Understanding Curriculum Learning in Policy Optimization for Solving Combinatorial Optimization Problems

Over the recent years, reinforcement learning (RL) has shown impressive performance in finding strategic solutions for game environments, and recently starts to show promising results in solving combinatorial optimization (CO) problems, inparticular when coupled with curriculum learning to facilitate training. Despite emerging empirical evidence, theoretical study on why RL helps is still at its early stage. This paper presents the first systematic study on policy optimization methods for solving CO problems. We show that CO problems can be naturally formulated as latent Markov Decision Processes (LMDPs), and prove convergence bounds on natural policy gradient (NPG) for solving LMDPs. Furthermore, our theory explains the benefit of curriculum learning: it can find a strong sampling policy and reduce the distribution shift, a critical quantity that governs the convergence rate in our theorem. For a canonical combinatorial problem, Secretary Problem, we formally prove that distribution shift is reduced exponentially with curriculum learning. Our theory also shows we can simplify the curriculum learning scheme used in prior work from multi-step to single-step. Lastly, we provide extensive experiments on Secretary Problem and Online Knapsack to empirically verify our findings.