Error Forgetting of Bregman Iteration

Abstract

This short article analyzes an interesting property of the Bregman iterative procedure, which is equivalent to the augmented Lagrangian method, for minimizing a convex piece-wise linear function J(x) subject to linear constraints Ax=b. The procedure obtains its solution by solving a sequence of unconstrained subproblems of minimizing $J(x)+\frac{1}{2}\|Ax-b^{k}\|_{2}^{2}$, where b ^k is iteratively updated. In practice, the subproblem at each iteration is solved at a relatively low accuracy. Let w ^k denote the error introduced by early stopping a subproblem solver at iteration k. We show that if all w ^k are sufficiently small so that Bregman iteration enters the optimal face, then while on the optimal face, Bregman iteration enjoys an interesting error-forgetting property: the distance between the current point $\bar{x}^{k}$ and the optimal solution set X ^∗ is bounded by ∥w ^k+1−w ^k∥, independent of the previous errors w ^k−1,w ^k−2,…,w ¹. This property partially explains why the Bregman iterative procedure works well for sparse optimization and, in particular, for ℓ ₁-minimization. The error-forgetting property is unique to J(x) that is a piece-wise linear function (also known as a polyhedral function), and the results of this article appear to be new to the literature of the augmented Lagrangian method.