  We revisit the fundamental problem of prediction with expert advice, in a
setting where the environment is benign and generates losses stochastically,
but the feedback observed by the learner is subject to a moderate adversarial
corruption. We prove that a variant of the classical Multiplicative Weights
algorithm with decreasing step sizes achieves constant regret in this setting
and performs optimally in a wide range of environments, regardless of the
magnitude of the injected corruption. Our results reveal a surprising disparity
between the often comparable Follow the Regularized Leader (FTRL) and Online
Mirror Descent (OMD) frameworks: we show that for experts in the corrupted
stochastic regime, the regret performance of OMD is in fact strictly inferior
to that of FTRL.
