Stochastic Mirror Descent Dynamics and Their Convergence in Monotone Variational Inequalities

Area of Science:

• Optimization Theory
• Stochastic Analysis
• Game Theory

Background:

• Monotone variational inequalities are fundamental in modeling equilibrium problems.
• Stochastic mirror descent offers a framework for analyzing dynamic systems with uncertainty.
• Understanding convergence properties is crucial for practical applications.

Purpose of the Study:

• To analyze stochastic mirror descent dynamics for monotone variational inequalities.
• To investigate the impact of controllable weight parameters on system convergence.
• To establish theoretical guarantees for convergence and deviation behavior.

Main Methods:

• Formulating dynamics as a stochastic differential equation driven by a monotone operator.
• Utilizing Brownian motion for stochastic perturbation.
• Analyzing convergence through ergodic properties and rate estimation.
• Establishing a large deviations principle for trajectory analysis.

Main Results:

• Demonstrated global convergence in the ergodic sense by tuning weight sequences.
• Estimated the average rate of convergence for the stochastic process.
• Established a large deviations principle, showing exponential concentration of trajectories.

Conclusions:

• The proposed stochastic mirror descent dynamics provide a robust method for solving monotone variational inequalities.
• Parameter tuning is key to achieving predictable and efficient convergence.
• The findings offer theoretical insights into the behavior of complex dynamic systems.