One-Shot Klein Cutting Planes for Lipschitz Geodesically Convex Optimization in Hyperbolic Space

Abstract

Motivated by the COLT 2023 open problem of Criscitiello, Mart\'inez-Rubio, and Boumal on deterministic first-order methods for Lipschitz geodesically convex optimization on Hadamard manifolds, we study hyperbolic space \[ \HH^d_{-\kappaC^2} =\{X\in\R^{d+1}:\ipL{X}{X}=-1,\ X_0>0\}, \qquad \ip{U}{V}_X=\kappaC^{-2}\ipL{U}{V}. \] For every geodesically convex $M$-Lipschitz function \[ f:\bar B_{\HH}(x_0,r)\to\R,\qquad s=\kappaC r, \] we give a one-shot Klein cutting-plane method returning a queried point $\hat x$ such that \[ f(\hat x)-\min_{\bar B_{\HH}(x_0,r)}f\le \eps Mr \] after at most \[ \left\lceil 2d(d+1)\log\!\left(\frac{16\sinh s\cosh s}{s\eps}\right) \right\rceil \] oracle calls. For $d\ge2$, each localization step costs $O(d^2)$ arithmetic operations; for $d=1$, an interval variant gives the same oracle bound. Hence \[ N=O\bigl(d^2(s+\log(e/\eps))\bigr) =O\bigl(d^2\zeta_s\log(e/\eps)\bigr), \qquad \zeta_s=s/\tanh s . \] Compared with the constant-curvature construction associated with the COLT problem, this replaces chained curvature--accuracy dependence by additive dependence. The proof does not rely on convexity of the Klein pullback, which is generally only quasiconvex. Instead, every Riemannian subgradient halfspace becomes an exact Euclidean central cut: for $\theta=\kappaC\dist(X,Y)$, \[ \ip{g}{\log_XY}_X =\frac{\theta}{\kappaC^2\sinh\theta}\ipL{g}{Y}, \] and tangency at $X$ converts $\ipL{g}{Y}\le0$ into \[ \gbar^{\mathsf T}(u-c)\le0,\qquad u=\Phi(Y),\ c=\Phi(X). \] Thus one fixed Euclidean ellipsoid localizes the hyperbolic ball, and curvature enters only through \[ \log\!\left(\frac{\sinh s\cosh s}{s\eps}\right) =\log(1/\eps)+2s-\log(4s)+O(e^{-4s}). \] The general Hadamard-manifold problem remains open.