Learning switched non-linear dynamical systems from a single trajectory

This paper addresses the challenge of learning the transition dynamics of a non-linear system that switches between $K$ different modes over time. While learning single-mode non-linear systems is well-studied, the added randomness of regime switching makes traditional analysis techniques difficult to apply. The authors seek to derive non-asymptotic convergence rates for empirical risk minimization (ERM) in this switched setting using only a single observed trajectory.

The researchers model the system as a non-linear autoregressive process where the transition function depends on a latent mode variable. To handle the complexity of the switched dynamics, they employ a Lyapunov drift condition to ensure system stability and localize the state space. Their proof strategy departs from standard approaches by using martingale-based tail bounds (specifically de la Peña-type arguments) instead of traditional Chernoff-based chaining. This allows them to decouple the fluctuations in regime occurrences from the martingale noise, providing a cleaner analysis of the prediction risk.

The study establishes that learning switched non-linear systems is feasible under mild stability and excitation assumptions. The derived risk bounds are expressed in terms of the metric entropy of the function class and the effective sample size ($Tp_i$), where $T$ is the trajectory length and $p_i$ is the probability of observing mode $i$. The authors instantiate these results for both linear and Hölder function classes, showing that the convergence rates depend on the frequency with which each mode is visited. Notably, the analysis remains valid for unbounded function classes, relaxing common constraints found in previous literature.

This work extends the theoretical foundation of non-linear system identification to a broader class of real-world problems, such as robotics, finance, and cyber-physical systems, where dynamics are rarely static. By providing the first non-asymptotic guarantees for switched non-linear systems, the paper offers a rigorous framework for practitioners to understand how mode-switching frequency and function complexity impact the reliability of learned models.