English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Tools annual

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Tools monthly

Global: Zendy Plus monthly

Presents the access of premium content as premium feature

Premium Content

Global: Zendy Plus annual

Global: Zendy Free Plan

We provide novel theoretical results regarding local optima of regularized$M$-estimators, allowing for nonconvexity in both loss and penalty functions.Under restricted strong convexity on the loss and suitable regularityconditions on the penalty, we prove that \emph{any stationary point} of thecomposite objective function will lie within statistical precision of theunderlying parameter vector. Our theory covers many nonconvex objectivefunctions of interest, including the corrected Lasso for errors-in-variableslinear models; regression for generalized linear models with nonconvexpenalties such as SCAD, MCP, and capped-$\ell_1$; and high-dimensionalgraphical model estimation. We quantify statistical accuracy by providingbounds on the $\ell_1$-, $\ell_2$-, and prediction error between stationarypoints and the population-level optimum. We also propose a simple modificationof composite gradient descent that may be used to obtain a near-global optimumwithin statistical precision $\epsilon$ in $\log(1/\epsilon)$ steps, which isthe fastest possible rate of any first-order method. We provide simulationstudies illustrating the sharpness of our theoretical results.

Regularized M-estimators with nonconvexity: Statistical and algorithmic theory for local optima