Understanding Curriculum Learning in Policy Optimization for Online Combinatorial Optimization
This work addresses the lack of theoretical understanding for why reinforcement learning helps in combinatorial optimization, particularly for researchers in machine learning and optimization theory.
This paper provides the first systematic theoretical study of policy optimization methods for online combinatorial optimization problems, showing they can be formulated as latent Markov Decision Processes and proving convergence bounds for natural policy gradient methods. The theory explains how curriculum learning reduces distribution shift and improves convergence, with experiments validating these findings on problems including the Best Choice Problem, Online Knapsack, and AdWords.
Over the recent years, reinforcement learning (RL) starts to show promising results in tackling combinatorial optimization (CO) problems, in particular 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 online CO problems. We show that online 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 online CO problem, the Best Choice Problem (BCP), we formally prove that distribution shift is reduced exponentially with curriculum learning even if the curriculum is a randomly generated BCP on a smaller scale. 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 the Best Choice Problem, Online Knapsack, and AdWords to verify our findings.