Regret without Optimal-Arm Feedback under Reward-Dependent Delays

We study stochastic bandits with reward-dependent delays, possibly infinite, whose reports omit the originating round. For each $c\in[0,1)$, Balanced Historical Optimism (BHO) has a delay term controlled by the largest suboptimal-arm quantile at level $1-cΔ_i$, without requiring optimal-arm reports. Lower bounds give an inverse-square slack cost in the logarithmic statistical regime and linear minimax regret at $c=1$. A family with growing $K$ forces worst-case regret $Ω(G\log K)$ for every learner, where $G=\max_iΔ_i h_i$ is the largest gap-weighted suboptimal cutoff. Pending-triggered balance preserves BHO's weighted-cutoff bound for $c<1$ and attains the historical-greedy exploration coefficient with deterministic rewards, a unique optimum and fixed bounded delays. Gap-weighted comparisons and experiments show when shared waiting helps and what balanced exploration costs.