Target Mirror Descent: A Unifying Framework for Solving Monotone Variational Inequalities

2026-04-20

AI Summary

This work addresses the instability of classical mirror descent in monotone variational inequality problems, where the method may diverge or cycle due to the absence of a stabilizing convergence mechanism. To overcome this limitation, the paper introduces the Targeted Mirror Descent (TMD) framework, which incorporates a target-point correction mechanism into the dual update to effectively stabilize the optimization dynamics. TMD unifies classical algorithms such as the proximal point method and extragradient method, revealing their underlying convergence principles. It also rectifies the equilibrium misalignment issue present in discounted mirror descent and enables geometric ensembling, allowing multiple heterogeneous mirror maps to operate collaboratively in parallel. The framework guarantees convergence for monotone variational inequalities and facilitates the construction of novel mirror maps, substantially enhancing algorithmic robustness and applicability.

Technology Category

• Multiagent Systems: Mechanism Design
• Game Theory and Economic Paradigms: Mechanism Design
• Search and Optimization: Mixed Discrete/Continuous Search

Application Category

• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• User Modeling, Personalization and Recommendation: User modeling for targeted and personalized online advertising
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

Abstract

It is well known that mirror descent may diverge or cycle on merely monotone variational inequalities. In this paper, we propose \emph{Target Mirror Descent} (TMD), a unified framework that stabilizes monotone flows via a target point correction mechanism in the dual update. By appropriate design choices, TMD recovers the proximal point algorithm, extragradient methods, splitting methods, Brown-von Neumann-Nash dynamics, forward-backward-forward dynamics, and discounted mirror descent as special cases. Thus, we establish a unified perspective on these landmark algorithms and their convergence. Beyond unification, we leverage the TMD framework to correct an equilibrium misalignment in discounted mirror descent and to generalize its higher-order extension beyond interior solutions. Moreover, a key structural feature of TMD is the explicit decoupling of the mirror map from the target determination, which enables \emph{geometric ensembles}: multiple algorithms solve the same problem in parallel using distinct mirror maps, while sharing a common dual update. We show that such an ensemble rigorously reduces to a single TMD with a synthesized mirror map, and thus inherits these convergence guarantees.

Problem

• monotone variational inequalities
• mirror descent
• divergence
• cycling
• stability

Innovation

• Target Mirror Descent
• monotone variational inequalities
• mirror map decoupling
• geometric ensembles
• unified algorithmic framework