Metropolis Diffusion Regularizer

1. Integration point: apply this to a probability distribution already produced by a model, such as a graph-attention vector over N nodes, a token-to-token attention row, or an MoE router distribution over graph-organized experts. For a first experiment, use a graph neural network and regularize each node's outgoing attention distribution. Construct a sparse connected proposal matrix Q from the graph: set q_ij=1/deg(i) for graph neighbors j, optionally adding a self-loop. Choose g as uniform, the empirical node-frequency distribution, or a learned positive target clipped below by epsilon. 2. Precompute the sparse MH matrix P using the acceptance formula. During each training step, let F be a batch of attention rows, set F_t=F, and run T sparse multiplications: F_t = F_t @ P; accumulate D += beta[t] * sqrt(sum((F_t-g)^2/g)). Add lambda_diff * D to the task loss. Keep P detached initially; gradients flow through F and the softmax logits but not through the graph construction. 3. The paper-derived computation is the target-preserving MH kernel, the finite-horizon convergence discrepancy, and the spectral contraction interpretation. Estimate the mixing factor empirically as rho_hat=max_t norm(F_t-g)/norm(F_{t-1}-g), or estimate the second eigenvalue of sparse P by power iteration on vectors orthogonal to g. 4. First cheap experiment: train a 2-layer GAT on Cora or a synthetic stochastic-block-model node-classification dataset. Compare cross-entropy plus ordinary attention entropy, one-hop Laplacian smoothness, and this regularizer. Sweep T in {1,2,4,8} and lambda in {1e-4,1e-3,1e-2}. Success means improved validation accuracy or calibration at equal parameter count, fewer isolated-node routing failures, and lower attention-concentration variance under graph perturbations. Failure is accuracy degradation without a robustness or stability gain.