✗ Failed on benchmark 2026

Feedback-preconditioned recurrent dynamics

Implementation & benchmark of arXiv:2607.11835 — Stabilize-then-optimize: Feedback transformations as preconditioners in optimal control

Usefulness7/10

Difficulty5/10

Novelty5/10

Dynamics Stability Accuracy Speedup Rnn Ssm Training dynamics Control theory Dynamical systems Linear algebra Functional analysis

Source paper: Stabilize-then-optimize: Feedback transformations as preconditioners in optimal control arXiv:2607.11835 ⓘ · analyzed Aug 30, 2026

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 Not peer-reviewed.

Idea description

Reparameterize a recurrent or state-space layer so that its hidden-state update contains an explicit stabilizing feedback controller, while the neural network learns only a residual control in the feedback coordinates. Choose K to reduce finite-horizon state-propagation amplification, suppressing exploding hidden states and gradients on long sequences.

Formulas

$$[\widetilde{\mathcal A},-\mathcal B]\coloneqq[\mathcal A-\mathcal B\mathcal K,-\mathcal B]$$

$$\begin{bmatrix}x\\u\end{bmatrix}=\begin{bmatrix}I&0\\\mathcal K&I\end{bmatrix}\begin{bmatrix}x\\v\end{bmatrix},\qquad u=\mathcal Kx+v$$

$$\sigma_0=\|\mathcal C\mathcal A^{-1}\mathcal B\|=\|\mathcal A^{-1}\|=\begin{cases}\mathcal O(1)&a<1,\\\mathcal O(N)&a=1,\\\mathcal O(a^N)&a>1,\end{cases}$$

$$\mathcal T_N(A_{\mathrm{cl}},B)_{ij}=\begin{cases}A_{\mathrm{cl}}^{\,i-j}B,&i\ge j,\\0,&i<j,\end{cases}\qquad A_{\mathrm{cl}}=A+BK,\qquad u_t=Kh_t+v_t$$

Mathematical statement

The paper replaces the constraint operator [A,-B] by [A-BK,-B], where A is the state operator, B maps controls into the state equation, and K maps states to feedback controls. The coordinate substitution is u=Kx+v, so v is the residual control and the physical control is u. If A is surjective, the transformed constraint remains surjective and the transformed optimization problem is equivalent to the original. For the discrete scalar system x_{k+1}=a x_k+u_k, the original finite-horizon control-to-state matrix T_N has entries a^{i-j} for i>=j, and its norm is O(1) for a<1, O(N) for a=1, and O(a^N) for a>1. Under u_k=Kx_k+v_k, the closed-loop multiplier is a_cl=a+K for this update convention; in the paper's operator convention it is A-BK. For a matrix recurrence h_{t+1}=Ah_t+Bu_t, use A_cl=A+BK and the block map T_N(A_cl,B)_{ij}=A_cl^{i-j}B for i>=j. Choosing rho(A_cl)<1 makes long-horizon propagation bounded instead of exponentially amplified.

Implementation notes

Integrate this into the recurrent transition of a vanilla RNN or linear state-space model. Keep hidden dimension d and input map B, but replace h_{t+1}=phi(Ah_t+Bu_t+c_t) with h_{t+1}=phi(A_cl h_t+Bv_t+c_t), where A_cl=A+BK and v_t is produced by the existing neural input/nonlinear pathway. Initialize A from the baseline model, then choose K so A_cl has target spectral radius rho_target, for example by setting K=-A+rho_target Q with Q orthogonal, or learn K with a penalty on an estimated spectral radius. Use pseudocode: h=init(); for t in range(T): v=control_net(x[t],h); h=phi(A_cl@h+B@v+b); y=readout(h). If the original model exposes a physical control, reconstruct it as u_t=K h_t+v_t. Estimate amplification by power iteration on the implicit map from a length-N sequence v to the resulting states: forward-propagate T_N q, apply the transpose recurrence backward, normalize, and record the dominant singular-value estimate. The paper supplies the feedback transformation and block structure; empirical measurements are the singular-value estimate, hidden-state RMS, and gradient norms. First test a 1-2 layer tanh RNN on adding, copy-memory, and permuted-MNIST at lengths 128, 512, and 2048. Compare with the identical baseline, orthogonal initialization, and spectral-radius clipping. Success means bounded hidden and gradient RMS, fewer NaNs, faster loss decrease at equal FLOPs, and less accuracy degradation as sequence length increases.

Verification

Failed on benchmark

Stage 1 · Toy mechanism gate: Passed ✓

Stage 2 · Mechanism transferred to benchmark: Not reproduced ✗

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 9/10

Built a scalar feedback-preconditioned recurrent experiment with u=Kh+v, verifying numerically that the transformed multiplier is exactly a_cl=a+K with maximum error 4.4e-16. Finite-horizon operator sweeps confirmed the predicted stable, critical, and unstable regimes: observed terminal norm ratios were 1.015 versus predicted 1, 1.992 versus 2, and 445.815 versus 445.792. Forced-state RMS fell from 93,760 at a_cl=1.1 to 1.66 at 0.8 and 1.16 at 0.5; the tiny training test also reduced final loss and late gradient RMS, so the mechanism manifested.

Agent confidence: 9/10

Baseline: a_cl=1.08: final loss 140.8649, late gradient RMS 18693.37, initial loss 773004.40
Idea: a_cl=0.80: final loss 115.3906, late gradient RMS 9.27, initial loss 133.42; feedback shift error 4.4e-16; operator ratios stable/critical/unstable 1.015/1.992/445.815

Limitations: The mini-experiment used a scalar linear recurrence and an online sum task rather than nonlinear multidimensional RNNs, copy-memory, adding, or permuted-MNIST. It did not compare orthogonal initialization, spectral-radius clipping, FLOPs, accuracy across multiple sequence lengths, or multiple random seeds; the baseline and feedback models use different fixed recurrent multipliers by design.

How to run: python3 experiment.py

Stage 2 — Benchmark vs. tuned baseline (arXiv:2607.11835) Sep 2, 2026

Significantly WORSE than the tuned baseline (+134.5%, p=0.0081, wins 0 / 8 paired seeds).

Benchmark: Dynamics forecasting
Model: rnn_small_feedback
Paired seeds: 8
Baseline mean: 0.0016
Idea mean: 0.0038
Effect (Δ): 0.0022 (+134.5%; negative = idea better)
Wins: 0 / 8 paired seeds
p-value: 0.0081 (permutation test, 20 000 shuffles)
Smallest detectable effect: ±47.9%
Mechanism: Not confirmed ✗
Practical effect: Hurts
Baseline tuning: swept over 9 configs

Limitations:

Only the registered dynamics track was tested. No comparison against the built-in GRU implementation, longer horizons, orthogonal initialization, spectral-radius clipping, or larger training budgets was run; the local intervention uses a fixed orthogonal A_cl rather than jointly learning K.

How to run: python3 bench_feedback.py

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

Stage 2 — Benchmark Aug 31, 2026 latest Failed ✗

Agent confidence: 9/10

Implemented and ran feedback-preconditioned recurrent dynamics on the registered built-in dynamics track with 8 paired seeds, canonical train_model training, and a tuned baseline sweep. The best baseline achieved MSE 0.0016241 versus 0.0038084 for the best feedback model; paired delta was +0.0021843 with permutation p=0.0081, so the idea was significantly worse. The trained-model signature measured the target operator norm but did not confirm the finite-horizon quantitative prediction.

Baseline: best_cfg={lr:0.006,rho:0.8}; mean=0.0016240773547906429; per_seed=[0.0018814491340890527,0.0011060595279559493,0.0010844746138900518,0.0026348030660301447,0.0019145066617056727,0.0013607223518192768,0.0019727598410099745,0.0010378436418250203]
Idea: best_cfg={lr:0.006,rho:0.6}; mean=0.003808425914030522; per_seed=[0.0037844234611839056,0.0025165914557874203,0.004247678443789482,0.006615340709686279,0.004185421857982874,0.0029389068949967623,0.003790728747844696,0.0023883157409727573]

Benchmark result