TY - JOUR
T1 - Alternating and Parallel Proximal Gradient Methods for Nonsmooth, Nonconvex Minimax
T2 - A Unified Convergence Analysis
AU - Cohen, Eyal
AU - Teboulle, Marc
N1 - Publisher Copyright:
© 2024 INFORMS.
PY - 2024
Y1 - 2024
N2 - There is growing interest in nonconvex minimax problems that is driven by an abundance of applications. Our focus is on nonsmooth, nonconvex-strongly concave minimax, thus departing from the more common weakly convex and smooth models assumed in the recent literature. We present proximal gradient schemes with either parallel or alternating steps. We show that both methods can be analyzed through a single scheme within a unified analysis that relies on expanding a general convergence mechanism used for analyzing nonconvex, nonsmooth optimization problems. In contrast to the current literature, which focuses on the complexity of obtaining nearly approximate stationary solutions, we prove subsequence convergence to a critical point of the primal objective and global convergence when the latter is semialgebraic. Furthermore, the complexity results we provide are with respect to approximate stationary solutions. Lastly, we expand the scope of problems that can be addressed by generalizing one of the steps with a Bregman proximal gradient update, and together with a few adjustments to the analysis, this allows us to extend the convergence and complexity results to this broader setting.
AB - There is growing interest in nonconvex minimax problems that is driven by an abundance of applications. Our focus is on nonsmooth, nonconvex-strongly concave minimax, thus departing from the more common weakly convex and smooth models assumed in the recent literature. We present proximal gradient schemes with either parallel or alternating steps. We show that both methods can be analyzed through a single scheme within a unified analysis that relies on expanding a general convergence mechanism used for analyzing nonconvex, nonsmooth optimization problems. In contrast to the current literature, which focuses on the complexity of obtaining nearly approximate stationary solutions, we prove subsequence convergence to a critical point of the primal objective and global convergence when the latter is semialgebraic. Furthermore, the complexity results we provide are with respect to approximate stationary solutions. Lastly, we expand the scope of problems that can be addressed by generalizing one of the steps with a Bregman proximal gradient update, and together with a few adjustments to the analysis, this allows us to extend the convergence and complexity results to this broader setting.
KW - Bregman distance
KW - Kurdyka–Lojasiewicz property
KW - convergence rate
KW - global convergence
KW - nonconvex nonsmooth minimax
KW - nonsmooth minimization-maximization
KW - proximal gradient method
UR - http://www.scopus.com/inward/record.url?scp=85185195821&partnerID=8YFLogxK
U2 - 10.1287/moor.2022.0294
DO - 10.1287/moor.2022.0294
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AN - SCOPUS:85185195821
SN - 0364-765X
SP - 1
EP - 28
JO - Mathematics of Operations Research
JF - Mathematics of Operations Research
ER -