# Gaussian Disturbance-Feedback Inference

- ID: 989
- Canonical URL: https://synthcore.org/idea/989/gaussian-disturbance-feedback-inference
- API JSON: https://synthcore.org/api/idea/989.json
- API Markdown: https://synthcore.org/api/idea/989.md
- Verification status: mech_ok_no_baseline
- Source: [arXiv:2607.15949](https://arxiv.org/abs/2607.15949)
- Category: dynamics
- Solves: stability, accuracy, generalization
- ML areas: rl, world-model, optimizer, ssm
- Math tags: probability, statistics, control-theory, optimization, linear-algebra
- Ratings: usefulness 8/10; difficulty 6/10; novelty 5/10

## 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.

## 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].

## Key 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]

## 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

- Status: mech_ok_no_baseline
- Mechanism evidence: yes
- Mechanism confirmed: yes
- Practical verdict: harms
- Verdict: 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%.

### Mechanism check

- Verdict: 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%.
- Confidence: 8/10
- 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.

### Practical benchmark

- Paired seeds: 8
- Baseline mean: 0
- Idea mean: 0
- p-value: 0.0081
- Paired wins: 0/8
- Benchmark verdict: idea worse (significant)

## Artifacts

- [bench_report.json](https://synthcore.org/code/178/bench_report.json)
- [experiment.py](https://synthcore.org/code/178/experiment.py)
- [report.md](https://synthcore.org/code/178/report.md)
- [report_bench_2026-08-30T193848.md](https://synthcore.org/code/178/report_bench_2026-08-30T193848.md)
- [results.json](https://synthcore.org/code/178/results.json)
- [stage2_bench.py](https://synthcore.org/code/178/stage2_bench.py)
- [stage2_official.py](https://synthcore.org/code/178/stage2_official.py)
- [Download all files as ZIP](https://synthcore.org/download/178)

## Disclaimer

AI-generated research hypothesis, automatically tested. Not peer-reviewed.