Riemannian Geometry of Optimal Rebalancing in Dynamic Weight Automated Market Makers
This work provides a theoretical foundation for optimizing rebalancing strategies in automated market makers, which is incremental as it builds on prior heuristics like the AM-GM method.
The paper tackles the problem of minimizing loss during weight rebalancing in dynamic-weight automated market makers by showing that the per-step log loss equals the KL divergence between successive weight vectors, leading to the Fisher-Rao metric as the natural Riemannian geometry. It finds that SLERP (Spherical Linear Interpolation) in Hellinger coordinates is the loss-minimizing interpolation under a leading-order expansion, with its relative sub-optimality proportional to the squared magnitude of weight change and inversely proportional to the square of interpolation steps, and identifies a finite optimal step count under LVR exposure.
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 $η_i = \sqrt{w_i}$: 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.