Decentralized Dynamic Smoothing with Partial Gradient Tracking on Compact Submanifolds

We study decentralized composite optimization on compact embedded submanifolds, where each local objective contains a smooth agent-specific loss and a common convex nonsmooth regularizer. Tracking the full smoothed gradient introduces a Lipschitz factor that diverges as the smoothing parameter decreases. We propose a decentralized smoothing method with partial gradient tracking (DSPGT), which tracks only the heterogeneous smooth gradients and evaluates the common smoothing gradient locally. A Lyapunov function containing the local regularizer values controls the latter term through the Jensen decrease induced by mixing. For a uniformly sampled tail iterate, the expected smoothed stationarity residual and consensus error are $\mathcal{O}(K^{-1/3})$. For Moreau smoothing, these bounds give a generalized $\epsilon$-stationary point of the original problem at every agent in $\mathcal{O}({\epsilon }^{-3})$ projection-oracle calls and communication rounds, matching the centralized order. Across the 18 reported settings on MNIST and CIFAR-10, DSPGT has the lowest median residual.