Shared-Observation Chaining for Hierarchical Adversarial Noisy Bandits

We study finite-time regret in adversarial noisy bandits with bounded conditional-mean feedback. Fixed-scale optimizer-cover scalars are not rate-complete: two finite-action classes agree on a nontrivial scale range yet have exponentially different regret. On a product routing model with $2^D$ templates, a known-horizon $O(D)$-time learner uses one route/reward observation to update all active levels and attains expected $O(\sqrt{T})$ regret against adaptive nonanticipating environments; an oblivious Bernoulli subclass matches this rate. Even the optimistic fixed-rate one-global-Shannon ExO expression, with arbitrary finite real-valued Borel scores, is $\Omega (\min \{T,\sqrt{DT}\})$. A simultaneous product packing matches the complete active-coordinate finite-horizon profile ${\sum }_{j\in J_{+}}{\lambda }_j\min \{{\alpha }_{j}T,\sqrt{T}\}$; in the information-limited regime this becomes $\sqrt{T}{\sum }_{j\in J_{+}}{\lambda }_j$, without a block-allocation depth loss. Under explicit margin, finite-range, matching, quadratic-regime, allocation, saturation, and terminal-control conditions, general optimizer hierarchies admit a conditional chain–packing comparison.