# Adaptive SOS Lyapunov Certificate Ladder

- ID: 2769
- Canonical URL: https://synthcore.org/idea/2769/adaptive-sos-lyapunov-certificate-ladder
- API JSON: https://synthcore.org/api/idea/2769.json
- API Markdown: https://synthcore.org/api/idea/2769.md
- Verification status: mechanism_failed
- Source: [arXiv:2608.27743](https://arxiv.org/abs/2608.27743)
- Category: dynamics
- Solves: stability, generalization
- ML areas: optimizer, training-dynamics, regularization, rnn, ssm
- Math tags: optimization, convex-analysis, linear-algebra, dynamical-systems, control-theory
- Ratings: usefulness 8/10; difficulty 7/10; novelty 7/10

## Idea description

Represent a small neural state-update map or optimizer update by polynomial constraints and certify decrease of a polynomial Lyapunov function on the nonnegative activation or state region using successive Parrilo SOS levels. Use the monotone shift-threshold construction to distinguish genuine instability from failure of a weak certificate, and raise the SOS level only when necessary.

## Mathematical statement

For a symmetric matrix A, Parrilo's level-r cone is \(\mathcal{K}^{(r)}_n=\{A:(\sum_i x_i^2)^r(x^{\circ 2})^TA(x^{\circ 2})\text{ is SOS}\}\), where \(x^{\circ 2}=(x_1^2,\ldots,x_n^2)\). The cones satisfy \(\mathcal{K}^{(r-1)}_n\subseteq\mathcal{K}^{(r)}_n\). Given an interior direction D, the threshold \(\varepsilon_r(M)=\inf\{\varepsilon\geq0:M+\varepsilon D\in\mathcal{K}^{(r)}_n\}\) is nonincreasing in r. For a neural update \(z^+=F_\theta(z)\) and Lyapunov polynomial \(V(z)\), define \(q_\theta(z)=V(z)-V(F_\theta(z))-\alpha\|z\|^{2d}\), with \(\alpha>0\). Certify decrease by requiring \((\sum_i z_i^2)^r q_\theta(z)\) to be SOS, represented by a positive-semidefinite Gram matrix \(Q_r\) satisfying \((\sum_i z_i^2)^r q_\theta(z)=m(z)^TQ_rm(z)\), where \(m(z)\) is the monomial vector of half the polynomial degree.

## Key formulas

- $$\mathcal{K}^{(r)}_n=\left\{A\in\mathcal{S}^n:\left(\sum_{i=1}^n x_i^2\right)^r (x^{\circ 2})^T A(x^{\circ 2})\ \text{is SOS}\right\},\qquad \mathcal{K}^{(r-1)}_n\subseteq\mathcal{K}^{(r)}_n.$$
- $$\varepsilon_r(M)=\inf\left\{\varepsilon\geq 0:M+\varepsilon D\in\mathcal{K}^{(r)}_n\right\},\qquad \varepsilon_r(M)\geq\varepsilon_{r+1}(M).$$
- $$q_\theta(z)=V(z)-V(F_\theta(z))-\alpha\|z\|^{2d},\qquad \left(\sum_{i=1}^n z_i^2\right)^r q_\theta(z)=m(z)^TQ_rm(z),\quad Q_r\succeq0.$$
- $$V(F_\theta(z))-V(z)\leq-\alpha\|z\|^{2d}\quad\Longrightarrow\quad V(z_t)\text{ decreases along certified trajectories}.$$

## Implementation notes

1. Integration point: start with a two- or three-dimensional polynomial RNN or a ReLU RNN whose activation regions are fixed during certification. Use \(z_{t+1}=F_\theta(z_t)\), choose \(V(z)=z^TPz\) with \(P\succ0\), and certify on \(z\geq0\) and \(\|z\|_2\leq R\). For ReLU networks, certify each active linear region or replace ReLU locally by a polynomial approximation. 2. Pseudocode: train for K steps; freeze \(\theta\); construct coefficients of \(q_\theta(z)=V(z)-V(F_\theta(z))-\alpha\|z\|^{2d}\); for r=0,1,2,... solve the SDP for \(Q_r\succeq0\), adding SOS multipliers for box constraints; accept the update if all Gram matrices have minimum eigenvalue above tolerance \(-\tau\). If infeasible, reduce the optimizer step, increase stability regularization, or try r+1. Estimate the threshold by binary-searching the smallest \(\varepsilon\) such that \(q_\theta+\varepsilon\|z\|^{2d}\) is certified at level r. 3. Computed quantities are SDP feasibility, Gram eigenvalues, and \(\varepsilon_r\); polynomial approximation error, region coverage, and numerical tolerance are empirical estimates. 4. First experiment: train a two-dimensional RNN on MNIST sequences, comparing SGD, spectral normalization, and SOS-controlled training. Sweep recurrent weights across an empirical rollout-divergence boundary. The prediction is \(\varepsilon_r\) decreases monotonically with r, and certified models exhibit geometric Lyapunov decay \(V(z_t)\lesssim(1-c)^tV(z_0)\). The first feasible level should predict the empirical stability boundary within 20%, while robust certificates should persist under small parameter perturbations.

## Verification

- Status: mechanism_failed
- Mechanism evidence: yes
- Mechanism confirmed: no
- Verdict: Built a small Parrilo-style SOS Lyapunov certificate ladder with coefficient-matching Gram matrices, PSD projection, shift-threshold sweeps, and a scalar rollout boundary check. The exact dynamics prediction gamma|a|=1 was recovered within 0.44% relative error, and the coarse threshold ordering was nonincreasing, but the approximate Gram solver produced large coefficient residuals and level 1 did not improve the certificate margin (-9.624 versus -6.531 at level 0). Therefore the central adaptive SOS mechanism was not numerically demonstrated and should not be considered a win.

### Mechanism check

- Verdict: Built a small Parrilo-style SOS Lyapunov certificate ladder with coefficient-matching Gram matrices, PSD projection, shift-threshold sweeps, and a scalar rollout boundary check. The exact dynamics prediction gamma|a|=1 was recovered within 0.44% relative error, and the coarse threshold ordering was nonincreasing, but the approximate Gram solver produced large coefficient residuals and level 1 did not improve the certificate margin (-9.624 versus -6.531 at level 0). Therefore the central adaptive SOS mechanism was not numerically demonstrated and should not be considered a win.
- Confidence: 9/10
- Limitations: No production SDP solver was available, so the Gram feasibility step used an approximate alternating-projection method with poor residuals. The neural/RNN experiment, constrained nonnegative-region multipliers, robust perturbation tests, training comparison, and geometric Lyapunov-decay evaluation were not tested.

## Artifacts

- [report.md](https://synthcore.org/code/1061/report.md)
- [results.json](https://synthcore.org/code/1061/results.json)
- [sos_ladder.py](https://synthcore.org/code/1061/sos_ladder.py)
- [Download all files as ZIP](https://synthcore.org/download/1061)

## Disclaimer

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