# Escaping Saddle Points in Heterogeneous Federated Learning via Distributed SGD with Communication Compression

arXiv: https://arxiv.org/abs/2310.19059 · HTML: https://mlsanity.com/paper/2310.19059/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- Builds on 31 of its 53 references that are in the map.
- First posted on arXiv 29 Oct 2023.

## Builds on (most cited first, 30 of 31)

- [Language Models are Few-Shot Learners](https://mlsanity.com/paper/2005.14165/) (2020)
- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [Advances and Open Problems in Federated Learning](https://mlsanity.com/paper/1912.04977/) (2019)
- [Federated Learning: Strategies for Improving Communication Efficiency](https://mlsanity.com/paper/1610.05492/) (2016)
- [signSGD: Compressed Optimisation for Non-Convex Problems](https://mlsanity.com/paper/1802.04434/) (2018)
- [Sparsified SGD with Memory](https://mlsanity.com/paper/1809.07599/) (2018)
- [A Unified Theory of Decentralized SGD with Changing Topology and Local Updates](https://mlsanity.com/paper/2003.10422/) (2020)
- [Error Feedback Fixes SignSGD and other Gradient Compression Schemes](https://mlsanity.com/paper/1901.09847/) (2019)
- [Identifying and attacking the saddle point problem in high-dimensional non-convex optimization](https://mlsanity.com/paper/1406.2572/) (2014)
- [How to Escape Saddle Points Efficiently](https://mlsanity.com/paper/1703.00887/) (2017)
- [Federated Learning for Internet of Things: A Comprehensive Survey](https://mlsanity.com/paper/2104.07914/) (2021)
- [Escaping From Saddle Points --- Online Stochastic Gradient for Tensor Decomposition](https://mlsanity.com/paper/1503.02101/) (2015)
- [Decentralized Stochastic Optimization and Gossip Algorithms with Compressed Communication](https://mlsanity.com/paper/1902.00340/) (2019)
- [Mini-batch Stochastic Approximation Methods for Nonconvex Stochastic Composite Optimization](https://mlsanity.com/paper/1308.6594/) (2013)
- [On Biased Compression for Distributed Learning](https://mlsanity.com/paper/2002.12410/) (2020)
- [EF21: A New, Simpler, Theoretically Better, and Practically Faster Error Feedback](https://mlsanity.com/paper/2106.05203/) (2021)
- [PAGE: A Simple and Optimal Probabilistic Gradient Estimator for Nonconvex Optimization](https://mlsanity.com/paper/2008.10898/) (2020)
- [Decentralized Deep Learning with Arbitrary Communication Compression](https://mlsanity.com/paper/1907.09356/) (2019)
- [Reliable Federated Learning for Mobile Networks](https://mlsanity.com/paper/1910.06837/) (2019)
- [Federated Learning with Cooperating Devices: A Consensus Approach for Massive IoT Networks](https://mlsanity.com/paper/1912.13163/) (2019)
- [Gradient Descent Can Take Exponential Time to Escape Saddle Points](https://mlsanity.com/paper/1705.10412/) (2017)
- [Escaping Saddles with Stochastic Gradients](https://mlsanity.com/paper/1803.05999/) (2018)
- [BEER: Fast O(1/T) Rate for Decentralized Nonconvex Optimization with Communication Compression](https://mlsanity.com/paper/2201.13320/) (2022)
- [SoteriaFL: A Unified Framework for Private Federated Learning with Communication Compression](https://mlsanity.com/paper/2206.09888/) (2022)
- [Lower Bounds and Nearly Optimal Algorithms in Distributed Learning with Communication Compression](https://mlsanity.com/paper/2206.03665/) (2022)
- [CSER: Communication-efficient SGD with Error Reset](https://mlsanity.com/paper/2007.13221/) (2020)
- [3PC: Three Point Compressors for Communication-Efficient Distributed Training and a Better Theory for Lazy Aggregation](https://mlsanity.com/paper/2202.00998/) (2022)
- [SSRGD: Simple Stochastic Recursive Gradient Descent for Escaping Saddle Points](https://mlsanity.com/paper/1904.09265/) (2019)
- [Stabilized SVRG: Simple Variance Reduction for Nonconvex Optimization](https://mlsanity.com/paper/1905.00529/) (2019)
- [Convergence and Privacy of Decentralized Nonconvex Optimization with Gradient Clipping and Communication Compression](https://mlsanity.com/paper/2305.09896/) (2023)

## Built on by (most cited first, 4 of 4)

- [LoCoDL: Communication-Efficient Distributed Learning with Local Training and Compression](https://mlsanity.com/paper/2403.04348/) (2024)
- [Review of Mathematical Optimization in Federated Learning](https://mlsanity.com/paper/2412.01630/) (2024)
- [Second-Order Convergence in Private Stochastic Non-Convex Optimization](https://mlsanity.com/paper/2505.15647/) (2025)
- [Analysis of Asynchronous Federated Learning: Unraveling the Interactions between Gradient Compression, Delay, and Data Heterogeneity](https://mlsanity.com/paper/2504.19903/) (2025)

## Abstract (arXiv)

We consider the problem of finding second-order stationary points of heterogeneous federated learning (FL). Previous works in FL mostly focus on first-order convergence guarantees, which do not rule out the scenario of unstable saddle points. Meanwhile, it is a key bottleneck of FL to achieve communication efficiency without compensating the learning accuracy, especially when local data are highly heterogeneous across different clients. Given this, we propose a novel algorithm Power-EF that only communicates compressed information via a novel error-feedback scheme. To our knowledge, Power-EF is the first distributed and compressed SGD algorithm that provably escapes saddle points in heterogeneous FL without any data homogeneity assumptions. In particular, Power-EF improves to second-order stationary points after visiting first-order (possibly saddle) points, using additional gradient queries and communication rounds only of almost the same order required by first-order convergence, and the convergence rate exhibits a linear speedup in terms of the number of workers. Our theory improves/recovers previous results, while extending to much more tolerant settings on the local data. Numerical experiments are provided to complement the theory.