Federated Matrix Factorization under Local Differential Privacy via Directional Noise

Abstract

Federated Recommender Systems (FRSs) enable on-device personalization while preserving user privacy, yet the gradients shared during model training can still leak sensitive information. Local differential privacy (LDP) offers a formal defense, but classic mechanisms—randomized response and Laplace perturbation—inject substantial noise into the high-dimensional gradient vectors of latent-factor models, severely degrading recommendation accuracy. In this paper, we systematically evaluate these traditional LDP strategies in federated matrix factorization under both explicit (MovieLens 100K and 1M) and implicit (Steam, LastFM) feedback benchmarks. We then propose a novel directional noise mechanism based on the von Mises–Fisher distribution, which preserves gradient magnitudes while randomizing directions. Through extensive experiments across a wide range of privacy budgets, we show that our vMF mechanism consistently outperforms Laplace and binary randomized-response, recovering up to more than 100 % of the non-private utility at ε = 1 and delivering higher ranking accuracy under moderate privacy levels (ε ≤ 2). We further demonstrate that directional perturbation reduces variance and remains robust under partial client participation. Our findings highlight the importance of data geometry in privacy–utility trade-offs and provide practical guidance for deploying LDP in high-dimensional federated recommendation settings.