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

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

13 Sep 2024arXiv:2409.08771archive 2025-07-28

Constantin Philippenko, Kevin Scaman, Laurent Massoulié

We analyze a distributed algorithm to compute a low-rank matrix factorization on N clients, each holding a local dataset 𝐒ⁱ ∈ℝ^(nᵢ ×d), mathematically, we seek to solve min_(𝐔ⁱ ∈ℝ^(nᵢ×r), 𝐕∈ℝ^(d ×r)) 1/2 ∑ᵢ₌₁ᴺ 𝐒ⁱ - 𝐔ⁱ 𝐕^⊤²_F. Considering a power initialization of 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, …, N}, we obtain a global 𝐕 in ℝ^(d ×r) common to all clients and a local variable 𝐔ⁱ in ℝ^(nᵢ ×r). We provide a linear rate of convergence of the excess loss which depends on σₘₐₓ / σᵣ, where σᵣ is the rᵗʰ singular value of the concatenation 𝐒 of the matrices (𝐒ⁱ)ᵢ₌₁ᴺ. This result improves the rates of convergence given in the literature, which depend on σₘₐₓ² / σₘᵢₙ². 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.

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