In-depth Analysis of Low-rank Matrix Factorisation in a Federated Setting

TL;DR

This paper presents a federated low-rank matrix factorization algorithm with improved convergence rates, requiring minimal communication, and provides theoretical guarantees and experimental validation on synthetic and real datasets.

Contribution

It introduces a novel federated approach that transforms a non-convex problem into a strongly convex one, achieving faster convergence and lower error bounds with minimal communication.

Findings

Linear convergence rate depending on singular value ratio

Improved convergence rates over existing methods

Effective on both synthetic and real data

Abstract

We analyze a distributed algorithm to compute a low-rank matrix factorization on $N$ clients, each holding a local dataset $\mathbf{S}^i \in \mathbb{R}^{n_i \times d}$, mathematically, we seek to solve $min_{\mathbf{U}^i \in \mathbb{R}^{n_i\times r}, \mathbf{V}\in \mathbb{R}^{d \times r} } \frac{1}{2} \sum_{i=1}^N \|\mathbf{S}^i - \mathbf{U}^i \mathbf{V}^\top\|^2_{\text{F}}$. Considering a power initialization of $\mathbf{V}$, we rewrite the previous smooth non-convex problem into a smooth strongly-convex problem that we solve using a parallel Nesterov gradient descent potentially requiring a single step of communication at the initialization step. For any client $i$ in $\{1, \dots, N\}$, we obtain a global $\mathbf{V}$ in $\mathbb{R}^{d \times r}$ common to all clients and a local variable $\mathbf{U}^i$ in $\mathbb{R}^{n_i \times r}$. We provide a linear rate of convergence of the excess loss which depends on $\sigma_{\max} / \sigma_{r}$, where $\sigma_{r}$ is the $r^{\mathrm{th}}$ singular value of the concatenation $\mathbf{S}$ of the matrices $(\mathbf{S}^i)_{i=1}^N$. This result improves the rates of convergence given in the literature, which depend on $\sigma_{\max}^2 / \sigma_{\min}^2$. We provide an upper bound on the Frobenius-norm error of reconstruction under the power initialization strategy. We complete our analysis with experiments on both synthetic and real data.