ADMMc function

Penalized precision matrix estimation via ADMM (c++)

Penalized precision matrix estimation using the ADMM algorithm

ADMMc(S, initOmega, initZ, initY, lam, alpha = 1, diagonal = FALSE,
  rho = 2, mu = 10, tau_inc = 2, tau_dec = 2, crit = "ADMM",
  tol_abs = 1e-04, tol_rel = 1e-04, maxit = 10000L)

Arguments

• S: pxp sample covariance matrix (denominator n).
• initOmega: initialization matrix for Omega
• initZ: initialization matrix for Z
• initY: initialization matrix for Y
• lam: postive tuning parameter for elastic net penalty.
• alpha: elastic net mixing parameter contained in [0, 1]. 0 = ridge, 1 = lasso. Defaults to alpha = 1.
• diagonal: option to penalize the diagonal elements of the estimated precision matrix ($\Omega$). Defaults to FALSE.
• rho: initial step size for ADMM algorithm.
• mu: factor for primal and residual norms in the ADMM algorithm. This will be used to adjust the step size rho after each iteration.
• tau_inc: factor in which to increase step size rho.
• tau_dec: factor in which to decrease step size rho.
• crit: criterion for convergence (ADMM or loglik). If crit = loglik then iterations will stop when the relative change in log-likelihood is less than tol.abs. Default is ADMM and follows the procedure outlined in Boyd, et al.
• tol_abs: absolute convergence tolerance. Defaults to 1e-4.
• tol_rel: relative convergence tolerance. Defaults to 1e-4.
• maxit: maximum number of iterations. Defaults to 1e4.

Returns

returns list of returns which includes: - Iterations: number of iterations.

• lam: optimal tuning parameters.

• alpha: optimal tuning parameter.

• Omega: estimated penalized precision matrix.

• Z2: estimated Z matrix.

• Y: estimated Y matrix.

• rho: estimated rho.

Details

For details on the implementation of 'ADMMsigma', see the vignette https://mgallow.github.io/ADMMsigma/.

References

• Boyd, Stephen, Neal Parikh, Eric Chu, Borja Peleato, Jonathan Eckstein, and others. 2011. 'Distributed Optimization and Statistical Learning via the Alternating Direction Method of Multipliers.' Foundations and Trends in Machine Learning 3 (1). Now Publishers, Inc.: 1-122. https://web.stanford.edu/~boyd/papers/pdf/admm_distr_stats.pdf
• Hu, Yue, Chi, Eric C, amd Allen, Genevera I. 2016. 'ADMM Algorithmic Regularization Paths for Sparse Statistical Machine Learning.' Splitting Methods in Communication, Imaging, Science, and Engineering. Springer: 433-459.
• Zou, Hui and Hastie, Trevor. 2005. "Regularization and Variable Selection via the Elastic Net." Journal of the Royal Statistial Society: Series B (Statistical Methodology) 67 (2). Wiley Online Library: 301-320.
• Rothman, Adam. 2017. 'STAT 8931 notes on an algorithm to compute the Lasso-penalized Gaussian likelihood precision matrix estimator.'

Author(s)

Matt Galloway gall0441@umn.edu