Paper: arXiv 2603.05326

Authors: Matthew Willetts

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 $η_i = \sqrt{w_i}$: 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.

Complexity vs Empirical Score

  • Math Complexity: 8.0/10
  • Empirical Rigor: 3.5/10
  • Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: The paper presents an elegant differential-geometric framework for AMM rebalancing, linking arbitrage loss to KL divergence and proving why the (AM+GM)/normalise heuristic works via SLERP on a Hellinger sphere. While highly novel and mathematically sound with strong theoretical derivations, empirical rigor is limited primarily to theoretical and numerical validations rather than extensive real-world market backtesting.

Research Flowchart

  flowchart TD
    A["Research Goal: Minimize arbitrage rebalancing loss in dynamic-weight AMMs"] --> B["Inputs: Initial/Target weights w, Arbitrage loss model, Price dynamics (GBM)"]
    B --> C["Establish Foundations: Prove log-loss = KL divergence & Fisher-Rao metric"]
    C --> D["Geodesic Optimization: Derive SLERP interpolation in Hellinger coordinates η = √w"]
    D --> E["Algorithmic Link: Prove SLERP midpoint = AM-GM heuristic & enable recursive bisection"]
    D --> F["LVR Analysis: Evaluate trade-off between KL divergence loss and LVR exposure"]
    E & F --> G["Key Outcomes: Constant-speed SLERP minimizes loss; optimal finite steps f* exist"]