DDLC Seminar - Prof. Sarah Dean

Title: Two-Layer Linear Auto-Regressive Models Estimate Latent States Abstract: Auto-regressive models, trained simply to predict future observations from past ones, have become a default tool for sequential data, from language to video to control-relevant time series. Their learned representations are increasingly repurposed for downstream planning, estimation, and control. Yet we lack rigorous understanding of what their internal representations capture, which matters if we want to use them in closed-loop, safety-critical systems. This talk addresses that gap in a canonical setting: partially observed linear dynamical systems, where the optimal recursive estimator is the Kalman filter. We study a two-layer linear auto-regressive model trained by empirical risk minimization on input-output data alone, with no knowledge of the dynamics, noise, or state. We show that this model provably learns a hidden representation matching the Kalman filter's state estimate, up to a similarity transformation. The result follows from three ingredients: (i) a bound showing the Kalman filter is well approximated by a finite-window predictor; (ii) a benign non-convex landscape, where every stationary point is a strict saddle or global minimum; and (iii) finite-sample guarantees on prediction error, parameter estimation error, and latent state recovery, stated in terms of model dimension and amount of training data. These results establish a foundation for finite-sample analysis of methods that use learned representations in adaptive, closed-loop decision-making systems. Based on joint work with Yahya Sattar, Yassir Jedra, Leo Maynard-Zhang, Sunmook Choi, Maryam Fazel.