model uses an encoder to represent observations and a predictor to forecast embeddings from frame histories and corresponding actions.The predictor uses temporal causal masking so each next-step prediction depends only on inputs at or before that time.Source: §3.1 · p. 3
  • Planning optimizes an action sequence through a frozen world model to minimize the distance between the predicted terminal latent and the encoded goal.The rollout spans horizon H.Source: §3.1 · p. 4

3.2 LEARNING CONTROL GEOMETRY BY PAIRWISE SIMILARITY MATCHING

  • CGS matches pairwise cosine similarities among actions to those among their corresponding latent differences, using action directions as the geometric reference.Aligned, orthogonal, and opposing actions encourage corresponding angular relationships between latent differences.Source: §3.2 · p. 4
  • When consecutive actions point in the same direction, CGS's adjacent term becomes the squared angular Temporal Straightening loss.Unlike that adjacent alignment case, CGS matches pairwise relations among actions and latent differences.Source: §3.2 · p. 4

3.3 TRAINING OBJECTIVE

  • Training combines the LeWM prediction and SIGReg anti-collapse objectives with the CGS loss.Source: §3.3 · p. 4
  • The CGS weight is ramped from zero to its final value over the first 5% of training updates, with default history length N = 3.The ramp is intended to preserve predictive geometry while gradually introducing control-geometry regularization.Source: §3.3 · p. 4

4 THEORY: FROM CGS TO SAMPLING-EFFICIENT PLANNING

Under linear latent dynamics, the analysis connects CGS to temporal straightening, balanced action effects, and more balanced terminal-cost curvature. These geometric properties yield results for population and finite-budget MPPI, with related local CEM and gradient-descent analyses.

4.1 LATENT CONTROL GEOMETRY LEARNED BY CGS

  • Under bounded-feature and joint-coverage assumptions, zero population CGS loss occurs exactly when A = I_d and B^⊤B = cI_da.The analysis assumes linear latent dynamics and initially considers invertible B with d_a = d.Source: §4.1 · p. 5, §4 · p. 5
  • With nonzero loss, CGS encourages temporal straightening, reduced drift–control coupling, and balanced action effects, though joint-training objectives may keep the loss nonzero.Source: §4.1 · p. 5

4.2 FROM FINITE-HORIZON PLANNING GEOMETRY TO FINITE-BUDGET MPPI

  • Under the CGS geometry A = I_d and B^⊤B = cI_da, the terminal planning Hessian has equal curvature across the active subspace of action-sequence perturbations constant across time.This is termed planning isotropy and makes population MPPI contract active directions equally under isotropic Gaussian proposals.Source: §4.2 · p. 5
  • For sufficiently small geometric deviations, the planning Hessian's effective condition number is bounded by (1 + ξ_H)/(1 − ξ_H).For fixed horizon and control-isotropy tolerance, CGS bounds drift–control coupling at O(ε), sharpening the Hessian perturbation from O_H(√ε) to O_H(ε) relative to temporal straightening.Source: §4.2 · pp. 5–6
  • With calibrated cost scale, CGS's more balanced curvature supports faster population MPPI contraction and higher one-step progress probability at the same sample budget in Cube probes.The finite-budget bound states that more updates suppress initial error and more samples reduce the sampling contribution, under its stated conditions.Source: §4.2 · p. 6, Figure 3 · p. 6
  • The same planning-Hessian projector geometry yields an approximately scalar local CEM mean update and gradient-descent planning-cost convergence bounds on leading controllable directions.Source: §4.2 · p. 6

5 MAIN RESULTS: PLANNING PERFORMANCE AND SAMPLE EFFICIENCY

CGS improves planning across multiple planners and environments, including under restricted sampling budgets. Its gains are planner- and environment-dependent, with a distinct pattern on Reacher.

5.1 EXPERIMENTAL SETUP

  • The evaluation uses four offline pixel-action environments, keeps base LeWM hyperparameters fixed, and varies only each added regularizer’s coefficient.MPPI and CEM use 30 optimization steps; GD uses 100 Adam steps and is independent of sampling budget K.Source: §5.1 · pp. 6–7

5.2 TOWARD EFFICIENT PLANNING WITH ACTION-STRUCTURED LATENT DYNAMICS

  • At K = 128, CGS improves over LeWM for all five planners on PushT, Cube, and TwoRooms; MPPI gains are 13.3 points on PushT and 20.0 on Cube.The table reports goal-reaching success rate and marks gains over LeWM in percentage points.Source: Table 1 · p. 7, §5.2 · p. 7
  • On PushT, CGS retains an MPPI advantage even with few samples; on Cube, its success reaches around 80% versus 60% for LeWM at K = 128–256.The Cube gap widens at larger sampling budgets.Source: §5.2 · p. 7
  • Reacher results vary by planner: at K = 128, CGS raises GD from 76.7% to 82.0% and MPPI from 65.3% to 66.7%, while CEM is slightly below LeWM.The passage attributes this pattern to fingertip motion depending strongly on current state beyond the executed action.Source: §5.2 · p. 8

6 CHARACTERIZING CGS: LATENT REPRESENTATIONS AND PLANNING

The section probes how latent differences encode physical motion and how CGS affects both motion representation and planning. The findings distinguish action-determined motion from state-dependent control effects.

Physical identification

  • On PushT, action alone nearly determines agent displacement (PA = PAS = 0.999); on Reacher, adding state raises R2 from −0.006 to 0.983.The result indicates that observed Reacher fingertip motion is only weakly specified by action alone.Source: §6 · p. 8

Motion decoding

  • On PushT, CGS raises linear motion-decoding R2 from 0.809 to 0.903 and nonlinear R2 from 0.888 to 0.959.On Reacher, nonlinear decoding retains nearly all motion information across representations, while linear scores vary more; TS reaches 0.974 and CGS 0.899.Source: §6 · p. 8

Control effects

  • Control-effect probes favor CGS on PushT (CA = 0.901, BAS = 0.902), while TS leads both measures on Reacher.The passage links the Reacher contrast to the same action producing different fingertip motion across configurations.Source: §6 · p. 9

CGS across representation architectures

  • On PushT, CGS improves planning with both DINO-WM CLS and Patch features, while Patch features plan better at substantially higher rollout cost than compact CLS features.LeWM jointly learns a compact global latent and predictive dynamics, retaining strong planning performance with cheaper repeated rollouts.Source: §6 · p. 9

Refinement efficiency and BC priors

  • CGS improves PushT success earlier in CEM/MPPI refinement and improves matched behavior-cloning priors before search begins.The priors achieve lower latent goal-distance ratios and higher prior-only success; Figure 6 separates refinement, prior distance, and prior-only success.Source: §6 · p. 9, Figure 6 · p. 9

7 CONCLUSION

The conclusion presents CGS as a local geometric regularizer that can shape full-horizon planning optimization without changing the planner at test time. The related work situates this objective among latent world models, planner-friendly representations, sampling-based MPC, and structural matching methods.

7 CONCLUSION

  • Across multiple tasks and planners, CGS improves planning success and refinement efficiency under limited budgets; extending it to state-dependent effects and richer dynamics remains a next step.The conclusion also identifies maintaining planner-friendly representations under distribution shift as a direction for future work.Source: §7 · p. 9

A.1 LATENT WORLD MODELS AND JEPA

  • JEPA and latent world-model work addresses predictive representations and collapse prevention, but accurate prediction and diverse features alone do not ensure an easily optimized planning objective.CGS adds a geometry objective applicable to end-to-end learned and pretrained representations.Source: §A.1 · p. 17

A.2 PLANNER-FRIENDLY REPRESENTATIONS

  • Prior approaches use local linear dynamics, differentiable planners, goal-derived supervision, or inverse-control signals; CGS instead matches action cosines to corresponding latent-difference cosines.Under the paper’s linear-dynamics assumptions, this relates temporal straightening and balanced action effects to terminal-cost curvature across the planning horizon.Source: §A.2 · pp. 17–18

A.3 SAMPLING-BASED MPC

  • Sampling-based MPC work studies proposal updates, sample reuse, and finite-budget convergence; the paper’s analysis connects representation geometry to finite-horizon curvature and planner guarantees.The reported guarantees include finite-budget results for MPPI, local contraction for CEM, and convergence bounds for gradient descent.Source: §A.3 · p. 18

A.4 RELATIONAL AND SIMILARITY-PRESERVING KNOWLEDGE DISTILLATION

  • Relational distillation matches structure across representations, whereas CGS uses observed actions as the reference geometry for organizing latent control effects.The analysis links this geometry to more balanced terminal-cost curvature and sampling-efficient optimization.Source: §A.4 · p. 18

B REFERENCED METHODS AND PLANNING ALGORITHMS

This material specifies the planning algorithms, temporal-straightening objective, experimental protocol, and additional sample-budget reporting. It describes how proposal updates and evaluation settings are kept comparable across representations and planners.

B.1 PLANNING ALGORITHMS

  • CEM refits a diagonal Gaussian to low-cost elites, while MPPI weights all candidates by cost and keeps its sampling standard deviation fixed.The algorithm description defines K as candidates per step and I as optimization steps; the candidate set includes the current proposal mean.Source: §B.1 · p. 19
  • Prior-guided planners initialize their proposal mean from a matched behavior-cloning model, whereas GD directly optimizes one action sequence through the frozen differentiable world model.The prior does not replace model-based refinement; GD uses Adam and is independent of sampling budget K.Source: §B.1 · pp. 19–20

B.2 TEMPORAL STRAIGHTENING

  • Temporal Straightening penalizes angular deviation between consecutive latent displacements, excluding pairs whose displacement norm is below 10^-6.The LeWM adaptation retains prediction and SIGReg terms; explicit TS can further reduce trajectory curvature, with clearer planning gains reported on Cube and TwoRooms.Source: §B.2 · p. 20

C.1.1 EXPERIMENT DETAILS

  • Experiments use four offline pixel-action environments and a shared LeWM backbone; each method’s checkpoint is fixed across planners and planning budgets.Evaluation uses a five-model-step horizon, executes 25 environment actions before replanning, and gives episodes at most two closed-loop solves.Source: §C.1.1 · p. 21

C.1.2 ADDITIONAL RESULTS

  • The sample-budget results report mean success rate and sample SD over three seeds, with 50 episodes per seed; planner-only timing excludes loading, environment stepping, and rendering.The nAUC metric trapezoidally integrates success over log2 K and divides by the horizontal span; the complete sweep covers K = 1–256.Source: §C.1.2 · pp. 21–23, Table 7 · p. 23

C.1.3 CUBE ABLATION OF REFERENCE GEOMETRY IN CGS.

The Cube ablation compares three action-reference geometries for CGS; grasp-active XYZ is strongest in most reported sweeps, while full-action geometry leads in one CEM sweep.

C.1.3 CUBE ABLATION OF REFERENCE GEOMETRY IN CGS.

  • Each Cube action has five dimensions, but the predictor receives a 25D block while reference geometries select either 15D XYZ channels or the full 25D block.The main setting uses grasp-active XYZ because XYZ commands most directly specify cube motion while grasped; grasp and arm-angle commands may not induce comparable object displacement.Source: §C.1.3 · pp. 23–24
  • The sample-budget sweep varies K = 1–256 and the refinement sweep varies I = 1–30, with nAUC summarizing each complete sweep.Higher nAUC is better; the reference is grasp-active XYZ15D, and both planners use no action prior.Source: Figure 7 · p. 24
  • Grasp-active XYZ has the highest nAUC for MPPI in both sweeps and for CEM in the refinement sweep; all-transition full-action leads for CEM in the sample-budget sweep.The full-action variant remains effective without reference selection, and Cube's grasp mask and XYZ reference use observed actions without extra state or task labels.Source: §C.1.3 · p. 24

C.1.4 STATE-INFORMED REFERENCE GEOMETRY AND OBSERVATION HISTORY ON REACHER

The Reacher ablation separates action-reference geometry from observation history, then examines how the learned representations organize action effects and motion. Results vary by planner and environment, with state-informed references particularly useful for MPPI and gradient descent when history supplies dynamical context.

C.1.4 STATE-INFORMED REFERENCE GEOMETRY AND OBSERVATION HISTORY ON REACHER

  • Transition-CGS uses observed joint-configuration displacements as a state-informed training reference, while action-CGS uses executed action blocks.Joint-angle labels construct the transition reference but are unavailable to the planner, which plans from images and actions; dataset-initialized history is a privileged-context diagnostic, unlike feasible online history after two model steps.Source: §C.1.4 · pp. 24–25
  • With dataset-initialized history, transition-CGS reaches 80.0% MPPI success at K = 128 versus 68.7% for LeWM and 72.7% for action-CGS; with online history it reaches 75.3% versus 69.3% and 68.7%.The benefit of a state-dependent reference becomes clearer for MPPI when short transition history makes current dynamics more observable, particularly at larger sampling budgets.Source: §C.1.4 · p. 26
  • CEM's K = 128 LeWM success rises from 83.3% with a single frame to 96.0% with dataset-initialized history and 92.7% with online history, leaving little headroom for mixed reference-variant gains.For gradient descent, transition-CGS reaches 85.3%, 94.0%, and 92.7% across single-frame, dataset-initialized, and online history, versus 76.7%, 84.0%, and 81.3% for LeWM.Source: §C.1.4 · p. 26

C.2.1 ACTION AND LATENT TRANSITION TRAJECTORIES

  • The trajectory visualization uses standardized five-step action blocks and latent differences, cumulatively sums their normalized directions, then projects aligned paths onto shared action-path principal components.CGS more faithfully follows action-reference turns than LeWM or TS on PushT and Cube; on Reacher the improvement is less consistent, while TwoRooms shows similar TS and CGS alignment.Source: §C.2.1 · p. 27

C.2.2 ACTION-PERTURBATION HEATMAPS AND PLANNING ISOTROPY

  • The perturbation heatmaps compare pairwise cosines of terminal latent responses to equal-norm circular action perturbations against an ideal unit-circle Gram matrix.CGS preserves the ideal bands more closely than LeWM and TS on PushT, Cube, and Reacher; TwoRooms shows similar TS and CGS patterns, but the diagnostic covers only a local tested plane, not full-space isotropy or the nonlinear residual Hessian.Source: §C.2.2 · p. 27

C.2.3 QUALITATIVE PUSHT PLANNING ROLLOUT

  • In a selected PushT rollout, LeWM fails to reach the target while LeWM+CGS succeeds, with decoded trajectories remaining qualitatively aligned to simulator evolution.The example illustrates task-relevant information retention for this rollout, not a population-level comparison.Source: §C.2.3 · p. 29, Figure 10 · p. 29

C.3.1 REPRESENTATION PROBES AMONG ENVIRONMENTS.

  • The physical-motion probes find Reacher fingertip displacement strongly configuration-dependent: action-only prediction has R2 = −0.006, while state-conditioned prediction reaches 0.983.Across representation probes, CGS improves several PushT and Cube motion-accessibility measures, but TS leads some object-agreement scores and is strongest on the reported TwoRooms diagnostics; Reacher motion remains accessible to nonlinear decoders but less linearly accessible from latent differences.Source: §C.3.1 · p. 32

C.4 MECHANISM-FIGURE PROTOCOL AND ADDITIONAL RESULTS

The mechanism figure tests how local Cube planning geometry relates to one-step MPPI progress. In a calibrated quadratic probe, CGS has lower population contraction and higher finite-sample progress probability than LeWM and TS.

C.4 MECHANISM-FIGURE PROTOCOL AND ADDITIONAL RESULTS

  • The probe perturbs XYZ commands across the 25-action horizon at 32 grasp-active anchors, using an orthonormal basis to map three-dimensional perturbations into the full action sequence.Trace normalization fixes average eigenvalue at one, enabling spectral-shape comparisons at a common scale.Source: §C.4 · pp. 32–33
  • The contraction comparison is calibrated to a three-dimensional quadratic probe, not the full nonlinear planning problem.The target lies along the weakest normalized XYZ-curvature direction; the measured random error ratio can exceed one.Source: §C.4 · p. 33
  • In the calibrated three-dimensional quadratic probe, at K = 32, LeWM, TS, and CGS had one-update progress probabilities of 75.1%, 80.3%, and 91.3%, with median q∞ values 0.841, 0.835, and 0.685.Source: §C.4 · p. 33

C.5 DINO-WM IMPLEMENTATION AND ADDITIONAL RESULTS

The appendix tests CGS across DINO-WM representations, training choices, MPPI temperatures, and refinement settings, and extends the theory with conditions linking CGS to transition geometry and planning curvature. Results vary by representation and planner, while refinement and prior analyses clarify where gains appear and how initialization interacts with search.

C.5 DINO-WM IMPLEMENTATION AND ADDITIONAL RESULTS

  • The DINO-WM comparisons use frozen DINOv2 features with either global CLS or spatial Patch representations, shared predictors and planning protocols, and trainable geometry components.The backbone stays frozen; the geometry objective is applied to global latent differences for CLS or an aggregation of patch-token differences for Patch.Source: §C.5 · pp. 33–34
  • On DINO-WM, CGS improves CLS results for CEM and GD on all four environments and MPPI on three; Patch results are more mixed.For Patch, CGS improves several planner–task combinations, but Base or TS is stronger in others.Source: §C.5 · p. 34, Table 13 · p. 34

C.6 TRAINING ABLATIONS

  • On PushT, the 5% ramp performs strongest overall across MPPI, CEM, and GD; SIGReg weight 0.03 substantially reduces success, while CGS-weight performance is broadly stable and declines at larger weights.The authors select λCGS = 0.01 for PushT; N = 3 is an economical history setting, with MPPI strongest at N = 3 and CEM and GD gaining slightly at N = 4.Source: Figure 11 · p. 35

C.6.1 CAN EXPLICIT TEMPORAL STRAIGHTENING FURTHER BENEFIT LEWM?

  • Explicit TS further raises PushT’s consecutive-displacement cosine from 0.548 to 0.613, while Cube and TwoRooms show larger straightening changes and planning gains.Reacher remains near zero cosine, with only a small additional straightening effect; the clearest planning gains are reported on Cube and TwoRooms.Source: §C.6.1 · p. 35

C.7 SHARED MPPI TEMPERATURE SELECTION

  • The shared MPPI temperatures are 4 on PushT, 64 on Cube, 128 on TwoRooms, and 64 on Reacher, chosen from broad stable regions where LeWM, TS, and CGS are competitive.Across temperature values, CGS shows stable improvement on PushT and Cube, TS and CGS comparable stable improvement on TwoRooms, and similar performance among all three on Reacher.Source: §C.7 · pp. 35–36

C.8 REFINEMENT EFFICIENCY AND MATCHED BC PRIORS ON OTHER ENVIRONMENTS

  • CGS improves both CEM and MPPI within the first few optimization steps on Cube; across refinement curves, gains are modest, nonmonotonic, and planner- and budget-dependent.On Reacher, CGS has the highest aggregate refinement nAUC for both planners; on TwoRooms, TS and CGS reach high success rapidly with little room for separation. On Cube, CGS also lowers the prior’s mean normalized goal-distance ratio from 0.262 for LeWM to 0.116, while on Reacher its matched prior starts weaker than LeWM’s and the differences narrow with refinement.Source: §C.8 · pp. 36, 38

D.1 FROM CGS LOSS TO TRANSITION GEOMETRY; D.1.1 PROOF OF PROPOSITION 4.2; D.1.3 COVERAGE AND DIMENSION ASSUMPTIONS

  • Under positive definite joint state–action coverage, zero CGS loss requires A = I and B⊤B = cI, and this geometry is attainable when d ≥ da.If joint second moments are singular, zero loss identifies geometry only on the data subspace and need not imply the separate block conditions or rollout conclusions; strong state–action correlation can also weaken the bounds by reducing the coverage lower bound.Source: §D.1.1 · p. 40, §D.1.3 · p. 41

D.1.2 PROOF OF COROLLARY 4.3; D.2 PLANNING HESSIAN AND DIMENSION CASES

  • CGS’s explicit mixed drift–control defect bound sharpens the planning-Hessian perturbation to O_H(ε), compared with the O_H(√ε) bound from temporal straightening.In the isotropic case, equal curvature across active action directions balances population MPPI contraction across those directions.Source: §D.1.2 · p. 40

D.2.1 PROOF OF THEOREM 4.4

The proofs bound how CGS shapes finite-horizon planning geometry and establish finite-sample, finite-update cost guarantees for MPPI and conditions for local contraction of the population CEM mean update under stated assumptions. They also identify limitations involving weakly conditioned extra controllable directions, nonlinear-rollout curvature, incomplete coordinate control, and the absence of a guarantee for full coupled mean–covariance contraction.

D.2.1 PROOF OF THEOREM 4.4

  • CGS bounds the normalized Gramian’s distance from the cumulative-action projector; directly controlling mixed drift–control error yields an O_H(ε) Hessian perturbation, versus generally unavoidable square-root order under the two weaker conditions.At H = 1, both bounds reduce to the control-isotropy error ε.Source: §D.2.1 · p. 42

D.2.2 SQUARE AND RECTANGULAR ACTION MAPS

  • For square and rectangular action maps, CGS bounds the leading da Gramian directions; guarantees cover all cost-relevant directions when rctrl = da, while additional directions need quantitative controllability for conditioning bounds.When rctrl > da, remaining positive eigenvalues can be arbitrarily weak even as the dominant-spectrum bound holds.Source: §D.2.2 · pp. 43–44

D.3.1 PLANNING COST AND NONLINEAR ROLLOUTS

  • For nonlinear terminal rollouts, a residual-weighted second-order Hessian term may be indefinite, so the analysis’s control-geometry guarantees focus on linear dynamics.The extra term can change the Hessian spectrum or introduce negative curvature.Source: §D.3.1 · p. 45

D.3.2 GAUSSIAN PROPOSALS AND UPDATE RULES

  • Proposal whitening makes the sampling-relevant Hessian depend on covariance; CEM analysis fixes the current covariance and does not claim coupled mean–covariance convergence.The MPPI and CEM bounds use isotropic covariance so Theorem 4.4 applies directly.Source: §D.3.2 · p. 45

D.3.3 A SHARED BOUND FOR REPEATED MEAN UPDATES

  • With probability at least 1 − ζ, repeated updates contract projected mean error up to sampling error while iterates remain within the stated exit condition; this does not bound exit probability or full-mean drift.The result requires a contracting population map and fresh-sample error control; changing dynamics, covariance, or replanning requires corresponding uniform bounds.Source: §D.3.3 · p. 46

D.4 MPPI: MEAN UPDATE AND PLANNING COST

  • MPPI’s population mean map is a resolvent contraction on the leading planning subspace, and its contraction depends on both Gramian shape and calibrated scale.CGS controls spectral spread but not c, so σ²_p and τ must make s = σ²_pcH/τ non-negligible.Source: §D.4.1 · p. 47

D.4.1 POPULATION MEAN UPDATE

  • On the leading eigenspace, MPPI’s contraction factor is bounded using ξH; for square action maps, this recovers the main-context result.The analysis uses the exact population map and its dominant-subspace spectral bounds.Source: §D.4.1 · p. 47

D.4.2 SAMPLING ERROR IN ONE UPDATE

  • MPPI’s finite-sample update admits a projected sampling-error bound, but coordinates outside the leading subspace still affect the importance-weight normalizer, which depends on nH − r.Combining this one-step bound with the repeated-update result gives a projected mean-error bound.Source: §D.4.2 · pp. 47–48

D.4.3 PLANNING COST GAP FOR MPPI

  • With probability at least 1 − ζ, finite-sample MPPI updates bound the planning-cost gap within the stated ranges, provided the update count does not exceed the exit time.The corollary supplies sufficient sample and update budgets for reaching the target cost gap under its conditions.Source: §D.4.3 · p. 48
  • When the retained subspace contains only the leading d_a eigenvectors, the MPPI argument bounds their cost contribution, but a full cost-gap bound requires control of all positive eigenvalues.The latter condition is provided by Corollary D.1.Source: §D.4.3 · p. 49

D.5.1 POPULATION MEAN UPDATE NEAR THE OPTIMUM

  • Near the optimum, the population CEM update has an exact local Jacobian and contracts the projected mean when ξ_H < 1/2 and β_α,r + ω_α,n_H,r(ξ_H) < 1.The top-r subspace is invariant; the ideal-projector contraction is adjusted by a continuity modulus, with no linear rate in ξ_H asserted.Source: §D.5.1 · pp. 49–50
  • CEM’s ideal mean contraction depends on α and r, while CGS controls deviation from a balanced active update; updating covariance requires a uniform proposal-whitened-matrix bound.The result does not assert a full coupled mean–covariance contraction.Source: §D.5.1 · p. 50

D.5.2 FINITE-SAMPLE AND REPEATED MEAN UPDATES

The appendix bounds finite-sample CEM mean updates and derives gradient-descent convergence under CGS geometry, with rates and conditions that depend on fixed-covariance and spectral assumptions.

D.5.2 FINITE-SAMPLE AND REPEATED MEAN UPDATES

  • The projected CEM mean update converges to its population map at the canonical K^-1/2 rate, up to logarithmic factors.The repeated-update analysis keeps proposal covariance fixed and bounds failure before exit; it does not condition on remaining in the full-mean neighborhood.Source: §D.5.2 · p. 51

D.6.1 CONVERGENCE FROM THE CGS SPECTRAL BOUNDS

  • Under the three ε-CGS bounds and ξH < 1, gradient descent with h = 1/(cH) satisfies the stated cost-convergence bound.The bound applies to the full planning cost when the active cost directions span the cost-bearing subspace.Source: §D.6.1 · pp. 51–52
  • When rctrl > da, the ξH factor controls only the dominant cost contribution; remaining positive eigenvalues can be arbitrarily small, and the experimental Adam optimizer needs separate analysis.A quantitative lower bound on all positive eigenvalues is needed to control convergence for the full cost.Source: §D.6.1 · p. 52

D.6.2 COMPARISON USING TEMPORAL STRAIGHTENING AND CONTROL ISOTROPY

  • In the square-action case, temporal-drift and control-isotropy bounds yield a conditioning comparison that is recovered under the ε-CGS bounds.The comparison assumes da = d, δA < 1, and δB < 1.Source: §D.6.2 · pp. 52–53

D.7 GEOMETRY UNDER OTHER TRAINING OBJECTIVES

Other training objectives jointly shape the encoder and predictor parameters that induce control geometry, so they may constrain how closely a model approaches ideal CGS geometry. Even with irreducible drift, the analysis identifies feasible control-map adjustments and corresponding planner trade-offs.

D.7 GEOMETRY UNDER OTHER TRAINING OBJECTIVES

  • Prediction and auxiliary objectives shape the encoder and predictor parameters jointly with CGS, but their preferred representations may differ from those minimizing CGS alone.A positive CGS loss floor arises when the other objectives keep the Gram defect uniformly away from zero; large temporal drift alone does not establish such a floor.Source: §D.7 · p. 53

D.7.1 EFFECT OF IRREDUCIBLE DRIFT ON THE CONTROL MAP

  • For fixed temporal drift, the defect-minimizing control map selects left singular directions from the eigenspaces with the da smallest drift eigenvalues.When the action map overlaps with non-removable temporal dynamics, CGS favors weaker drift directions; unequal drift eigenvalues can yield unequal positive singular values.Source: §D.7.1 · p. 54
  • Stronger drift lowers preferred action sensitivity along the corresponding latent direction, without generally determining whether the largest or smallest singular value changes more.The spectral spread can shrink mainly by suppressing strong action sensitivities while the weakest sensitivity changes little.Source: §D.7.1 · p. 54
  • Reducing excessive curvature can improve gradient-based conditioning more clearly than MPPI contraction, while CEM gains may remain limited.The analysis relates this distinction to Reacher, where straightening methods improve more under GD than under sampling-based planners.Source: §D.7.1 · p. 54
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