# Decentralized EM to Learn Gaussian Mixtures from Datasets Distributed by Features

arXiv: https://arxiv.org/abs/2201.09965 · HTML: https://mlsanity.com/paper/2201.09965/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 10 of its 49 references that are in the map.
- First posted on arXiv 24 Jan 2022.

## Builds on (most cited first, 10 of 10)

- [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)
- [The Future of Digital Health with Federated Learning](https://mlsanity.com/paper/2003.08119/) (2020)
- [Can Decentralized Algorithms Outperform Centralized Algorithms? A Case Study for Decentralized Parallel Stochastic Gradient Descent](https://mlsanity.com/paper/1705.09056/) (2017)
- [Federated Evaluation of On-device Personalization](https://mlsanity.com/paper/1910.10252/) (2019)
- [Statistical guarantees for the EM algorithm: From population to sample-based analysis](https://mlsanity.com/paper/1408.2156/) (2014)
- [Ten Steps of EM Suffice for Mixtures of Two Gaussians](https://mlsanity.com/paper/1609.00368/) (2016)
- [Homeomorphic-Invariance of EM: Non-Asymptotic Convergence in KL Divergence for Exponential Families via Mirror Descent](https://mlsanity.com/paper/2011.01170/) (2020)
- [Supervised Learning Under Distributed Features](https://mlsanity.com/paper/1805.11384/) (2018)
- [Distributed Variational Bayesian Algorithms Over Sensor Networks](https://mlsanity.com/paper/2011.13600/) (2020)

## Abstract (arXiv)

Expectation Maximization (EM) is the standard method to learn Gaussian mixtures. Yet its classic, centralized form is often infeasible, due to privacy concerns and computational and communication bottlenecks. Prior work dealt with data distributed by examples, horizontal partitioning, but we lack a counterpart for data scattered by features, an increasingly common scheme (e.g. user profiling with data from multiple entities). To fill this gap, we provide an EM-based algorithm to fit Gaussian mixtures to Vertically Partitioned data (VP-EM). In federated learning setups, our algorithm matches the centralized EM fitting of Gaussian mixtures constrained to a subspace. In arbitrary communication graphs, consensus averaging allows VP-EM to run on large peer-to-peer networks as an EM approximation. This mismatch comes from consensus error only, which vanishes exponentially fast with the number of consensus rounds. We demonstrate VP-EM on various topologies for both synthetic and real data, evaluating its approximation of centralized EM and seeing that it outperforms the available benchmark.