# FedSGM: A Unified Framework for Constraint Aware, Bidirectionally Compressed, Multi-Step Federated Optimization

arXiv: https://arxiv.org/abs/2601.16897 · HTML: https://mlsanity.com/paper/2601.16897/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 23 Jan 2026.

## Built on by (most cited first, 3 of 3)

- [Lower Bounds and Proximally Anchored SGD for Non-Convex Minimization Under Unbounded Variance](https://mlsanity.com/paper/2604.16620/) (2026)
- [First-Order Softmax Weighted Switching Gradient Method for Distributed Stochastic Minimax Optimization with Stochastic Constraints](https://mlsanity.com/paper/2603.05774/) (2026)
- [Unified High-Probability Analysis of Stochastic Variance-Reduced Estimation](https://mlsanity.com/paper/2605.15388/) (2026)

## Abstract (arXiv)

We introduce FedSGM, a unified framework for federated constrained optimization that addresses four major challenges in federated learning (FL): functional constraints, communication bottlenecks, local updates, and partial client participation. Building on the switching gradient method, FedSGM provides projection-free, primal-only updates, avoiding expensive dual-variable tuning or inner solvers. To handle communication limits, FedSGM incorporates bi-directional error feedback, correcting the bias introduced by compression while explicitly understanding the interaction between compression noise and multi-step local updates. We derive convergence guarantees showing that the averaged iterate achieves the canonical boldsymbolO(1/√T) rate, with additional high-probability bounds that decouple optimization progress from sampling noise due to partial participation. Additionally, we introduce a soft switching version of FedSGM to stabilize updates near the feasibility boundary. To our knowledge, FedSGM is the first framework to unify functional constraints, compression, multiple local updates, and partial client participation, establishing a theoretically grounded foundation for constrained federated learning. Finally, we validate the theoretical guarantees of FedSGM via experimentation on Neyman-Pearson classification and constrained Markov decision process (CMDP) tasks.