The Statistical Price of Unknown Transient Amplitude in Best-Asymptote Identification

When several learning or optimization runs share a budget of updates, the run with the smallest current loss need not be the run with the smallest limiting loss because the runs are still improving. We study how to identify the run with the best limit (best-asymptote identification) when the $s$th update of run $i$ produces $Y_{i,s}={\mu }_i+{\beta }_i{s}^{-r}+{\epsilon }_{i,s}$ with Gaussian noise: each pull advances only the arm it is spent on, the decay exponent $r$ and the noise variance ${\sigma }^2$ are known, and the limits ${\mu }_i$ and the transient amplitudes ${\beta }_i$ are not. The question is how much not knowing the amplitudes costs compared with an oracle that reveals them. We give three exact answers. For a single arm, the sharp asymptotic variance for estimating ${\mu }_i$ is ${\sigma }^2\kappa (r)/n$ instead of the oracle's ${\sigma }^2/n$, where $\kappa (r)=(1-r{)}^2/r^2$ for $0<r<1/2$ and $\kappa (r)=1$ for $r\ge 1/2$; unknown amplitude, therefore, has a first-order cost only when the transient decays slowly. For two arms whose limits differ by $c/\sqrt{T}$, with amplitudes in any fixed compact subinterval of $(0,\infty )$ and a budget cap ${\tau }_T\le CT$ almost surely, the exact local asymptotic minimax misidentification error over all parameter-free policies with predictable, history-dependent allocation and adapted stopping is $\Phi (-c\sqrt{C}/[2\sigma \sqrt{\kappa (r)}])$; it is attained by deterministic balanced sampling with full-prefix least squares, and the oracle value is the same formula with $\kappa (r)=1$. Hence, no adaptive allocation or stopping rule can remove the nuisance penalty: matching the oracle requires $\kappa (r)$ times as much first-order budget. For $K$ arms, when at most one challenger may approach the best arm at the $T^{-1/2}$ scale and every other arm keeps a fixed margin, a sublinear screening pilot followed by an independent-suffix comparison attains the two-arm value, and the optimized deterministic all-arm-prefix budget is $K/2$ times larger. Exact finite-prefix risk calculations and Gaussian simulations illustrate the boundary at $r=1/2$ and the screening mechanism.