The Complexity of Sequential Prediction in Dynamical Systems
This work addresses fundamental limits in sequential prediction for dynamical systems, providing theoretical insights for researchers in learning theory and control, but it is incremental as it builds on existing regret frameworks.
The paper tackles the problem of learning to predict the next state in dynamical systems without parametric assumptions, showing that in the realizable setting, mistakes can grow arbitrarily with time, while in the agnostic setting, regret rates are limited to Θ(T) or Ϙ(√T).
We study the problem of learning to predict the next state of a dynamical system when the underlying evolution function is unknown. Unlike previous work, we place no parametric assumptions on the dynamical system, and study the problem from a learning theory perspective. We define new combinatorial measures and dimensions and show that they quantify the optimal mistake and regret bounds in the realizable and agnostic settings respectively. By doing so, we find that in the realizable setting, the total number of mistakes can grow according to \emph{any} increasing function of the time horizon $T$. In contrast, we show that in the agnostic setting under the commonly studied notion of Markovian regret, the only possible rates are $Θ(T)$ and $\tildeΘ(\sqrt{T})$.