# Robust guarantees for learning an autoregressive filter

arXiv: https://arxiv.org/abs/1905.09897 · HTML: https://mlsanity.com/paper/1905.09897/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 23 May 2019.

## Built on by (most cited first, 2 of 2)

- [Deep inference of latent dynamics with spatio-temporal super-resolution using selective backpropagation through time](https://mlsanity.com/paper/2111.00070/) (2021)
- [Dynamic Structure Estimation from Bandit Feedback using Nonvanishing Exponential Sums](https://mlsanity.com/paper/2206.00861/) (2022)

## Abstract (arXiv)

The optimal predictor for a linear dynamical system (with hidden state and Gaussian noise) takes the form of an autoregressive linear filter, namely the Kalman filter. However, a fundamental problem in reinforcement learning and control theory is to make optimal predictions in an unknown dynamical system. To this end, we take the approach of directly learning an autoregressive filter for time-series prediction under unknown dynamics. Our analysis differs from previous statistical analyses in that we regress not only on the inputs to the dynamical system, but also the outputs, which is essential to dealing with process noise. The main challenge is to estimate the filter under worst case input (in mathcal H_∞ norm), for which we use an L^∞-based objective rather than ordinary least-squares. For learning an autoregressive model, our algorithm has optimal sample complexity in terms of the rollout length, which does not seem to be attained by naive least-squares.