Online Monotone Optimization

This paper presents a new framework for analyzing and designing noregret algorithms for dynamic (possibly adversarial) systems. The proposed framework generalizes the popular online convex optimization framework and extends it to its natural limit allowing it to capture a notion of regret that is intuitive for more general problems such as those encountered in game theory and variational inequalities. The framework hinges on a special choice of a system-wide loss function we have developed. Using this framework, we prove that a simple update scheme provides a no-regret algorithm for monotone systems. While previous results in game theory prove individual agents can enjoy unilateral no-regret guarantees, our result proves monotonicity sufficient for guaranteeing no-regret when considering the adjustments of multiple agent strategies in parallel. Furthermore, to our knowledge, this is the first framework to provide a suitable notion of regret for variational inequalities. Most importantly, our proposed framework ensures monotonicity a sufficient condition for employing multiple online learners safely in parallel. 0.