We present an algorithm for learning switched linear dynamical systems in discrete time from noisy observations of the system's full state or output. Switched linear systems use multiple linear dynamical modes to fit the data within some desired tolerance. They arise quite naturally in applications to robotics and cyberphysical systems. Learning switched systems from data is a NP-hard problem that is nearly identical to the k-linear regression problem of fitting k > 1 linear models to the data. A direct mixed-integer linear programming (MILP) approach yields time complexity that is exponential in the number of data points. In this paper, we modify the problem formulation to yield an algorithm that is linear in the size of the data while remaining exponential in the number of state variables and the desired number of modes. To do so, we combine classic ideas from the ellipsoidal method for solving convex optimization problems, and well-known oracle separation results in non-smooth optimization. We demonstrate our approach on a set of microbenchmarks and a few interesting real-world problems. Our evaluation suggests that the benefits of this algorithm can be made practical even against highly optimized off-the-shelf MILP solvers.

Neural Information Processing Systems http://nips.cc/

We establish upper bounds on the regrets of the problem with respect to time horizon, manifold curvature, and manifold dimension.

There exists an online convex optimization problem where Adam (and RMSprop) has non-zero average regret, and one of the problem is in the form ft(x)= ( Px, if t mod P =1 x, Otherwise x [ 1,1], P N,P 3 (1) Proof. See [1] Thm.1 for proof. For the problem defined above, there's a threshold of β2 above which RMSprop converge. For the problem defined by Eq. (1), ACProp algorithm converges β1,β2 (0,1), P N,P 3. Proof. We analyze the limit behavior of ACProp algorithm.

Neural Information Processing Systems http://nips.cc/