Generalizing the Regret: an Analysis of Lower and Upper Bounds

The (expected cumulative) regret is the customary index to judge the performance of online sequential decision-making algorithms. In the traditional form, it is defined as the expected sum over a learning horizon T of the sub-optimality gaps triangle It (i.e., expected instantaneous regret) the agent suffers when playing arm It at round t. In this paper, we propose and investigate a generalization of this notion, named g-(expected cumulative) regret, obtained by applying a transformation function g to the sub-optimality gaps, making the agent suffer g(triangle(It)) instead of just triangle(It). Intuitively, function g embeds the "perception" that the agent manifests when performing a sub-optimal decision. We first show that sublinear g-regret is not achievable for a generic transformation function g. Then, we introduce a mild condition on g and provide instance-dependent and worst-case (i.e., minimax) lower bounds for the g-regret. Finally, we show that state-of-the-art stochastic bandit algorithms with no modification surprisingly display optimal performances for the g-regret. Specifically, we prove that UCB1 matches (up to constant factors) the instance-dependent lower bound regardless of function g and that MOSS matches (up to constant factors) the minimax lower bound at least for a wide class of transformation functions.