Lower Bounds for $γ$-Regret via the Decision-Estimation Coefficient

In this note, we give a new lower bound for the $\gamma$-regret in bandit problems, the regret which arises when comparing against a benchmark that is $\gamma$ times the optimal solution, i.e., $\mathsf{Reg}_{\gamma}(T) = \sum_{t = 1}^T \gamma \max_{\pi} f(\pi) - f(\pi_t)$. The $\gamma$-regret arises in structured bandit problems where finding an exact optimum of $f$ is intractable. Our lower bound is given in terms of a modification of the constrained Decision-Estimation Coefficient (DEC) of~\citet{foster2023tight} (and closely related to the original offset DEC of \citet{foster2021statistical}), which we term the $\gamma$-DEC. When restricted to the traditional regret setting where $\gamma = 1$, our result removes the logarithmic factors in the lower bound of \citet{foster2023tight}.