Composite and Stochastic Smoothed Decentralized Bregman Proximal-Gradient Methods

Abstract

Smoothed decentralized Bregman proximal-gradient (SDBPG) methods combine relative-smoothness geometry with edge-dual consensus. We extend SDBPG to node-specific convex nonsmooth regularizers and unbiased stochastic local gradients while retaining its Bregman primal-dual smoothing mechanism. The analysis isolates temporal subgradient motion and aggregation mismatch as two defect sequences. For trajectories in a forward-invariant compact subset of the kernel domain and admissible parameters, finite accumulated defects yield an $O(T^{-1})$ bound on the squared best-iterate proximal residual over $T$ iterations. The stochastic bound separates accumulated oracle variance from expected defects and attains the same order when the variance bounds and expected defects are summable. Common active-structure identification, together with local subgradient conditions, gives sufficient finite-defect conditions for ${\ell }_1$ and group ${\ell }_{2,1}$ regularization. Experiments with common regularizers and heterogeneous sensing operators relate defect decay to observed support stabilization and compare reconstruction and communication with decentralized proximal and gradient-tracking methods. A separate phase-retrieval study evaluates the effect of a quartic Bregman kernel. Code and experiment scripts are available at https://github.com/shengze-xu/composite-stochastic-sdbpg .