Dropping Convexity for Faster Semi-definite Optimization
29th Annual Conference on Learning Theory, PMLR 49:530-582, 2016.
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^⊤), with U ∈\mathbbR^n \times r and r ≤n. We study the performance of gradient descent on g—which we refer to as Factored Gradient Descent (\textscFgd)—under standard assumptions on the \em original function f. We provide a rule for selecting the step size and, with this choice, show that the \emphlocal convergence rate of \textscFgd mirrors that of standard gradient descent on the original f: \emphi.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 \textscFgd 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 \emphglobal convergence guarantees. \textscFgd 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.