# One-Shot Federated Ridge Regression: Exact Recovery via Sufficient Statistic Aggregation

arXiv: https://arxiv.org/abs/2601.08216 · HTML: https://mlsanity.com/paper/2601.08216/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 13 of its 29 references that are in the map.
- First posted on arXiv 13 Jan 2026.

## Builds on (most cited first, 13 of 13)

- [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)
- [Deep Learning with Differential Privacy](https://mlsanity.com/paper/1607.00133/) (2016)
- [Neural Tangent Kernel: Convergence and Generalization in Neural Networks](https://mlsanity.com/paper/1806.07572/) (2018)
- [Federated Learning: Strategies for Improving Communication Efficiency](https://mlsanity.com/paper/1610.05492/) (2016)
- [Measuring the Effects of Non-Identical Data Distribution for Federated Visual Classification](https://mlsanity.com/paper/1909.06335/) (2019)
- [Local SGD Converges Fast and Communicates Little](https://mlsanity.com/paper/1805.09767/) (2018)
- [Differentially Private Federated Learning: A Client Level Perspective](https://mlsanity.com/paper/1712.07557/) (2017)
- [Privacy Amplification by Subsampling: Tight Analyses via Couplings and Divergences](https://mlsanity.com/paper/1807.01647/) (2018)
- [One-Shot Federated Learning](https://mlsanity.com/paper/1902.11175/) (2019)
- [One-Shot Federated Learning: Theoretical Limits and Algorithms to Achieve Them](https://mlsanity.com/paper/1905.04634/) (2019)
- [DUAL-LOCO: Distributing Statistical Estimation Using Random Projections](https://mlsanity.com/paper/1506.02554/) (2015)

## Built on by (most cited first, 1 of 1)

- [Recovering Governing Dynamics from Distributed Observations via Exact Spline Merging](https://mlsanity.com/paper/2609.16579/) (2026)

## Abstract (arXiv)

Federated learning protocols require repeated synchronization between clients and a central server, with convergence rates depending on learning rates, data heterogeneity, and client sampling. This paper asks whether iterative communication is necessary for distributed linear regression. We show it is not. We formulate federated ridge regression as a distributed equilibrium problem where each client computes local sufficient statistics -- the Gram matrix and moment vector -- and transmits them once. The server reconstructs the global solution through a single matrix inversion. We prove exact recovery: under a coverage condition on client feature matrices, one-shot aggregation yields the centralized ridge solution, not an approximation. For heterogeneous distributions violating coverage, we derive non-asymptotic error bounds depending on spectral properties of the aggregated Gram matrix. Communication reduces from O(Rd) in iterative methods to O(d2) total; for high-dimensional settings, we propose and experimentally validate random projection techniques reducing this to O(m2) where m ll d. We establish differential privacy guarantees where noise is injected once per client, eliminating the composition penalty that degrades privacy in multi-round protocols. We further address practical considerations including client dropout robustness, federated cross-validation for hyperparameter selection, and comparison with gradient-based alternatives. Comprehensive experiments on synthetic heterogeneous regression demonstrate that one-shot fusion matches FedAvg accuracy while requiring up to 38× less communication. The framework applies to kernel methods and random feature models but not to general nonlinear architectures.