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).

Variants of the coordinate descent approach for minimizing a nonlinear function are distinguished in part by the order in which coordinates are considered for relaxation. Three common orderings are cyclic (CCD), in which we cycle through the components of x in order; randomized (RCD), in which the component to update is selected randomly and independently at each iteration; and random-permutations cyclic (RPCD), which differs from CCD only in that a random permutation is applied to the variables at the start of each cycle. Known convergence guarantees are weaker for CCD and RPCD than for RCD, though in most practical cases, computational performance is similar among all these variants. There is a certain type of quadratic function for which CCD is significantly slower than for RCD; a recent paper by Sun & Ye (2016) has explored the poor behavior of CCD on functions of this type. The RPCD approach performs well on these functions, even better than RCD in a certain regime. This paper explains the good behavior of RPCD with a tight analysis.

Prediction of compounds that are active against a desired biological target is a common step in drug discovery efforts. Virtual screening methods seek some active-enriched fraction of a library for experimental testing. Where data are too scarce to train supervised learning models for compound prioritization, initial screening must provide the necessary data. Commonly, such an initial library is selected on the basis of chemical diversity by some pseudo-random process (for example, the first few plates of a larger library) or by selecting an entire smaller library. These approaches may not produce a sufficient number or diversity of actives. An alternative approach is to select an informer set of screening compounds on the basis of chemogenomic information from previous testing of compounds against a large number of targets. We compare different ways of using chemogenomic data to choose a small informer set of compounds based on previously measured bioactivity data. We develop this Informer-Based-Ranking (IBR) approach using the Published Kinase Inhibitor Sets (PKIS) as the chemogenomic data to select the informer sets. We test the informer compounds on a target that is not part of the chemogenomic data, then predict the activity of the remaining compounds based on the experimental informer data and the chemogenomic data. Through new chemical screening experiments, we demonstrate the utility of IBR strategies in a prospective test on three kinase targets not included in the PKIS.

We consider coordinate descent methods on convex quadratic problems, in which exact line searches are performed at each iteration. (This algorithm is identical to Gauss-Seidel on the equivalent symmetric positive definite linear system.) We describe a class of convex quadratic problems for which the random-permutations version of cyclic coordinate descent (RPCD) outperforms the standard cyclic coordinate descent (CCD) approach, yielding convergence behavior similar to the fully-random variant (RCD). A convergence analysis is developed to explain the empirical observations.