Convergence of Alternating Gradient Descent for Matrix Factorization
This provides a theoretical guarantee for a nonconvex optimization method in machine learning, but it is incremental as it builds on existing analyses with a new initialization and proof technique.
The paper tackles the convergence of alternating gradient descent for matrix factorization, showing that with an atypical random initialization, it achieves an ε-optimal factorization in T = C (σ₁(A)/σᵣ(A))² log(1/ε) iterations with high probability, and experiments indicate this initialization improves practical convergence rates.
We consider alternating gradient descent (AGD) with fixed step size applied to the asymmetric matrix factorization objective. We show that, for a rank-$r$ matrix $\mathbf{A} \in \mathbb{R}^{m \times n}$, $T = C (\frac{σ_1(\mathbf{A})}{σ_r(\mathbf{A})})^2 \log(1/ε)$ iterations of alternating gradient descent suffice to reach an $ε$-optimal factorization $\| \mathbf{A} - \mathbf{X} \mathbf{Y}^{T} \|^2 \leq ε\| \mathbf{A}\|^2$ with high probability starting from an atypical random initialization. The factors have rank $d \geq r$ so that $\mathbf{X}_{T}\in\mathbb{R}^{m \times d}$ and $\mathbf{Y}_{T} \in\mathbb{R}^{n \times d}$, and mild overparameterization suffices for the constant $C$ in the iteration complexity $T$ to be an absolute constant. Experiments suggest that our proposed initialization is not merely of theoretical benefit, but rather significantly improves the convergence rate of gradient descent in practice. Our proof is conceptually simple: a uniform Polyak-Łojasiewicz (PL) inequality and uniform Lipschitz smoothness constant are guaranteed for a sufficient number of iterations, starting from our random initialization. Our proof method should be useful for extending and simplifying convergence analyses for a broader class of nonconvex low-rank factorization problems.