# Robust Lyapunov Training Under Model Error

- ID: 2824
- Canonical URL: https://synthcore.org/idea/2824/robust-lyapunov-training-under-model-error
- API JSON: https://synthcore.org/api/idea/2824.json
- API Markdown: https://synthcore.org/api/idea/2824.md
- Verification status: unverified
- Source: [arXiv:2608.29303](https://arxiv.org/abs/2608.29303)
- Category: dynamics
- Solves: stability, generalization
- ML areas: rl, world-model, regularization, training-dynamics
- Math tags: control-theory, dynamical-systems, probability, optimization
- Ratings: usefulness 7/10; difficulty 6/10; novelty 7/10

## Idea description

Require Lyapunov decrease not only under the nominal learned transition, but throughout a bounded uncertainty set around that transition. The policy is therefore optimized against identification error and distribution shift rather than trusting a potentially overconfident world model.

## Mathematical statement

Let the true transition be $f(x,u)=\hat f_{\phi}(x,u)+e(x,u)$, where the unknown model error satisfies $\|e(x,u)\|_{2}\leq\delta(x,u)$. Let $L_{V}(z)$ be a bound on the local Lipschitz constant of $V_{\psi}$, meaning $|V_{\psi}(z+e)-V_{\psi}(z)|\leq L_{V}(z)\|e\|_{2}$. A sufficient robust decrease condition is $V_{\psi}(\hat f_{\phi}(x,\pi_{\theta}(x)))-V_{\psi}(x)+L_{V}(\hat f_{\phi}(x,\pi_{\theta}(x)))\delta(x,\pi_{\theta}(x))\leq-c_{2}\|x-x^{\star}\|_{2}^{2}$. The uncertainty term is the worst-case increase in Lyapunov energy caused by identification error. The certificate is meaningful only where the residual bound $\delta$ is calibrated on held-out transitions or obtained from a statistical confidence bound.

## Key formulas

- $$f(x,u)=\hat f_{\phi}(x,u)+e(x,u),\qquad \|e(x,u)\|_{2}\leq\delta(x,u),$$
- $$|V_{\psi}(z+e)-V_{\psi}(z)|\leq L_{V}(z)\|e\|_{2},\qquad z=\hat f_{\phi}(x,\pi_{\theta}(x)),$$
- $$V_{\psi}(z)-V_{\psi}(x)+L_{V}(z)\delta(x,\pi_{\theta}(x))\leq-c_{2}\|x-x^{\star}\|_{2}^{2},$$
- $$\mathcal{L}_{\mathrm{rob}}=\mathcal{L}_{\mathrm{task}}+\lambda\,\operatorname{softplus}\!\left(\Delta V+L_{V}(z)\delta(x,u)+c_{2}\|x-x^{\star}\|_{2}^{2}\right).$$

## Implementation notes

1. Integration point: place the robust penalty in model-based RL or closed-loop sequence training after the learned dynamics model predicts the next state. Use an ensemble of dynamics models or residual statistics to estimate uncertainty. For each replay transition, compute $z=\hat f_{\phi}(x,u)$ and a residual radius $\delta(x,u)$; compute the Lyapunov decrease and add the uncertainty margin before backpropagation into the policy and Lyapunov networks. 2. Pseudocode: fit $M$ dynamics ensembles on the training split; at each batch, evaluate their next-state predictions, set $\delta(x,u)$ to a high quantile such as the empirical 95th-percentile ensemble deviation plus held-out residual error, compute $L_{V}(z)$ either by automatic differentiation $\|\nabla V(z)\|_{2}$ or by a local finite-difference maximum, and minimize the displayed robust loss. Initially stop gradients through $\delta$ to prevent the policy from gaming the uncertainty estimator. Periodically evaluate the robust inequality on held-out transitions. 3. The robust inequality and Lipschitz inflation are the analytic mechanism; $\delta$ and $L_{V}$ are estimated quantities. Calibrate $\delta$ so that at least 95% of held-out residuals fall inside the claimed ball. 4. First cheap experiment: use a two-dimensional nonlinear stabilization system, train with 20%, 50%, and 100% of the transition data, and compare ordinary Lyapunov regularization with uncertainty inflation. Measure the minimum robust margin $m=\min_x[-\Delta V-L_{V}\delta-c_{2}\|x-x^{\star}\|^{2}]$ on held-out states and the fraction of real rollouts that converge. The quantitative prediction is that the robust policy maintains convergence as data are removed until $m$ crosses zero; empirical failures should concentrate near this predicted boundary, while the nominal certificate can remain positive on the learned model but fail on real transitions.

## Disclaimer

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