Kullback-Leibler Mirror-Prox for Measure-Valued Variational Inequalities and Mean-Field Equilibria

What the math gives to ML

The paper recasts mean-field equilibrium computation as a variational inequality over probability measures and replaces Euclidean extragradient projections with explicit KL-proximal exponential reweighting. The transferable asset is an entropy-aware Mirror-Prox update that preserves positivity and normalization while using a predictor-corrector evaluation of a generally non-gradient operator, which is directly relevant to attention distributions, MoE routing probabilities, and other simplex-valued neural parameters. A practical adaptation is to apply the update to logits or routing probabilities when the objective contains strategic coupling between examples, tokens, experts, or agents; the Minty residual supplies a measurable convergence diagnostic. Its KL Tikhonov variant additionally provides a principled way to prevent probability collapse while selecting the solution closest in relative entropy to a reference distribution.