Vector-Lyapunov Optimizer Controller

1. Exact integration point: augment Adam or momentum SGD with a scalar control that rescales the nominal parameter update. Let $V_1$ be a normalized loss gap, $V_2=\|m_t\|^2$ be momentum energy, and define the observable optimizer state as $x=(\theta,m)$. Use the nominal Adam step as $f$ and the scalar rescaling or damping correction as $u$. 2. Pseudocode: maintain EMA estimates of $V_1$ and $V_2$; estimate $\dot V$ and $\ddot V$ from consecutive minibatches or a differentiable one-step lookahead; set $\mathbf V=(V,\dot V)$; choose $\Lambda=[[-\lambda_1,1],[-\lambda_2,-\lambda_3]]$ with positive gains and verify negative-real-part eigenvalues; compute $a=\ddot V-[-\lambda_2,-\lambda_3]\mathbf V$ and $b=\partial\ddot V/\partial u$; apply $u=-(a+\sqrt{a^2+b^4})b/(b^2+\varepsilon)$; clip $u$ to a trust region and update parameters with $u$ times the nominal step. 3. Taken from the paper: the vector comparison inequality, explicit feedback, and Hurwitz criterion. Estimated empirically: Lyapunov derivatives, control sensitivity $b$, minibatch noise, and the observed decay slope. 4. First experiment: train a small MLP on MNIST using SGD-momentum, Adam, and controlled Adam across learning rates and batch sizes. The predicted boundary is that the discrete comparison system becomes unstable when $\rho(I+\Delta t\Lambda)\ge1$, where $\rho$ is spectral radius and $\Delta t$ is the optimizer step interval. Controlled runs should remain stable below estimation error. In the stable regime, $V_1+V_2$ should decay approximately exponentially, with measured log-decay slope agreeing within 20 percent of $\lambda_{\max}(\bar\Lambda)$.