Riemannian Bandit Convex Optimization with Self-Concordant Geometry

We study adversarial bandit convex optimization on Riemannian manifolds without a global affine structure. Our intrinsic barrier-proximal method combines exponential-map Dikin exploration with single-step updates based on scalar logarithmic surrogates. A squared-distance analysis avoids aggregating tangent vectors across base points or assuming surrogate convexity. We bound the curvature-induced estimator-to-smoothing support error by $C_{\mathrm{sm}}{\delta }^2$ under $C^2$ loss regularity. Conditional on compact-interior iterates and uniform smooth normal-tube data, we establish $\tilde{\mathcal{O}}(T^{2/3})$ one-point regret against nonanticipating adversaries and provide a two-point extension. Covariance-tracking experiments on symmetric positive definite manifolds demonstrate effective parameter transfer and query-loss improvements over feedback-matched baselines under both one- and two-point feedback.