Understanding Limitation of Two Symmetrized Orders by Worst-case Complexity
This work addresses the convergence limitations of deterministic update orders in optimization algorithms, which is incremental as it extends prior findings on cyclic order to symmetrized variants.
The paper shows that two symmetrized update orders, Gaussian back substitution and symmetric Gauss-Seidel, suffer from the same worst-case slowdown as cyclic order in coordinate descent algorithms, being O(n^2) times slower than randomized versions for unconstrained problems and empirically for linearly constrained ones.
Update order is one of the major design choices of block decomposition algorithms. There are at least two classes of deterministic update orders: nonsymmetric (e.g. cyclic order) and symmetric (e.g. Gaussian back substitution or symmetric Gauss-Seidel). Recently, Coordinate Descent (CD) with cyclic order was shown to be $O(n^2)$ times slower than randomized versions in the worst-case. A natural question arises: can the symmetrized orders achieve faster convergence rates than the cyclic order, or even getting close to the randomized versions? In this paper, we give a negative answer to this question. We show that both Gaussian back substitution (GBS) and symmetric Gauss-Seidel (sGS) suffer from the same slow convergence issue as the cyclic order in the worst case. In particular, we prove that for unconstrained problems, both GBS-CD and sGS-CD can be $O(n^2)$ times slower than R-CD. Despite unconstrained problems, we also empirically study linearly constrained problems with quadratic objective: we empirically demonstrate that the convergence speed of GBS-ADMM and sGS-ADMM can be roughly $O(n^2)$ times slower than randomly permuted ADMM.