We consider a stochastic linear bandit problem in which the rewards are not only subject to random noise, but also adversarial attacks subject to a suitable budget $C$ (i.e., an upper bound on the sum of corruption magnitudes across the time horizon). We provide two variants of a Robust Phased Elimination algorithm, one that knows $C$ and one that does not. Both variants are shown to attain near-optimal regret in the non-corrupted case $C=0$, while incurring additional additive terms respectively having a linear and quadratic dependency on $C$ in general. We present algorithm-independent lower bounds showing that these additive terms are near-optimal. In addition, in a contextual setting, we revisit a setup of diverse contexts, and show that a simple greedy algorithm is provably robust with a near-optimal additive regret term, despite performing no explicit exploration and not knowing $C$.

@inproceedings{bogunovic2020stochastic,
	author = {Ilija Bogunovic and Apran Losalka and Andreas Krause and Jonathan Scarlett},
	booktitle = {Proc. International Conference on Artificial Intelligence and Statistics (AISTATS)},
	month = {March},
	title = {Stochastic Linear Bandits Robust to Adversarial Attacks},
	year = {2021}}