Convergence Analysis of Stochastic Saddle Point Mirror Descent Algorithm: A Projected Dynamical Viewpoint

Saddle point problems, ubiquitous in optimization, extend beyond game theory to diverse domains like power networks and reinforcement learning. We present novel approaches to tackle saddle point problem, with a focus on continuous-time contexts. A continuous time dynamics is proposed to tackle saddl...

Celý popis

Uložené v:
Podrobná bibliografia
Vydané v:IEEE transactions on automatic control s. 1 - 16
Hlavní autori: Paul, Anik Kumar, Mahindrakar, Arun D, Kalaimani, Rachel K
Médium: Journal Article
Jazyk:English
Vydavateľské údaje: IEEE 2025
Predmet:
Almost sure convergence
Approximation algorithms
Asymptotic stability
Convergence
Dynamical systems
Heuristic algorithms
mirror descent algorithm
Mirrors
Optimization
projected dynamical system
Signal processing algorithms
Smoothing methods
Vectors
ISSN:0018-9286, 1558-2523
On-line prístup:Získať plný text
Tagy: Pridať tag
Žiadne tagy, Buďte prvý, kto otaguje tento záznam!
Popis
Shrnutí:Saddle point problems, ubiquitous in optimization, extend beyond game theory to diverse domains like power networks and reinforcement learning. We present novel approaches to tackle saddle point problem, with a focus on continuous-time contexts. A continuous time dynamics is proposed to tackle saddle point problem utilizing a projected dynamical system in a non-Euclidean domain. This involves computing the (sub/super) gradient of the min-max function within a Riemannian metric. Additionally, we establish viable Carathéodory solutions and prove the Lyapunov stability and asymptotic set stability of the proposed continuous time dynamical system. Next, we present the Stochastic Saddle Point Mirror Descent (SSPMD) algorithm and establish its equivalence with the proposed continuous-time dynamics. Leveraging stability analysis of the continuous-time dynamics, we demonstrate the almost sure convergence of the algorithm's iterates. Furthermore, we introduce the Stochastic Zeroth-Order Saddle Point Mirror Descent (SZSPMD) algorithm, which approximates gradients using Gaussian Approximation, showcasing convergence to a neighbourhood about saddle points. The analysis in this paper provides geometric insights into the mirror descent algorithm. It demonstrates how these insights offer theoretical foundations for analysis of different variants of stochastic mirror descent algorithm.
ISSN:0018-9286
1558-2523
DOI:10.1109/TAC.2025.3586762