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Many emerging applications involve control of systems with unknown dynamics. As a result, model-free random search techniques that directly search over the space of parameters have become popular. These algorithms often exhibit a competitive sample complexity compared to state-of- the-art techniques. However, due to the nonconvex nature of the underlying optimization problems, the convergence behavior and statistical properties of these approaches are poorly understood. In this paper, we examine the standard linear quadratic regulator problem for continuous-time systems with unknown state-space parameters. We establish theoretical bounds on the sample complexity and prove the linear convergence rate of the random search method. 

Several problems in modeling and control of stochastically-driven dynamical systems can be cast as regularized semi-definite programs. We examine two such representative problems and show that they can be formulated in a similar manner. The first, in statistical modeling, seeks to reconcile observed statistics by suitably and minimally perturbing prior dynamics. The second seeks to optimally select a subset of available sensors and actuators for control purposes. To address modeling and control of large-scale systems we develop a unified algorithmic framework using proximal methods. Our customized algorithms exploit problem structure and allow handling statistical modeling, as well as sensor and actuator selection, for substantially larger scales than what is amenable to current general-purpose solvers. We establish linear convergence of the proximal gradient algorithm, draw contrast between the proposed proximal algorithms and alternating direction method of multipliers, and provide examples that illustrate the merits and effectiveness of our framework. 

— In large-scale and model-free settings, first-order algorithms are often used in an attempt to find the optimal control action without identifying the underlying dynamics. The convergence properties of these algorithms remain poorly understood because of nonconvexity. In this paper, we revisit the continuous-time linear quadratic regulator problem and take a step towards demystifying the efficiency of gradient-based strategies. Despite the lack of convexity, we establish a linear rate of convergence to the globally optimal solution for the gradient descent algorithm. The key component of our analysis is that we relate the gradient-flow dynamics associated with the nonconvex formulation to that of a convex reparameterization. This allows us to provide convergence guarantees for the nonconvex approach from its convex counterpart.