Differentiable Multiple Shooting Layers

The paper’s Appendix A.1 proves the O(η_p^2) one-step Newton tracking bound by a second-order Taylor (2-jet) expansion of g(B,θ)=0 at (B*_p,θ_{p+1}), using that Dg=I−Dγ is invertible thanks to the strictly lower-triangular (nilpotent) structure of Dγ, and bounding the inverse via a finite Neumann series; this yields an explicit constant M in Eq. (A.8) . The candidate solution follows the same strategy: (i) represent the direct Newton update using the block lower–bidiagonal Jacobian, (ii) apply a second-order Taylor remainder identity, and (iii) control ΔB* via Lipschitz dependence on θ (via variational equations or an implicit function theorem argument). Differences are stylistic: the paper phrases bounds in terms of Lipschitz constants m^θ_γ, m^z_γ, while the model derives them from standard ODE flow sensitivities; both yield the same O(η_p^2) conclusion and rely on the same structural facts about the Jacobian and Newton’s method. The direct Newton iteration used by both matches Eq. (3.2) in the paper , and the training context and assumptions preceding Theorem 1 align with the model’s conditions (fixed z0, Lipschitz loss, γ Lipschitz in z and θ) .