# FedNL: Making Newton-Type Methods Applicable to Federated Learning

arXiv: https://arxiv.org/abs/2106.02969 · HTML: https://mlsanity.com/paper/2106.02969/

## Key facts

- Cited by 52 papers in the map (data through 14 Oct 2026).
- Builds on 36 of its 43 references that are in the map.
- First posted on arXiv 5 Jun 2021.

## Builds on (most cited first, 30 of 36)

- [Communication-Efficient Learning of Deep Networks from Decentralized Data](https://mlsanity.com/paper/1602.05629/) (2016)
- [Advances and Open Problems in Federated Learning](https://mlsanity.com/paper/1912.04977/) (2019)
- [Federated Optimization in Heterogeneous Networks](https://mlsanity.com/paper/1812.06127/) (2018)
- [Federated Learning: Challenges, Methods, and Future Directions](https://mlsanity.com/paper/1908.07873/) (2019)
- [Federated Learning: Strategies for Improving Communication Efficiency](https://mlsanity.com/paper/1610.05492/) (2016)
- [Adaptive Federated Optimization](https://mlsanity.com/paper/2003.00295/) (2020)
- [Local SGD Converges Fast and Communicates Little](https://mlsanity.com/paper/1805.09767/) (2018)
- [Federated Optimization: Distributed Machine Learning for On-Device Intelligence](https://mlsanity.com/paper/1610.02527/) (2016)
- [Sparsified SGD with Memory](https://mlsanity.com/paper/1809.07599/) (2018)
- [Error Feedback Fixes SignSGD and other Gradient Compression Schemes](https://mlsanity.com/paper/1901.09847/) (2019)
- [Tighter Theory for Local SGD on Identical and Heterogeneous Data](https://mlsanity.com/paper/1909.04746/) (2019)
- [PowerSGD: Practical Low-Rank Gradient Compression for Distributed Optimization](https://mlsanity.com/paper/1905.13727/) (2019)
- [SGD: General Analysis and Improved Rates](https://mlsanity.com/paper/1901.09401/) (2019)
- [SCAFFOLD: Stochastic Controlled Averaging for Federated Learning](https://mlsanity.com/paper/1910.06378/) (2019)
- [QSGD: Communication-Efficient SGD via Gradient Quantization and Encoding](https://mlsanity.com/paper/1610.02132/) (2016)
- [On Biased Compression for Distributed Learning](https://mlsanity.com/paper/2002.12410/) (2020)
- [Communication Efficient Distributed Optimization using an Approximate Newton-type Method](https://mlsanity.com/paper/1312.7853/) (2013)
- [Optimal Client Sampling for Federated Learning](https://mlsanity.com/paper/2010.13723/) (2020)
- [Distributed Learning with Compressed Gradient Differences](https://mlsanity.com/paper/1901.09269/) (2019)
- [Acceleration for Compressed Gradient Descent in Distributed and Federated Optimization](https://mlsanity.com/paper/2002.11364/) (2020)
- [Local SGD: Unified Theory and New Efficient Methods](https://mlsanity.com/paper/2011.02828/) (2020)
- [A Unified Theory of SGD: Variance Reduction, Sampling, Quantization and Coordinate Descent](https://mlsanity.com/paper/1905.11261/) (2019)
- [Adaptive Gradient Descent without Descent](https://mlsanity.com/paper/1910.09529/) (2019)
- [MARINA: Faster Non-Convex Distributed Learning with Compression](https://mlsanity.com/paper/2102.07845/) (2021)
- [Stochastic Distributed Learning with Gradient Quantization and Variance Reduction](https://mlsanity.com/paper/1904.05115/) (2019)
- [Linearly Converging Error Compensated SGD](https://mlsanity.com/paper/2010.12292/) (2020)
- [Distributed learning with compressed gradients](https://mlsanity.com/paper/1806.06573/) (2018)
- [Bidirectional compression in heterogeneous settings for distributed or federated learning with partial participation: tight convergence guarantees](https://mlsanity.com/paper/2006.14591/) (2020)
- [GIANT: Globally Improved Approximate Newton Method for Distributed Optimization](https://mlsanity.com/paper/1709.03528/) (2017)
- [AIDE: Fast and Communication Efficient Distributed Optimization](https://mlsanity.com/paper/1608.06879/) (2016)

## Built on by (most cited first, 30 of 52)

- [Momentum Provably Improves Error Feedback!](https://mlsanity.com/paper/2305.15155/) (2023)
- [3PC: Three Point Compressors for Communication-Efficient Distributed Training and a Better Theory for Lazy Aggregation](https://mlsanity.com/paper/2202.00998/) (2022)
- [Matrix Compression via Randomized Low Rank and Low Precision Factorization](https://mlsanity.com/paper/2310.11028/) (2023)
- [FedNew: A Communication-Efficient and Privacy-Preserving Newton-Type Method for Federated Learning](https://mlsanity.com/paper/2206.08829/) (2022)
- [Efficient Federated Learning via Local Adaptive Amended Optimizer with Linear Speedup](https://mlsanity.com/paper/2308.00522/) (2023)
- [Improved Communication Efficiency in Federated Natural Policy Gradient via ADMM-based Gradient Updates](https://mlsanity.com/paper/2310.19807/) (2023)
- [Communication Acceleration of Local Gradient Methods via an Accelerated Primal-Dual Algorithm with Inexact Prox](https://mlsanity.com/paper/2207.03957/) (2022)
- [AsGrad: A Sharp Unified Analysis of Asynchronous-SGD Algorithms](https://mlsanity.com/paper/2310.20452/) (2023)
- [Basis Matters: Better Communication-Efficient Second Order Methods for Federated Learning](https://mlsanity.com/paper/2111.01847/) (2021)
- [EControl: Fast Distributed Optimization with Compression and Error Control](https://mlsanity.com/paper/2311.05645/) (2023)
- [Hessian Averaging in Stochastic Newton Methods Achieves Superlinear Convergence](https://mlsanity.com/paper/2204.09266/) (2022)
- [Communication Compression for Byzantine Robust Learning: New Efficient Algorithms and Improved Rates](https://mlsanity.com/paper/2310.09804/) (2023)
- [SHED: A Newton-type algorithm for federated learning based on incremental Hessian eigenvector sharing](https://mlsanity.com/paper/2202.05800/) (2022)
- [Distributed Newton-Type Methods with Communication Compression and Bernoulli Aggregation](https://mlsanity.com/paper/2206.03588/) (2022)
- [FL_PyTorch: optimization research simulator for federated learning](https://mlsanity.com/paper/2202.03099/) (2022)
- [On Second-order Optimization Methods for Federated Learning](https://mlsanity.com/paper/2109.02388/) (2021)
- [EF21 with Bells & Whistles: Six Algorithmic Extensions of Modern Error Feedback](https://mlsanity.com/paper/2110.03294/) (2021)
- [Advancements in Federated Learning: Models, Methods, and Privacy](https://mlsanity.com/paper/2302.11466/) (2023)
- [Shifted Compression Framework: Generalizations and Improvements](https://mlsanity.com/paper/2206.10452/) (2022)
- [Towards a Better Theoretical Understanding of Independent Subnetwork Training](https://mlsanity.com/paper/2306.16484/) (2023)
- [PersA-FL: Personalized Asynchronous Federated Learning](https://mlsanity.com/paper/2210.01176/) (2022)
- [Byzantine Robustness and Partial Participation Can Be Achieved at Once: Just Clip Gradient Differences](https://mlsanity.com/paper/2311.14127/) (2023)
- [Towards Faster Decentralized Stochastic Optimization with Communication Compression](https://mlsanity.com/paper/2405.20114/) (2024)
- [Fed-Sophia: A Communication-Efficient Second-Order Federated Learning Algorithm](https://mlsanity.com/paper/2406.06655/) (2024)
- [Distributed Quasi-Newton Method for Fair and Fast Federated Learning](https://mlsanity.com/paper/2501.10877/) (2025)
- [Unbiased and Sign Compression in Distributed Learning: Comparing Noise Resilience via SDEs](https://mlsanity.com/paper/2502.17009/) (2025)
- [FedZeN: Towards superlinear zeroth-order federated learning via incremental Hessian estimation](https://mlsanity.com/paper/2309.17174/) (2023)
- [Error Feedback under (L0,L1)-Smoothness: Normalization and Momentum](https://mlsanity.com/paper/2410.16871/) (2024)
- [Accelerated Distributed Optimization with Compression and Error Feedback](https://mlsanity.com/paper/2503.08427/) (2025)
- [FedSSO: A Federated Server-Side Second-Order Optimization Algorithm](https://mlsanity.com/paper/2206.09576/) (2022)

## Abstract (arXiv)

Inspired by recent work of Islamov et al (2021), we propose a family of Federated Newton Learn (FedNL) methods, which we believe is a marked step in the direction of making second-order methods applicable to FL. In contrast to the aforementioned work, FedNL employs a different Hessian learning technique which i) enhances privacy as it does not rely on the training data to be revealed to the coordinating server, ii) makes it applicable beyond generalized linear models, and iii) provably works with general contractive compression operators for compressing the local Hessians, such as Top-K or Rank-R, which are vastly superior in practice. Notably, we do not need to rely on error feedback for our methods to work with contractive compressors. Moreover, we develop FedNL-PP, FedNL-CR and FedNL-LS, which are variants of FedNL that support partial participation, and globalization via cubic regularization and line search, respectively, and FedNL-BC, which is a variant that can further benefit from bidirectional compression of gradients and models, i.e., smart uplink gradient and smart downlink model compression. We prove local convergence rates that are independent of the condition number, the number of training data points, and compression variance. Our communication efficient Hessian learning technique provably learns the Hessian at the optimum. Finally, we perform a variety of numerical experiments that show that our FedNL methods have state-of-the-art communication complexity when compared to key baselines.