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/

In this work, we study the convergence in high probability of clipped gradient methods when the noise distribution has heavy tails, i.e., with bounded p th moments, for some 1 < p 2 . Prior works in this setting follow the same recipe of using concentration inequalities and an inductive argument with union bound to bound the iterates across all iterations. This method results in an increase in the failure probability by a factor of T, where T is the number of iterations. We instead propose a new analysis approach based on bounding the moment generating function of a well chosen supermartingale sequence. We improve the dependency on T in the convergence guarantee for a wide range of algorithms with clipped gradients, including stochastic (accelerated) mirror descent for convex objectives and stochastic gradient descent for nonconvex objectives. Our high probability bounds achieve the optimal convergence rates and match the best currently known in-expectation bounds. Our approach naturally allows the algorithms to use time-varying step sizes and clipping parameters when the time horizon is unknown, which appears difficult or even impossible using existing techniques from prior works. Furthermore, we show that in the case of clipped stochastic mirror descent, several problem constants, including the initial distance to the optimum, are not required when setting step sizes and clipping parameters.

Moreoever, we develop estimation theory providing tight control on the error of any estimated trajectory, indicating that the representations could even be used in quite noise-sensitive follow-on analyses.