Variance-Dependent Regret Lower Bounds for Contextual Bandits

Variance-dependent regret bounds for linear contextual bandits, which improve upon the classical tildeO(d√K) regret bound to tildeO(d√sumk=1Kσk2), where d is the context dimension, K is the number of rounds, and σ2k is the noise variance in round k, has been widely studied in recent years. However, most existing works focus on the regret upper bounds instead of lower bounds. To our knowledge, the only lower bound is from Jia et al. (2024), which proved that for any eluder dimension delu and total variance budget Lambda, there exists an instance with sumk=1Kσk2≤ Lambda for which any algorithm incurs a variance-dependent lower bound of Omega(√deluLambda). However, this lower bound has a √d gap with existing upper bounds. Moreover, it only considers a fixed total variance budget Lambda and does not apply to a general variance sequence σ12,ldots,σK2. In this paper, to overcome the limitations of Jia et al. (2024), we consider the general variance sequence under two settings. For a prefixed sequence, where the entire variance sequence is revealed to the learner at the beginning of the learning process, we establish a variance-dependent lower bound of Omega(d √sumk=1Kσk2 /log K) for linear contextual bandits. For an adaptive sequence, where an adversary can generate the variance σk2 in each round k based on historical observations, we show that when the adversary must generate σk2 before observing the decision set Dk, a similar lower bound of Omega(d√ sumk=1Kσk2 /log6(dK)) holds. In both settings, our results match the upper bounds of the SAVE algorithm (Zhao et al., 2023) up to logarithmic factors.