Information-aware finite-horizon optimizer

1. Integration point: wrap the controller around SGD at the minibatch-update level. Let $\theta_n$ be network parameters, $g_n$ the minibatch gradient, $\mu_n$ an exponential moving average of gradients, and $\Sigma_n$ a diagonal exponential moving average of $(g_n-\mu_n)^2$. Use a short macro-horizon of $N$ updates and a target checkpoint $\theta_\star$, such as the best validation checkpoint or a scheduled endpoint. Candidate actions are step sizes $\eta$ from a small discrete set, with $a_n=-\eta\mu_n$. 2. Pseudocode: construct a grid over normalized distance $d_n$ and uncertainty $u_n=\operatorname{mean}(\Sigma_n)$. Set terminal values $V_N(d,u)=\Phi(d)$, for example $\Phi(d)=d^2$. Perform backward dynamic programming over grid states and candidate step sizes. For each candidate, estimate the next state using $\theta' = \theta-\eta\mu$, $d'=\|\theta'-\theta_\star\|$, and the EMA covariance update; evaluate transport cost, information gain $\lambda_I[\log\det\Sigma-\log\det\Sigma']$, fixed cost $\kappa$, and future value. During training, quantize the observed $(d_n,u_n)$ and apply the stored minimizing $\eta$. 3. Computed versus estimated: the equilibration-to-belief-state reduction, Bellman structure, and intervention threshold are transferred from the paper; gradient moments, transition noise, $\tau$, $\lambda_I$, and $\kappa$ are empirical hyperparameters. 4. First cheap experiment: train a two-layer MLP on MNIST with SGD, comparing fixed learning rate and cosine decay against this controller at equal minibatch budgets. Use $N=20$ stages, 8 candidate learning rates, and a 16-by-16 state grid. Sweep $\kappa$. The falsifiable prediction is a continuation boundary: the controller should stop or reduce interventions when measured expected loss reduction per update falls below $\kappa$, and the selected update count should decrease monotonically with $\kappa$. High-variance early states should receive larger steps than low-variance late states. The empirical continuation boundary should predict held-out one-step loss reduction within 20%.