Manifold-Aware Information Gain and Lower Bounds for Gaussian-Process Bandits on Riemannian Quotient Spaces

We prove a regret lower bound for Gaussian-process bandits on a smooth compact Riemannian manifold $\text{M}$ of dimension $d$ with intrinsic Matérn-$\nu$ kernel ($\nu >d\mathrm{/}2$) that exposes how the geometry of the arm space enters the constant. For any algorithm and time horizon $T$ exceeding an explicit threshold, the worst-case expected regret over the RKHS-ball $\mathrm{\parallel }f{\mathrm{\parallel }}_{{\text{Hil}}_{k_{\nu }}}\text{\hspace{0.17em}⁣}\le \text{\hspace{0.17em}⁣}B$ satisfies \begin{multline*} \E[R_T(f)];\ge;c_(d,ν),B^{d/(2ν+d)},σ_n^{2ν/(2ν+d)} \ \cdot,\vol_g(\M)^{ν/(2ν+d)},T^{(ν+d)/(2ν+d)}(\log T)^{ν/(2ν+d)}. \end{multline} The exponent matches the Vakili–Khezeli–Picheny upper bound \cite{vakili2021information}; the ${\text{vol}}_g(\text{M}{)}^{\nu \mathrm{/}(2\nu +d)}$ factor is, to our knowledge, the first explicit volume-dependent geometric constant in a manifold GP-bandit lower bound. We extend the analysis in five directions: (i)~a companion Assouad-style proof gives a different lower bound with a strictly smaller $T$-exponent $(2\nu +3d)\mathrm{/}(4(\nu +d))$ but with a polylog factor of the form $1\mathrm{/}(\log \log T{)}^{(2\nu +d)\mathrm{/}(4(\nu +d))}$, sharpening the $(\log T{)}^{\nu \mathrm{/}(2\nu +d)}$ Fano polylog of Theorem~\ref{thm:main}; (ii)~we prove a $\mathrm{\mid }G{\mathrm{\mid }}^{1\mathrm{/}2}$ upper bound on the regret of an extrinsic-kernel GP-UCB algorithm on a quotient space $\text{M}=\text{Mt}\mathrm{/}G$, plus a bracketing theorem (Theorem~\ref{thm:gauge-bracket}); the precise constant is conjectured to take the modulated form $(1+(\mathrm{\mid }G\mathrm{\mid }-1)h(\text{rinj}\mathrm{/}\kappa ){)}^{1\mathrm{/}2}$ (Conjecture~\ref{conj:gauge-modulated}), validated numerically on $\text{SO}(3)$; (iii)~we write the leading constant $c_{\ast }(d,\nu )$ out fully; (iv)~we extract a curvature dependence $1+O(K{\text{eps}}_T^2)$ via Bishop–Gromov; (v)~we transfer the bound to the Bayesian regret framework via the Yang–Barron / Castillo et al.\ Bayesian-Fano transfer.