Online learning for min-max discrete problems

We study various discrete nonlinear combinatorial optimization problems in an online learning framework. In the first part, we address computational complexity of designing vanishing regret (and approximate regret) algorithms. provide a general reduction showing that many (min-max) polynomial time solvable not only do have regret, but also no approximation α-regret, for some α, unless NP=RP. particular, min-max version vertex cover problem, which is offline case, our implies there (2−ϵ)-regret randomized algorithm Unique Game RP. Besides, prove bound tight by providing efficient based on gradient descent method. second turn attention to algorithms are oracle that, given set multiple instances able compute optimum static solution performs best overall. show several (nonlinear) problems, it strongly NP-hard solve oracle, even can be solved single-instance case (e.g. cover, perfect matching, etc.). This provides useful insight into connection between non-linear nature and drastic change their hardness when moved setting.