MMD CONVERGENCE

Magnetic Mirror Descent and Last-Iterate Convergence

Takeaway: MMD with a fixed magnetic anchor achieves linear convergence to equilibrium on zero-sum games (F1–F3).

The Finding: On symmetric zero-sum games (Matching Pennies, Rock-Paper-Scissors), Magnetic Mirror Descent with a fixed-point anchor exhibits linear (geometric) last-iterate convergence to equilibrium while simultaneous gradient descent cycles, bounded away.

The Numbers (F1–F3):

Matching Pennies: MMD log-linear fit R²=0.9948 (linear convergence), GDA NashConv 1.93±[1.90,1.96] (cycling)

Rock-Paper-Scissors: MMD R²=0.9015, GDA NashConv 1.76±[1.62,1.88]

Regularized Nash Dynamics with periodic resets: reaches NashConv < 0.05 on both symmetric and asymmetric games

How It Works: MMD adds a "magnetic" term that pulls the player strategy toward a reference distribution (the anchor). On symmetric games, this anchor is the uniform distribution; on asymmetric games, it becomes context-dependent. The magnetic pull prevents the oscillatory behavior of GDA and forces convergence. The physics is straightforward: the fixed point becomes attractive instead of a saddle point.

Asymmetric Games (F2): The uniform-anchor MMD dynamics do NOT converge to the logit-QRE; instead, they reach a distinct context-dependent attractor. This is a discovery finding: the natural equilibrium notion for MMD differs from QRE on asymmetric settings.

Implication for Training: If preference learning or multi-model aggregation is a zero-sum game between model and reference, MMD provides convergence where Adam + GDA would cycle. The magnetic pull can be tuned (via τ, the temperature) to control exploration vs. exploitation at training time.