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

Abstract

In this note, we give a new lower bound for the γ-regret in bandit problems, the regret which arises when comparing against a benchmark that is γ times the optimal solution, i.e., Reg γ (T ) = T t=1 γ maxπ f (π) − f (πt). The γ-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 Foster et al. [2023] (and closely related to the original offset DEC of Foster et al. [2021]), which we term the γ-DEC. When restricted to the traditional regret setting where γ = 1, our result removes the logarithmic factors in the lower bound of Foster et al. [2023].