AbstractWe study minimization of a structured objective function, being the sum of a smooth function and a composition of a weakly convex function with a linear operator. Applications include image reconstruction problems with regularizers that introduce less bias than the standard convex regularizers. We develop a variable smoothing algorithm, based on the Moreau envelope with a decreasing sequence of smoothing parameters, and prove a complexity of $${\mathcal {O}}(\epsilon ^{-3})$$
O
(
ϵ
-
3
)
to achieve an $$\epsilon $$
ϵ
-approximate solution. This bound interpolates between the $${\mathcal {O}}(\epsilon ^{-2})$$
O
(
ϵ
-
2
)
bound for the smooth case and the $${\mathcal {O}}(\epsilon ^{-4})$$
O
(
ϵ
-
4
)
bound for the subgradient method. Our complexity bound is in line with other works that deal with structured nonsmoothness of weakly convex functions.
Proximal Methods Avoid Active Strict Saddles of Weakly Convex Functions
