Discounted Online Convex Optimization with an Unknown Discount: Sharp Rates and Prefix–Endpoint Evaluation

We study online convex optimization evaluated with a geometric discount $\lambda \in (0,1)$ that is unknown to the learner, so one decision sequence must work for every discount. With known horizon and curvature parameter, bounded subgradients, and possibly nonsmooth losses, we use the known-discount benchmark $B_T^{\mathrm{or}}(\lambda )$. This benchmark is attainable and minimax-correct up to absolute constants on a normalized family. We introduce Sharp-BDF, which queries one genuine subgradient and no function values per round, while matching the benchmark's leading constant simultaneously for all $\lambda$. The method recycles the quadratic surplus of each base learner through a shared compensation coefficient and transfers the guarantee from a finite forgetting grid to the full discount interval without an $\epsilon$-net. The price of not knowing $\lambda$ depends on the evaluation protocol. The worst-case additive excess is $\Theta (\log N)$ when every prefix must be served, but $\Theta (\log \log N)$ when only a single fixed endpoint is evaluated. Matching lower bounds hold even for randomized full-feedback algorithms and persist after subtracting the true minimax value on the steady-state discount set. A hybrid variant has constant adaptation at both scale extremes.