Idea description

Use the Gaussian trajectory predictor inside an inference-time planner or model-based reinforcement-learning policy, optimizing a nominal action sequence together with affine feedback gains against predicted disturbances. The resulting controller reacts to realized model residuals rather than relying on open-loop neural rollouts, while preserving a convex quadratic structure when the prediction map and covariance are frozen.

Formulas

y = Mw + b + e, e ~ N(0, Σe)

u = v + Ke, K[t,s] = 0 for s ≥ t

y = ȳ(v) + G(K)e, ȳ(v) = Mw w + Mv v + b, G(K) = I + Mu K

E[yᵀQy] = ȳᵀQȳ + tr(QG(K)ΣeG(K)ᵀ), E_r = {e : eᵀΣe⁻¹e ≤ r²}, r² = χ²[d_e, 1 − δ]

Mathematical statement

Let the learned trajectory model be y = Mw + b + e, where w contains context and future controls, M is fixed during one planning call, and e is Gaussian with covariance Σe. Parameterize controls by disturbance feedback u = v + Ke, where v is nominal control and K is strictly block lower triangular, ensuring causality. Then y = ȳ(v) + G(K)e, with ȳ(v) affine in v and G(K) affine in K. For positive semidefinite state cost Q, the expected quadratic cost equals ȳᵀQȳ + tr(QG(K)ΣeG(K)ᵀ), which is convex in v and K. A Gaussian high-probability uncertainty ellipsoid uses radius r² = χ²[d_e, 1 − δ].

Implementation notes

Integrate the method at inference time for a model-predictive-control policy, world-model agent, or trajectory optimizer. Train a neural dynamics model first, then collect validation residual trajectories e = y − ŷ over a fixed prediction horizon and estimate their joint covariance Σ̂e. At every environment step, freeze the network and covariance, construct the affine prediction y = ȳ(v) + G(K)e, and solve for nominal controls v and strictly causal gain blocks K. Use a quadratic tracking objective and action regularization. For safety constraints, either sample the ellipsoid E_r or use a second-order-cone robust approximation for linear state and action limits. Apply only the first action, observe the next state, compute the realized innovation relative to the nominal prediction, and update the remaining action sequence with u[t] = v[t] + K[t]e[0:t−1]. Pseudocode is: estimate Σ̂e; choose r = sqrt(χ²[d_e, 1 − δ]); solve min over v,K of expected cost subject to robust constraints for all e in E_r; execute the first feedback-corrected action. The affine disturbance-feedback construction and Gaussian radius are the paper mechanism; neural residual covariance, horizon, covariance inflation, and solver tolerances are empirical choices. The first cheap experiment is noisy cart-pole or pendulum control using a small GRU dynamics model, comparing open-loop shooting, nominal MPC, and disturbance-feedback MPC. The predicted signature is a covariance-coverage boundary: when Σ̄e = αΣ̂e is large enough to contain validation residuals, empirical constraint violations should remain near or below δ; below that inflation threshold, violations should increase sharply. Also verify numerically that the optimized quadratic objective has nonnegative curvature in v and K.

Verification

Mechanism confirmed, baseline not beaten

Stage 1 · Toy mechanism gate: Passed ✓

Stage 2 · Mechanism transferred to benchmark: Confirmed ✓

Stage 2 · Practical benchmark result: Significantly worse ✗

Methodology: 8 paired seeds, permutation test (p<0.05) against a learning-rate-tuned baseline. How verification works

Stage 1 — Mechanism check agent confidence 8/10

Built a lifted linear Gaussian disturbance-feedback controller with strictly causal gains, quadratic optimization, covariance inflation sweeps, chi-square coverage testing, and Monte Carlo verification of the expected-cost identity. The mechanism manifested: Hessian minimum eigenvalues stayed positive across covariance scales, stochastic cost scaled linearly with covariance, the chi-square ellipsoid achieved approximately 95% coverage at alpha=1, and the analytic quadratic expectation matched Monte Carlo closely. The secondary control signal was real but small: disturbance feedback reduced mean squared state error by about 0.11%.

Agent confidence: 8/10

Baseline: Open-loop nominal controller MSE 5.4075056915
Idea: Disturbance-feedback controller MSE 5.4016308242; 0.11% lower. Curvature minimum eigenvalue ranged from 2.0e-5 to 1.017e-3 over covariance scales 0–16; covariance-cost linear-fit maximum relative deviation was 1.71e-5; quadratic identity relative Monte Carlo error was 1.37e-4; calibrated ellipsoid coverage at alpha=1 was 0.94975.

Limitations: This is a small lifted linear toy system, not a trained GRU, nonlinear cart-pole, or full receding-horizon environment. It does not test robust safety constraints, online covariance estimation, model mismatch beyond sampled Gaussian residuals, computational speed, or long-horizon closed-loop stability.

How to run: python3 experiment.py

Stage 2 — Benchmark vs. tuned baseline (arXiv:2607.15949) Sep 2, 2026

Significantly WORSE than the tuned baseline (+682.4%, p=0.0081, wins 0 / 8 paired seeds).

Benchmark: Dynamics forecasting
Model: rnn_small
Paired seeds: 8
Baseline mean: 0.0005
Idea mean: 0.0043
Effect (Δ): 0.0037 (+682.4%; negative = idea better)
Wins: 0 / 8 paired seeds
p-value: 0.0081 (permutation test, 20 000 shuffles)
Smallest detectable effect: ±161.9%
Mechanism: Confirmed ✓
Practical effect: Hurts
Baseline tuning: swept over 3 configs

Limitations:

The built-in dynamics track is scalar supervised horizon prediction rather than a closed-loop environment rollout. The implementation uses a simplified causal residual correction from the observed final transition, not a full multi-step convex optimization over nominal controls and strictly lower-triangular gain matrices; robust safety constraints, covariance estimation from a separate validation residual trajectory set, and receding-horizon execution were not tested.

How to run: python3 stage2_official.py

Verdict computed by deterministic test code from paired-seed statistics — not by the language model.

Stage 2 — Benchmark Aug 30, 2026 latest Failed ✗

Agent confidence: 10/10

Implemented Gaussian disturbance-feedback inference as an inference-time residual correction around the official dynamics/rnn_small benchmark model, using the canonical bench.train_model path and identical systems otherwise. The mechanism signature held: innovation and forecast residuals correlated at 0.2264, confirming a measurable residual signal. Nevertheless, the best idea configuration significantly worsened MSE by +0.0037401 with p=0.0081, yielding verdict "idea worse (significant)"; therefore worked=false.

Baseline: Registered track dynamics, model rnn_small; tuned best lr=0.01; full 8-seed mean MSE 0.0005480458312
Comparison: delta_mean=+0.0037401357440, idea_wins=0/8, p_value=0.0081, verdict=idea worse (significant)
Idea: Best idea lr=0.01, gain=0.25; full 8-seed mean MSE 0.0042881815752
Mechanism signature: predicted innovation variance=0.0711328760, observed forecast residual variance=0.0005430718, correlation=0.2264259217, confirmed=true

Benchmark result