Mean-Square Proximal Relaxation Optimizer

1. Integration point: replace the standard optimizer update in a small or medium neural network with blockwise updates. Blocks can be attention projections, MLP layers, convolutional filters, or expert parameters. For each block, compute a minibatch loss while holding the other blocks fixed, take one or a few proximal-gradient steps toward the block response, and then apply global Krasnoselskii relaxation. A cheap approximation to the proximal response is T_hat_n=w_n-r_n g_hat_n; use the exact proximal subproblem only for small blocks. 2. Pseudocode: sample a minibatch of size B_k; compute each block gradient g_hat_n; set T_hat_n=w_n-r_n g_hat_n; form w_n <- (1-alpha_k)w_n+alpha_k T_hat_n. Estimate gradient-noise variance from two independent half-minibatches, sigma_hat_k^2, and increase B_k or decrease alpha_k when alpha_k^2 sigma_hat_k^2/B_k exceeds a chosen fraction of (1-rho_hat_k^2)||w-w_ema||^2. 3. Computed from the paper's mechanism: proximal response, variable sample size, relaxation, and mean-square Lyapunov recursion. Estimated empirically: contraction q, noise variance sigma_k^2, Jacobian J_k, and optional matrix P. Estimate q from ratios of response differences on two nearby parameter vectors, and solve a small semidefinite program for P using sampled low-dimensional block Jacobians. 4. First experiment: train a 3-layer MLP on MNIST and a small Transformer on Shakespeare, comparing Adam, unrelaxed proximal-gradient, and this method at matched examples processed. Sweep alpha and B. The predicted signature is a sharp rise in long-run parameter variance and occasional loss divergence when rho^2+alpha^2 sigma^2/(B||w-w^star||^2) approaches 1. The empirical stability boundary should move toward smaller alpha as B decreases, approximately following B proportional to alpha^2 divided by 1-[1-alpha(1-q)]^2, with qualitative agreement and preferably 20 percent relative error.