Recently, chaotic phenomena in laser dynamics have attracted much attention to its applied aspects, and a synchronization phenomenon, leader-laggard relationship, in time-delay coupled lasers has been used in reinforcement learning. In the present paper, we discuss the possibility of capturing the essential stochasticity of the leader-laggard relationship; in nonlinear science, it is known that coarse-graining allows one to derive stochastic models from deterministic systems. We derive stochastic models with the aid of the Koopman operator approach, and we clarify that the low-pass filtered data is enough to recover the essential features of the original deterministic chaos, such as peak shifts in the distribution of being the leader and a power-law behavior in the distribution of switching-time intervals. We also confirm that the derived stochastic model works well in reinforcement learning tasks, i.e., multi-armed bandit problems, as with the original laser chaos system.

11 pages, 6 figures