Target Mirror Descent: A Unifying Framework for Solving Monotone Variational Inequalities
Provides a unified theoretical framework for multiple landmark algorithms in optimization, with potential to simplify analysis and inspire new methods.
Target Mirror Descent (TMD) unifies several algorithms for monotone variational inequalities, including extragradient and proximal point methods, and corrects equilibrium misalignment in discounted mirror descent. The framework enables geometric ensembles that inherit convergence guarantees.
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.