On the Linear Convergence of a Proximal Gradient Method for a Class of Nonsmooth Convex Minimization Problems

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Linear Convergence of Gradient and Proximal-Gradient Methods Under the Polyak-Łojasiewicz Condition (2016, background): “shown in Appendix G, the proximal-PL inequality also holds for other problems where a linear convergence rate has been show like group L1-regularization [Tseng, 2010], sparse group L1-regularization [Zhang et al., 2013], nuclear-norm regularization [Hou et al., 2013], and other classes of functions ”
A unified approach to error bounds for structured convex optimization problems (2015, unknown): “([38, Theorem 2]; cf. [44, Theorem 1]) E = R n , f takes the form f ( x ) = h ( Ax ), where A ∈ R m × n and h : R m → ( , + ] are as in scenario (S2), and P is the grouped LASSO regularizer; i.e. , P ( x ) = J ∈J ω J x J 2 , where J is a partition of the index set { 1 , . . . , n , x J ∈ R | J | is…”
On the linear convergence of the alternating direction method of multipliers (2012, unknown): “4) and under assumptions (a)–(g), the work of [34,45,52] suggests that the size of the proximal gradient”
On the Linear Convergence of the Proximal Gradient Method for Trace Norm Regularization (2013, methodology): “Prior to this work, Lipschitzian error bounds for composite function minimization are available for cases where the non–smooth part either has a polyhedral epigraph (such as the l1–norm) [23] or is the (sparse) group LASSO regularization [22, 25].”