Online Inverse Integer Linear Optimization via Small-Gradient Skipping: Constant Regret and Finite Mistakes

In online inverse linear optimization, the learner predicts a weight at each round, observes the optimal action of the agent, and updates its prediction. In the general setting, the gap of log T between the regret upper bound O(d log T) and the lower bound Omega(d) is unresolved (here T is the total number of rounds and d is the dimension). When the action set is M-convex, the regret is known to be bounded by O(d log d), but the method attaining it computes a center of gravity at every round. This paper therefore proposes Small-Gradient Skipping (SGS), a mechanism that skips the update at rounds without a mistake, and applies it to online gradient descent, the online Newton step, and MetaGrad. When the forward problem is an integer linear program with a unique optimal solution, the number of mistakes is bounded, for all three, by a quantity independent of T; and for the online Newton step and for MetaGrad with SGS, the dimension dependence of the regret becomes O(d2), that is, the factor log T is removed. Moreover, when the action set is M-convex, the regret is bounded efficiently without computing a center of gravity.