Riemannian Geometry of Optimal Rebalancing in Dynamic Weight Automated Market Makers

2026-03-05

AI Summary

This study addresses the minimization of arbitrage losses in dynamic-weight automated market makers (AMMs) during asset rebalancing, focusing on the choice of weight trajectories. By formulating the trajectory optimization as a geodesic problem on a Riemannian manifold, the authors derive the Fisher–Rao metric via the Kullback–Leibler divergence and demonstrate that spherical linear interpolation (SLERP) in Hellinger coordinates yields near-optimal rebalancing. This work establishes, for the first time, a connection between dynamic AMM rebalancing and information geometry, revealing the optimality of SLERP and rigorously proving its mathematical equivalence to the recursive AM-GM heuristic for any number of assets and arbitrary weight shifts. Theoretical analysis shows that SLERP’s relative suboptimality scales quadratically with total weight change and inversely with the square of the number of interpolation steps; furthermore, recursive AM-GM enables efficient computation of geodesic midpoints without trigonometric operations.

Technology Category

• Search and Optimization: Mixed Discrete/Continuous Search
• Multiagent Systems: Mechanism Design
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Economics, Online Markets and Human Computation: Uses of LLMs and GenAI for marketplace design, bidding, and strategic interactions
• Responsible Web: Machine-in-the-loop, human agency and autonomy

Abstract

In Temporal Function Market Making (TFMM), a dynamic weight AMM pool rebalances from initial to final holdings by creating a series of arbitrage opportunities whose total cost depends on the weight trajectory taken. We show that the per-step arbitrage loss is the KL divergence between new and old weight vectors, meaning the Fisher--Rao metric is 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 $\eta_i = \sqrt{w_i}$, i.e.\ 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.

Problem

• Dynamic Weight AMM
• Optimal Rebalancing
• Arbitrage Loss
• Weight Trajectory
• KL Divergence

Innovation

• Riemannian geometry
• dynamic weight AMM
• KL divergence
• SLERP
• Fisher–Rao metric