Follow the Perturbed Approximate Leader for Solving Semi-bandit Combinatorial Optimization

Combinatorial optimization in the face of uncertainty is a challenge in both operational research and machine learning. In this paper, we consider a special and important class called the adversarial online combinatorial optimization with semi-bandit feedback, in which a player makes combinatorial decisions and gets the corresponding feedback repeatedly. While existing algorithms focus on the regret guarantee or assume there exists an efficient offline oracle, it is still a challenge to solve this problem efficiently if the offline counterpart is NP-hard. In this paper, we propose a variant of the Follow-the-Perturbed-Leader (FPL) algorithm to solve this problem. Unlike the existing FPL approach, our method employs an approximation algorithm as an offline oracle and perturbs the collected data by adding nonnegative random variables. Our approach is simple and computationally efficient. Moreover, it can guarantee a sublinear (1+ε)-scaled regret of order O(T^{2/3})  for any small ε > 0 for an important class of combinatorial optimization problems that admit an FPTAS (fully polynomial time approximation scheme), in which T is the number of rounds of the learning process. In addition to the theoretical analysis, we also conduct a series of experiments to demonstrate the performance of our algorithm.