Optimal adversarial strategies in learning with expert advice

We propose an adversarial setting for the framework of learning with expert advice in which one of the experts has the intention to compromise the recommendation system by providing wrong recommendations. The problem is formulated as a Markov Decision Process (MDP) and solved by dynamic programming. Somewhat surprisingly, we prove that, in the case of logarithmic loss, the optimal strategy for the malicious expert is the greedy policy of lying at every step. Furthermore, a sufficient condition on the loss function is provided that guarantees the optimality of the greedy policy. Our experimental results, however, show that the condition is not necessary since the greedy policy is also optimal when the square loss is used, even though the square loss does not satisfy the condition. Moreover, the experimental results suggest that, for absolute loss, the optimal policy is a threshold one.