Dropping Convexity for Faster Semi-definite Optimization

We study the minimization of a convex function $f(X)$ over the set of $n\times n$ positive semi-definite matrices, but when the problem is recast as $\min_U g(U) := f(UU^\top)$, with $U \in \mathbb{R}^{n \times r}$ and $r \leq n$. We study the performance of gradient descent on $g$---which we refer to as Factored Gradient Descent (FGD)---under standard assumptions on the original function $f$. We provide a rule for selecting the step size and, with this choice, show that the local convergence rate of FGD mirrors that of standard gradient descent on the original $f$: i.e., after $k$ steps, the error is $O(1/k)$ for smooth $f$, and exponentially small in $k$ when $f$ is (restricted) strongly convex. In addition, we provide a procedure to initialize FGD for (restricted) strongly convex objectives and when one only has access to $f$ via a first-order oracle; for several problem instances, such proper initialization leads to global convergence guarantees. FGD and similar procedures are widely used in practice for problems that can be posed as matrix factorization. To the best of our knowledge, this is the first paper to provide precise convergence rate guarantees for general convex functions under standard convex assumptions.