Observer-Corrected Robust Optimizer

1. Exact integration point: replace the velocity update in SGD with an observer-corrected update. Maintain parameters \(\theta_k\), optimizer velocity \(v_k\), an estimated slow disturbance \(\hat d_k\), and an optional local curvature regime \(\rho_k\). Use \(x_k=[\theta_k-\theta_{k-1};v_k]\); the absolute parameter offset is unnecessary, so the first component can be the parameter increment. Estimate \(A(\rho_k)\) online from the momentum equations or use a small set of fixed local linear models indexed by an EMA of gradient curvature. 2. Pseudocode: compute minibatch gradient \(g_k\); form the nominal update \(u_k^{\rm nom}\); predict \(\hat x^-_{k+1}=A(\rho_k)\hat x_k+B u_k^{\rm nom}+E\hat d_k\); measure \(y_{k+1}=x_{k+1}\) from the actual parameter and velocity transition; set \(r_{k+1}=y_{k+1}-C\hat x^-_{k+1}\); update \(\hat x_{k+1}=\hat x^-_{k+1}+Lr_{k+1}\) and \(\hat d_{k+1}=\hat d_k+L_d r_{k+1}\); apply \(u_{k+1}=u^{\rm nom}_{k+1}-K\hat x_{k+1}-E\hat d_{k+1}\). Clip \(\hat d\) to a robust norm bound. 3. Computed versus estimated: parameters, velocities, gradients, and transition residuals are directly measured; \(A\), \(L\), \(L_d\), and \(K\) are estimated or designed from a quadratic toy problem, local Hessian-vector products, or short system-identification windows. The paper contributes the observer/feedforward/robust-correction decomposition; the neural implementation should not assume the AUV-specific input matrix. 4. First cheap experiment: train a two-layer MLP on MNIST or CIFAR-10 with SGD-momentum, Adam, and the observer-corrected variant, while also running a two-dimensional quadratic with Hessian eigenvalues \(\lambda_1,\lambda_2\). Sweep learning rate \(h\) and observer gain. The falsifiable prediction is a sharp divergence boundary when the measured closed-loop spectral radius crosses one; the observed critical step size should agree with \(\rho_{\mathrm{spec}}(I+hA_c)=1\) within 20 percent. On a quadratic with injected slowly varying gradient bias, the corrected method should reduce low-frequency residual variance approximately in proportion to \(\|d-\hat d\|^2\), while high-frequency minibatch noise should not improve and may increase with excessive observer gain. Also report the break-even point: correction is beneficial only when \(\operatorname{RMS}(d-\hat d)<\operatorname{RMS}(d)\) after accounting for the extra control-update norm.