# Relaxed proximal message passing

- ID: 1131
- Canonical URL: https://synthcore.org/idea/1131/relaxed-proximal-message-passing
- API JSON: https://synthcore.org/api/idea/1131.json
- API Markdown: https://synthcore.org/api/idea/1131.md
- Verification status: unverified
- Source: [arXiv:2607.18932](https://arxiv.org/abs/2607.18932)
- Category: dynamics
- Solves: speedup, stability, scalability
- ML areas: optimizer, training-dynamics, federated, inference-speedup
- Math tags: optimization, dynamical-systems, convex-analysis
- Ratings: usefulness 6/10; difficulty 5/10; novelty 6/10

## Idea description

Use the paper's prediction-relaxation decomposition to build a pipelined optimizer in which workers compute local proximal or gradient predictions as soon as parent messages arrive, then apply independently tunable relaxation to primal and dual states. This provides a controlled alternative to undamped stale updates and can overlap communication with local computation.

## Mathematical statement

The paper represents each local update as a prediction followed by diagonal relaxation. For primal state $z_{i_q}$ and dual state $s_{j_q}$, the update is $[z_{i_q}^{k+1};s_{j_q}^{k+1}]=[z_{i_q}^{k};s_{j_q}^{k}]+\operatorname{diag}(\theta_{i_q},\zeta_{j_q})[\tilde z_{i_q}^{k}-z_{i_q}^{k};\tilde s_{j_q}^{k}-s_{j_q}^{k}]$, where $\tilde z$ and $\tilde s$ are predictions and $\theta_{i_q},\zeta_{j_q}$ are independent relaxation parameters. The paper's level order is root primal state, root dual states, level-1 primal state, level-1 dual and edge states, and so on. The useful transferable property is that local operators produce predictions while relaxation damps the fixed-point residual; primal and dual damping need not be equal. In the neural optimizer, $z_i$ is a parameter block, $s_i$ is a consensus or gradient-tracking dual variable, and $w_i$ is an edge memory state.

## Key formulas

- $$\begin{bmatrix}z_{i_q}^{k+1}\\s_{j_q}^{k+1}\end{bmatrix}=\begin{bmatrix}z_{i_q}^{k}\\s_{j_q}^{k}\end{bmatrix}+\begin{bmatrix}\theta_{i_q}&0\\0&\zeta_{j_q}\end{bmatrix}\begin{bmatrix}\tilde z_{i_q}^{k}-{z}_{i_q}^{k}\\\tilde s_{j_q}^{k}-{s}_{j_q}^{k}\end{bmatrix}.$$
- $$w_i^{k+1}=w_i^k-u_i^k+u_{i-1}^{k+1}.$$
- $$u_i^{k+1}=u_i^k+\theta_i(\tilde u_i^{k+1}-u_i^k),\qquad s_i^{k+1}=s_i^k+\zeta_i\rho_i(u_i^{k+1}-u_{parent(i)}^{k+1}).$$

## Implementation notes

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.

## Disclaimer

AI-generated research hypothesis, automatically tested. Not peer-reviewed.