This paper introduces the Nyström PCG algorithm for solving a symmetric positivedefinite linear system. The algorithm applies the randomized Nyström method to form a low-rank approximation of the matrix, which leads to an efficient preconditioner that can be deployed with the conjugate gradient algorithm. Theoretical analysis shows that preconditioned system has constant condition number as soon as the rank of the approximation is comparable with the number of effective degrees of freedom in the matrix. The paper also develops adaptive methods for achieving similar performance without knowledge of the effective dimension. Numerical tests show that Nyström PCG can rapidly solve large linear systems that arise in data analysis problems, and it surpasses several competing methods from the literature.

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This paper presents GeNI-ADMM, a framework for large-scale composite convex optimization, that facilitates theoretical analysis of both existing and new approximate ADMM schemes. GeNI-ADMM encompasses any ADMM algorithm that solves a first-or second-order approximation to the ADMM subproblem inexactly. GeNI-ADMM exhibits the usual O(1/t)-convergence rate under standard hypotheses and converges linearly under additional hypotheses such as strong convexity. Further, the GeNI-ADMM framework provides explicit convergence rates for ADMM variants accelerated with randomized linear algebra, such as NysADMM and sketchand-solve ADMM, resolving an important open question on the convergence of these methods. This analysis quantifies the benefit of improved approximations and can inspire the design of new ADMM variants with faster convergence.

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This paper develops a scalable new algorithm, called NysADMM, to minimize a smooth convex loss function with a convex regularizer. NysADMM accelerates the inexact Alternating Direction Method of Multipliers (ADMM) by constructing a preconditioner for the ADMM subproblem from a randomized low-rank Nyström approximation. NysADMM comes with strong theoretical guarantees: it solves the ADMM subproblem in a constant number of iterations when the rank of the Nyström approximation is the effective dimension of the subproblem regularized Gram matrix. In practice, ranks much smaller than the effective dimension can succeed, so NysADMM uses an adaptive strategy to choose the rank that enjoys analogous guarantees. Numerical experiments on real-world datasets demonstrate that NysADMM can solve important applications, such as the lasso, logistic regression, and support vector machines, in half the time (or less) required by standard solvers. The breadth of problems on which NysADMM beats standard solvers is a surprise: it suggests that ADMM is a dominant paradigm for numerical optimization across a wide range of statistical learning problems that are usually solved with bespoke methods.