Adding Fisher information regularization to a gradient flow should help. Fisher information penalizes distributions with sharp features, encouraging smooth descent toward equilibrium. It adds geometric structure — the Fisher metric — to the optimization landscape. More structure should mean better behavior.
Farmer, Kochar, and Lee show it can make things temporarily worse.
The Fisher-regularized Wasserstein gradient flow contains a cross-dissipation term whose sign depends on the state. When the distribution's width falls below a critical scale, this term turns positive. The Fisher channel — the geometric regularizer meant to assist descent — actively resists the decrease of the baseline free energy. The free energy still decreases overall (the baseline term dominates), but the descent is slower than it would be without the regularization.
On the Gaussian manifold, where exact analysis is possible, the variance equation has closed-form solutions revealing three regimes: initial Fisher-assisted descent, a paradoxical interval where the geometric term opposes the flow, and eventual convergence to a shifted equilibrium. The equilibrium itself is displaced — the logarithmic barrier from the Fisher term moves the minimum, so the system doesn't converge to the same point it would reach without regularization.
The interference duration correlates with the initial information distance between the starting distribution and the target. Distributions that start far away (high KL divergence) experience longer paradoxical intervals. The geometric structure that helps near equilibrium actively interferes far from it.
The result generalizes beyond Gaussians: bimodal and Laplace initial conditions show the same paradoxical slowdown in finite-difference simulations. The effect is not an artifact of the restricted manifold but a property of the cross-dissipation term itself.
The lesson is about the cost of structure. Geometric regularization is not free — it introduces an auxiliary dissipation channel that can flow in the wrong direction.