Riemannian Geometry of Optimal Rebalancing in Dynamic Weight Automated Market Makers
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| Format: | Preprint |
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2026
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| author | Willetts, Matthew |
| author_facet | Willetts, Matthew |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2603_05326 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Riemannian Geometry of Optimal Rebalancing in Dynamic Weight Automated Market Makers Willetts, Matthew Mathematical Finance Information Theory Differential Geometry Trading and Market Microstructure 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. |
| title | Riemannian Geometry of Optimal Rebalancing in Dynamic Weight Automated Market Makers |
| topic | Mathematical Finance Information Theory Differential Geometry Trading and Market Microstructure |
| url | https://arxiv.org/abs/2603.05326 |