A mathematical model for the two-learners problem

Objective. We present the first generic theoretical formulation of the co-adaptive learning problem and give a simple example of two interacting linear learning systems, a human and a machine. Approach. After the description of the training protocol of the two learning systems, we define a simple linear model where the two learning agents are coupled by a joint loss function. The simplicity of the model allows us to find learning rules for both human and machine that permit computing theoretical simulations. Main results. As seen in simulations, an astonishingly rich structure is found for this eco-system of learners. While the co-adaptive learners are shown to easily stall or get out of sync for some parameter settings, we can find a broad sweet spot of parameters where the learning system can converge quickly. It is defined by mid-range learning rates on the side of the learning machine, quite independent of the human in the loop. Despite its simplistic assumptions the theoretical study could be confirmed by a real-world experimental study where human and machine co-adapt to perform cursor control under distortion. Also in this practical setting the mid-range learning rates yield the best performance and behavioral ratings. Significance. The results presented in this mathematical study allow the computation of simple theoretical simulations and performance of real experimental paradigms. Additionally, they are nicely in line with previous results in the BCI literature.