Co-adaptive learning over a countable space

Co-adaptation is a special form of on-line learning where an algorithm $\mathcal{A}$ must assist an unknown algorithm $\mathcal{B}$ to perform some task. This is a general framework and has applications in recommendation systems, search, education, and much more. Today, the most common use of co-adaptive algorithms is in brain-computer interfacing (BCI), where algorithms help patients gain and maintain control over prosthetic devices. While previous studies have shown strong empirical results Kowalski et al. (2013); Orsborn et al. (2014) or have been studied in specific examples Merel et al. (2013, 2015), there is no general analysis of the co-adaptive learning problem. Here we will study the co-adaptive learning problem in the online, closed-loop setting. We will prove that, with high probability, co-adaptive learning is guaranteed to outperform learning with a fixed decoder as long as a particular condition is met.