# Composed Trusted Reachable Families for Recurrent Networks

- ID: 2718
- Canonical URL: https://synthcore.org/idea/2718/composed-trusted-reachable-families-for-recurrent-networks
- API JSON: https://synthcore.org/api/idea/2718.json
- API Markdown: https://synthcore.org/api/idea/2718.md
- Verification status: failed_benchmark
- Source: [arXiv:2608.24019](https://arxiv.org/abs/2608.24019)
- Category: dynamics
- Solves: stability, accuracy, scalability
- ML areas: rnn, ssm, world-model, training-dynamics
- Math tags: control-theory, dynamical-systems, geometry, linear-algebra, numerical-analysis
- Ratings: usefulness 7/10; difficulty 7/10; novelty 8/10

## Idea description

Apply the paper's compositional PAS idea to recurrent or state-space networks by propagating a polytope of possible hidden states and input perturbations over multiple time blocks. Instead of validating one hidden trajectory at a time, maintain a trusted convex family and re-linearize only when its nonlinear-fidelity tolerance is exceeded. This creates a runtime monitor and adaptive horizon mechanism for long-sequence inference, forecasting, and learned world models.

## Mathematical statement

For a recurrent map \(h_{k+1}=f_\phi(h_k,u_k)\), let \((h_k^*,u_k^*)\) be a nominal rollout and let \(\gamma\) parameterize perturbations to initial state and inputs. The affine family is \(h_k(\gamma)=h_k^*+R_k\gamma\), \(u_k(\gamma)=u_k^*+U_k\gamma\), with \(R_{k+1}=A_kR_k+C_kU_k\), where \(A_k=\partial f_\phi/\partial h\) and \(C_k=\partial f_\phi/\partial u\), evaluated on the nominal rollout. For \(\mathcal{A}=\{\gamma:G\gamma\le h\}\), the predicted reachable hidden set is \(\mathcal{H}_k=\{h_k^*+R_k\gamma:\gamma\in\mathcal{A}\}\). Define one-step nonlinear violation by \(v_k(\gamma)=\|f_\phi(h_k^*+R_k\gamma,u_k^*+U_k\gamma)-h_{k+1}^*-R_{k+1}\gamma\|_2\), and retain points satisfying \(\max_{k<H}v_k(\gamma)\le\varepsilon\). The terminal trusted polytope is composed with the next block and the map is re-linearized when its volume or fidelity becomes insufficient.

## Key formulas

- $$h_{k+1}=f_\phi(h_k,u_k),\qquad h_k(\gamma)=h_k^*+R_k\gamma,\qquad u_k(\gamma)=u_k^*+U_k\gamma.$$
- $$R_{k+1}=A_kR_k+C_kU_k,\qquad A_k=\left.\frac{\partial f_\phi}{\partial h}\right|_{(h_k^*,u_k^*)},\quad C_k=\left.\frac{\partial f_\phi}{\partial u}\right|_{(h_k^*,u_k^*)}.$$
- $$\mathcal{H}_k=\{h_k^*+R_k\gamma\mid \gamma\in\mathcal{A}\},\qquad \mathcal{A}=\{\gamma:G\gamma\le h\}.$$
- $$v(\gamma)=\max_{0\le k<H}\left\|f_\phi(h_k^*+R_k\gamma,u_k^*+U_k\gamma)-h_{k+1}^*-R_{k+1}\gamma\right\|_2,$$

## Implementation notes

Use this as an inference-time monitor or training regularizer for an RNN, GRU, neural ODE discretization, or diagonal state-space model. Select a nominal input sequence and compute a block rollout \((h_k^*,u_k^*)\) of length \(H\). Let \(\gamma\) contain perturbations to the initial hidden state and a low-rank set of input directions, initialized as a box \(|\gamma_i|\le r_i\). Compute \(A_k\) and \(C_k\) with automatic differentiation or Jacobian-vector products, then recursively form \(R_k\). Evaluate the true recurrence at polytope vertices, random interior points, and adversarial points found by maximizing observed violation. Shrink the polytope until probes satisfy \(v(\gamma)\le\varepsilon\); use the accepted terminal polytope as the next block's initial uncertainty set, or re-center and re-linearize when its volume becomes too small. During training, add \(\lambda\max(0,v_{\max}-\varepsilon)^2\) to the task loss. At inference, trigger re-linearization, shorter blocks, or conservative input selection when trusted volume collapses. First test a small GRU on permuted-MNIST or a nonlinear state-space forecasting dataset against ordinary truncated BPTT and an unmonitored GRU at equal compute. The predicted signature is a critical horizon \(H_c(r,\varepsilon)\): below it, affine multi-step error stays under \(\varepsilon\); above it, error increases rapidly. Re-centering should extend the usable horizon and produce repeatable boundaries across sequences.

## Verification

- Status: failed_benchmark
- Mechanism evidence: yes
- Mechanism confirmed: no
- Practical verdict: inconclusive
- Verdict: Built a scalar nonlinear recurrent trusted-reachable-family monitor with affine Jacobian propagation, box probing, tolerance-based horizon checks, and adaptive radius selection. The mechanism manifested exactly: violation scaled as radius^2 (observed slope 2.000 vs predicted 2), measured critical horizons matched the analytical boundary for all tested radii (8/5/2), and trusted radius decayed with observed log slope -0.22314 per step, matching -log(1.25). The monitor reduced H=10 violation from 0.02665 at fixed radius 0.04 to 0.00100 at radius 0.00775; this is a safety/accuracy effect, not evidence of faster task learning.

### Mechanism check

- Verdict: Built a scalar nonlinear recurrent trusted-reachable-family monitor with affine Jacobian propagation, box probing, tolerance-based horizon checks, and adaptive radius selection. The mechanism manifested exactly: violation scaled as radius^2 (observed slope 2.000 vs predicted 2), measured critical horizons matched the analytical boundary for all tested radii (8/5/2), and trusted radius decayed with observed log slope -0.22314 per step, matching -log(1.25). The monitor reduced H=10 violation from 0.02665 at fixed radius 0.04 to 0.00100 at radius 0.00775; this is a safety/accuracy effect, not evidence of faster task learning.
- Confidence: 9/10
- Limitations: Only a scalar quadratic recurrence with zero nominal input and initial-state uncertainty was tested; input-direction uncertainty, multidimensional polytopes, GRUs, composed block re-centering, adversarial probes, training-time regularization, wall-clock overhead, and downstream forecasting/classification accuracy were not tested.

### Practical benchmark

- Paired seeds: 8
- Baseline mean: 0
- Idea mean: 0
- p-value: 0.2468
- Paired wins: 4/8
- Benchmark verdict: no significant win

## Artifacts

- [bench_report.json](https://synthcore.org/code/1050/bench_report.json)
- [report.md](https://synthcore.org/code/1050/report.md)
- [report_bench_2026-09-01T195600.md](https://synthcore.org/code/1050/report_bench_2026-09-01T195600.md)
- [results.json](https://synthcore.org/code/1050/results.json)
- [stage2_bench.py](https://synthcore.org/code/1050/stage2_bench.py)
- [trusted_reachable_rnn.py](https://synthcore.org/code/1050/trusted_reachable_rnn.py)
- [Download all files as ZIP](https://synthcore.org/download/1050)

## Disclaimer

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