Spectral Learning of Predictive State Representations with Insufficient Statistics

Predictive state representations (PSRs) are models of dynamical systems that represent state as a vector of predictions about future observable events (tests) conditioned on past observed events (histories). If a practitioner selects finite sets of tests and histories that are known to be sufficient to completely capture the system, an exact PSR can be learned in polynomial time using spectral methods. However, most real-world systems are complex, and in practice computational constraints limit us to small sets of tests and histories which are therefore never truly sufficient. How, then, should we choose these sets? Existing theory offers little guidance here, and yet we show that the choice is highly consequential -- tests and histories selected at random or by a naive rule significantly underperform the best sets. In this paper we approach the problem both theoretically and empirically. While any fixed system can be represented by an infinite number of equivalent but distinct PSRs, we show that in the computationally unconstrained setting, where existing theory guarantees accurate predictions, the PSRs learned by spectral methods always satisfy a particular spectral bound. Adapting this idea, we propose a simple algorithmic technique to search for sets of tests and histories that approximately satisfy the bound while respecting computational limits. Empirically, our method significantly reduces prediction errors compared to standard spectral learning approaches.