Convex-mixture graph Langevin optimizer

Integrate this into the optimizer update, not the forward pass. Select one or more large weight matrices, flatten a small subset of their scalar coordinates, and create a sparse parameter graph \mathcal{E}: connect neighboring coordinates in each output row, adjacent channels, or corresponding coordinates across two consecutive layers. Define edge differences d_e= heta_i- heta_j. Use a finite mixture of convex edge potentials, for example V_1(s)=s^2/2 and V_2(s)=\sqrt{s^2+\delta^2}-\delta+s^2/(2R), with \delta>0 and R chosen so V_2''(s)>0; sample component k independently for each training step from probabilities \pi_k. A concrete update is: (1) compute the minibatch gradient g=
abla_ heta\mathcal{L}; (2) compute the graph force r_i=\sum_{e=(i,j)}\partial_{ heta_i}V_k(d_e), with opposite signs at j; (3) set h=g+\beta r; (4) update heta\leftarrow heta-\eta h+\sqrt{2\eta T}\,\xi, where \xi\sim\mathcal{N}(0,I) and T is a small optimizer temperature. Estimate the component curvature bounds m_k and upper bounds C_k analytically from V_k; estimate the empirical Poincare proxy by measuring Var(F) divided by the graph Dirichlet energy over recent checkpoints. Do not modify Adam's moments initially; test SGD or AdamW plus the extra force and noise. Start with a 2-layer MLP on MNIST or CIFAR-10, using 1-5% of parameters in the graph, against AdamW with matched parameter-decay and injected Gaussian noise. The first success signal is lower run-to-run variance and fewer exploding-loss runs at the same learning rate; a stronger signal is improved validation accuracy or faster recovery after deliberately increasing the learning rate, without increasing wall-clock cost by more than 10%. Ablate graph coupling, mixture sampling, and Langevin noise separately.