Exact graceful degradation of ISM performance

Characterize the exact graceful degradation of the Importance Weights for Shifting Means (ISM) algorithm’s performance as the adversary’s available shifting range decreases relative to the mean gaps, beyond the constant-factor sample-complexity bounds established for the Shifting Means setting.

Background

The paper derives upper and lower bounds on the inverse maximin drift rate $1/D^*(\boldsymbol\Delta)$, showing that ISM has sample complexity scaling with $\sigma^2+U^2$ in the regime where the adversary can substantially shift the means. The lower bound in Theorem 2 requires the gaps to be sufficiently small relative to the mean range, whereas the general upper and lower bounds do not coincide exactly across all parameter regimes.

The unresolved issue is to determine precisely how ISM’s performance transitions between the strongly shifted regime and the stationary best-arm-identification regime, rather than bounding this transition only up to universal constant factors.