Review for NeurIPS paper: A Bandit Learning Algorithm and Applications to Auction Design

Additional Feedback: The paper studies the problem of online convex optimization problem, except that the functions that arrive online are not really concave. They are "close to" concave, formalized by the paper as (lambda, mu) concavity. The idea is that in many problems of interest where the input functions are not concave, the paper discretizes the function and consider the multilinear extension of the discretized function, which happens to be (lambda, mu) concave for reasonable values of lambda and mu. The paper presents three applications to illustrate the value of their approach. The first of these is the analysis of adaptive dynamics on (lambda, mu) smooth games where previously high welfare was known to be guaranteed (i.e., average welfare of playing the dynamics over time is at least lambda/mu of the optimal welfare) only for dynamics that had vanishing regret for each player.