(A) An environment is characterized not only by its current state, but also by its fluctuation structure, such as variances and correlations of fluctuating environmental parameters. In this work, we consider an environment undergoing epochs that differ in their fluctuation structure. Epochs are long compared to the physiological timescale, but switch faster than the evolutionary timescale. (B) The fluctuation structure can inform the fitness-maximizing strategy, but cannot be sensed directly. Instead, it would need to be learned from past observations, and used to inform future behavior. (C) To formalize the problem, we consider a situation where some internal physiological quantities must track fluctuating external factors undergoing a random walk. Since it is impossible to react instantaneously, always lags behind . The dashed ellipse illustrates the fluctuation structure of (encoded in parameters and Γ, see text), and changes on a slower timescale than the fluctuations of . (D, E) The optimal behavior in the two-dimensional version of our problem, under a constrained maximal rate of change . For a given current (blue dot), the optimal control strategy would steer any current (green arrows) toward the best guess of the future , which depends on the fluctuation structure (red ellipse: (D) fluctuations are uncorrelated and isotropic; (E) fluctuations have a preferred direction). The optimal strategy is derived using control theory (Appendix 1, section 'Control theory calculation').