Relaxed proximal message passing

Integrate this into a pipeline-parallel or federated training loop at the optimizer-state update, rather than modifying the forward network. Partition parameters into tree-owned blocks $u_i$, retain one lifted edge tensor $w_i$ per child, and retain a dual tensor $s_i$ for each parent-child consensus relation. When a parent block arrives, each worker immediately forms a prediction: compute a minibatch gradient $g_i=\nabla f_i(u_i)$, calculate $\tilde u_i=\operatorname{prox}_{\eta_i g_i}(u_{parent}^{new}+w_i^{old}-\eta_i g_i)$, and calculate $\tilde s_i=s_i^{old}+\rho_i(\tilde u_i-u_{parent}^{new})$. Update the edge memory using $w_i\leftarrow w_i-u_i^{old}+u_{parent}^{new}$. Apply separate relaxation, $u_i\leftarrow u_i+\theta_i(\tilde u_i-u_i)$ and $s_i\leftarrow s_i+\zeta_i(\tilde s_i-s_i)$, then send the relaxed block and correction vector $L_i^*(s_i^{new}-s_i^{old})$ downstream. Begin with $\theta_i=\zeta_i=1$ as the synchronous baseline, then sweep $(\theta,\zeta)$ over $(0.25,0.25)$, $(0.5,0.5)$, $(0.75,0.5)$, and $(1,0.5)$. Run one-message-in-flight pipeline execution and log gradient staleness, correction norms, consensus residuals, throughput, and validation loss. The paper supplies the prediction/relaxation structure; gradient minibatches and step sizes are empirical. First test on CIFAR-10 with a four-stage MLP or CNN, assigning one stage per worker and comparing ordinary synchronous pipeline SGD. Success means lower wall-clock time to a fixed validation loss, reduced worker idle time, and no divergence; specifically, dual relaxation $\zeta<1$ should reduce oscillations caused by delayed parent parameters without eliminating the throughput gain.