Block-coordinate descent (BCD) is a popular method for large-scale regularized optimization problems with block-separable structure. However, existing analyses require either a fixed second-order approximation of the smooth part, or restrictions on the subproblem solutions, such as exactness or termination conditions that are difficult to verify except in simple cases. These assumptions essentially confine the quadratic term in the approximation of the smooth part to being diagonal, so the benefits of second-order approximation are mostly lost. Moreover, in contrast to the smooth case, non-uniform sampling has not yet been shown to improve the convergence rate for regularized problems. In this work, we propose an inexact randomized BCD method based on a regularized quadratic subproblem, in which the quadratic term can vary from iteration to iteration (and is thus known as a variable metric'). We provide a detailed convergence analysis. When specialized to the non-regularized case, Nesterov's proposal to improve convergence rate by sampling proportional to the blockwise Lipschitz constants is covered in our framework. Empirical results also show that significant benefits are attainable when a variable quadratic term is used in the subproblem, rather than a fixed term.
Successive quadratic approximations, or second-order proximal methods, are useful for minimizing functions that are a sum of a smooth part and a convex, possibly nonsmooth part that promotes regularization. Most analyses of iteration complexity focus on the special case of proximal gradient method, or accelerated variants thereof. There have been only a few studies of methods that use a second-order approximation to the smooth part, due in part to the difficulty of obtaining closed-form solutions to the subproblems at each iteration. In fact, iterative algorithms may need to be used to find inexact solutions to these subproblems. In this work, we present global analysis of the iteration complexity of inexact successive quadratic approximation methods, showing that an inexact solution of the subproblem that is within a fixed multiplicative precision of optimality suffices to guarantee the same order of convergence rate as the exact version, with complexity related in an intuitive way to the measure of inexactness. Our result allows flexible choices of the second-order term, including Newton and quasi-Newton choices, and does not necessarily require increasing precision of the subproblem solution on later iterations. For problems exhibiting a property related to strong convexity, the algorithms converge at global linear rates. For general convex problems, the convergence rate is linear in early stages, while the overall rate is O(1 k). For nonconvex problems, a first-order optimality criterion converges to zero at a rate of (O(1 k ) ).