Convergence of Gradient Descent with Small Initialization for Unregularized Matrix Completion

We study the problem of symmetric matrix completion, where the goal is to reconstruct a positive semidefinite matrix $X^\star \in \mathbb{R}^{d\times d}$ of rank-$r$, parameterized by $UU^{\top}$, from only a subset of its observed entries. We show that the vanilla gradient descent (GD) with small initialization provably converges to the ground truth $X^\star$ without requiring any explicit regularization. This convergence result holds true even in the over-parameterized scenario, where the true rank $r$ is unknown and conservatively over-estimated by a search rank $r’\gg r$. The existing results for this problem either require explicit regularization, a sufficiently accurate initial point, or exact knowledge of the true rank $r$. In the over-parameterized regime where $r’\geq r$, we show that, with $\widetilde\Omega(dr^9)$ observations, GD with an initial point $\|U_0\| \leq O(\epsilon)$ converges near-linearly to an $\epsilon$-neighborhood of $X^\star$. Consequently, smaller initial points result in increasingly accurate solutions. Surprisingly, neither the convergence rate nor the final accuracy depends on the over-parameterized search rank $r’$, and they are only governed by the true rank $r$. In the exactly-parameterized regime where $r’=r$, we further enhance this result by proving that GD converges at a faster rate to achieve an arbitrarily small accuracy $\epsilon>0$, provided the initial point satisfies $\|U_0\| = O(1/d)$. At the crux of our method lies a novel weakly-coupled leave-one-out analysis, which allows us to establish the global convergence of GD, extending beyond what was previously possible using the classical leave-one-out analysis.