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The optimal predictor for a linear dynamical system (with hidden state andGaussian noise) takes the form of an autoregressive linear filter, namely theKalman filter. However, a fundamental problem in reinforcement learning andcontrol theory is to make optimal predictions in an unknown dynamical system.To this end, we take the approach of directly learning an autoregressive filterfor time-series prediction under unknown dynamics. Our analysis differs fromprevious statistical analyses in that we regress not only on the inputs to thedynamical system, but also the outputs, which is essential to dealing withprocess noise. The main challenge is to estimate the filter under worst caseinput (in $\mathcal H_\infty$ norm), for which we use an $L^\infty$-basedobjective rather than ordinary least-squares. For learning an autoregressivemodel, our algorithm has optimal sample complexity in terms of the rolloutlength, which does not seem to be attained by naive least-squares.

Robust guarantees for learning an autoregressive filter