  We present an efficient and practical (polynomial time) algorithm for online
prediction in unknown and partially observed linear dynamical systems (LDS)
under stochastic noise. When the system parameters are known, the optimal
linear predictor is the Kalman filter. However, the performance of existing
predictive models is poor in important classes of LDS that are only marginally
stable and exhibit long-term forecast memory. We tackle this problem through
bounding the generalized Kolmogorov width of the Kalman filter model by
spectral methods and conducting tight convex relaxation. We provide a
finite-sample analysis, showing that our algorithm competes with Kalman filter
in hindsight with only logarithmic regret. Our regret analysis relies on
Mendelson's small-ball method, providing sharp error bounds without
concentration, boundedness, or exponential forgetting assumptions. We also give
experimental results demonstrating that our algorithm outperforms
state-of-the-art methods. Our theoretical and experimental results shed light
on the conditions required for efficient probably approximately correct (PAC)
learning of the Kalman filter from partially observed data.
