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.
The case in which is solved exactly is discussed in @cite_8 . This paper uses the same boundedness condition for @math as ours, and the blocks are selected either under a cyclic manner (with an arbitrary order), or a Gauss-Southwell fashion. For the cyclic variant, The convergence rate of the special case in which @math forms an upper bound of the objective improvement is further sharpened by @cite_22 @cite_5 . The relaxation to approximate subproblem solutions, with an inexactness criterion different from ours, is analyzed in @cite_18 . In the latter paper, the coefficients in their convergence rates are unclear, only linear or a certain type of sublinear convergence is shown, the relation between convergence rates and either the measure of inexactness or the choice @math is unclear. We note to that the cyclic ordering of blocks is inefficient in certain cases: @cite_25 showed that the worst case of cyclic BCD is @math times slower than the expected rate for randomized BCD.
@cite_18 @cite_22 @cite_8 @cite_5 @cite_25
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In this paper, we consider conic programming problems whose constraints consist of linear equalities, linear inequalities, a nonpolyhedral cone, and a polyhedral cone. A convenient way for solving this class of problems is to apply the directly extended alternating direction method of multipliers (ADMM) to its dual problem, which has been observed to perform well in numerical computations but may diverge in theory. Ideally, one should find a convergent variant which is at least as efficient as the directly extended ADMM in practice. We achieve this goal by designing a convergent semiproximal ADMM (called sPADMM3c for convenience) for convex programming problems having three separable blocks in the objective function with the third part being linear. At each iteration, the proposed sPADMM3c takes one special block coordinate descent (BCD) cycle with the order @math , instead of the usual @math Gauss--Seidel BCD cycle used in the nonconvergent directly extended 3-block ADMM, for updating the variable blocks. Our numerical experiments demonstrate that the convergent method is at least 20 faster than the directly extended ADMM with unit step-length for the vast majority of about 550 large-scale doubly nonnegative semidefinite programming problems with linear equality and or inequality constraints. This confirms that at least for conic convex programming, one can design a convergent and efficient ADMM with a special BCD cycle of updating the variable blocks. This paper concerns the worst-case complexity of Gauss-Seidel method for solving a positive semi-definite linear system; or equivalently, that of cyclic coordinate descent (C-CD) for minimizing a convex quadratic function. The known provable complexity of C-CD can be @math times slower than gradient descent (GD) and @math times slower than randomized coordinate descent (R-CD). However, these gaps seem rather puzzling since so far they have not been observed in practice; in fact, C-CD usually converges much faster than GD and sometimes comparable to R-CD. Thus some researchers believe the gaps are due to the weakness of the proof, but not that of the C-CD algorithm itself. In this paper we show that the gaps indeed exist. We prove that there exists an example for which C-CD takes at least @math or @math operations, where @math is the condition number, @math is a well-studied quantity that determines the convergence rate of R-CD, and @math hides the dependency on @math . This implies that C-CD can indeed be @math times slower than GD, and @math times slower than R-CD in the worst case. Our result establishes one of the few examples in continuous optimization that demonstrates a large gap between a deterministic algorithm and its randomized counterpart. Based on the example, we establish several almost tight complexity bounds of C-CD for quadratic problems. One difficulty with the analysis of the constructed example is that the spectral radius of a non-symmetric iteration matrix does not necessarily constitutes a lower bound for the convergence rate. The block coordinate descent (BCD) method is widely used for minimizing a continuous function @math of several block variables. At each iteration of this method, a single block of variables is optimized, while the remaining variables are held fixed. To ensure the convergence of the BCD method, the subproblem of each block variable needs to be solved to its unique global optimal. Unfortunately, this requirement is often too restrictive for many practical scenarios. In this paper, we study an alternative inexact BCD approach which updates the variable blocks by successively minimizing a sequence of approximations of @math which are either locally tight upper bounds of @math or strictly convex local approximations of @math . The main contributions of this work include the characterizations of the convergence conditions for a fairly wide class of such methods, especially for the cases where the objective functions are either nondifferentiable or nonconvex. Our results unify and extend the existing convergence results ... Augmented Lagrangian method (ALM) has been popularly used for solving constrained optimization problems. Practically, subproblems for updating primal variables in the framework of ALM usually can only be solved inexactly. The convergence and local convergence speed of ALM have been extensively studied. However, the global convergence rate of the inexact ALM is still open for problems with nonlinear inequality constraints. In this paper, we work on general convex programs with both equality and inequality constraints. For these problems, we establish the global convergence rate of the inexact ALM and estimate its iteration complexity in terms of the number of gradient evaluations to produce a primal and or primal-dual solution with a specified accuracy. We first establish an ergodic convergence rate result of the inexact ALM that uses constant penalty parameters or geometrically increasing penalty parameters. Based on the convergence rate result, we then apply Nesterov’s optimal first-order method on each primal subproblem and estimate the iteration complexity of the inexact ALM. We show that if the objective is convex, then (O( ^ -1 ) ) gradient evaluations are sufficient to guarantee a primal ( )-solution in terms of both primal objective and feasibility violation. If the objective is strongly convex, the result can be improved to (O( ^ - 1 2 | |) ). To produce a primal-dual ( )-solution, more gradient evaluations are needed for convex case, and the number is (O( ^ - 4 3 ) ), while for strongly convex case, the number is still (O( ^ - 1 2 | |) ). Finally, we establish a nonergodic convergence rate result of the inexact ALM that uses geometrically increasing penalty parameters. This result is established only for the primal problem. We show that the nonergodic iteration complexity result is in the same order as that for the ergodic result. Numerical experiments on quadratically constrained quadratic programming are conducted to compare the performance of the inexact ALM with different settings.