Riemannian Geometry of Optimal Rebalancing in Dynamic Weight Automated Market Makers

TL;DR

This paper explores the Riemannian geometry underlying optimal rebalancing in dynamic-weight AMMs, revealing that KL divergence defines the cost and SLERP provides an efficient interpolation method.

Contribution

It establishes the Fisher-Rao metric as natural for weight rebalancing and connects SLERP to geodesics on the weight simplex, extending prior heuristics.

Findings

KL divergence measures per-step log loss in rebalancing.

SLERP interpolates weights along geodesics, matching prior heuristics.

SLERP's sub-optimality scales with weight change magnitude and number of steps.

Abstract

We show that when a dynamic-weight AMM rebalances by creating arbitrage opportunities, the per-step log loss is the KL divergence between successive weight vectors. The Fisher-Rao metric is therefore the natural Riemannian metric on the weight simplex. The loss-minimising interpolation under the leading-order expansion of this KL cost is SLERP (Spherical Linear Interpolation) in the Hellinger coordinates $\eta_i = \sqrt{w_i}$: a geodesic on the positive orthant of the unit sphere, traversed at constant speed. The SLERP midpoint equals the (AM+GM)/normalise heuristic of prior work (Willetts & Harrington, 2024), so the heuristic lies on the geodesic. This identity holds for any number of tokens and any magnitude of weight change; using this link, all dyadic points on the geodesic can be reached by recursive AM-GM bisection without trigonometric functions. SLERP's relative sub-optimality on the full KL cost is proportional to the squared magnitude of the overall weight change and to $1/f^2$, where $f$ is the number of interpolation steps. Under driftless GBM prices, the fractional value loss from each weight update is price-independent, and the cross term between weight and price changes telescopes, so the constant-price geometry carries over. LVR exposure introduces a finite optimal step count $f^*$, which lies in the perturbative regime where SLERP remains near-optimal.