The vast majority of zeroth order optimization methods try to imitate first order methods via some smooth approximation of the gradient. Here, the smaller the smoothing parameter, the smaller the gradient approximation error. We show that for the majority of zeroth order methods this smoothing parameter can however not be chosen arbitrarily small as numerical cancellation errors will dominate. As such, theoretical and numerical performance could differ significantly. Using classical tools from numerical differentiation we will propose a new smoothed approximation of the gradient that can be integrated into general zeroth order algorithmic frameworks. Since the proposed smoothed approximation does not suffer from cancellation error, the smoothing parameter (and hence the approximation error) can be made arbitrarily small. Sublinear convergence rates for algorithms based on our smoothed approximation are proved. Numerical experiments are also presented to demonstrate the superiority of algorithms based on the proposed approximation.

We propose a principled method for projecting an arbitrary square matrix to the nonconvex set of asymptotically stable matrices. Leveraging ideas from large deviations theory, we show that this projection is optimal in an information-theoretic sense and that it simply amounts to shifting the initial matrix by an optimal linear quadratic feedback gain, which can be computed exactly and highly efficiently by solving a standard linear quadratic regulator problem. The proposed approach allows us to learn the system matrix of a stable linear dynamical system from a single trajectory of correlated state observations. The resulting estimator is guaranteed to be stable and offers explicit statistical bounds on the estimation error.

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We consider the problem of controlling an unknown stochastic linear dynamical system subject to an infinitehorizon discounted quadratic cost. Existing approaches for handling the corresponding robust optimal control problem resort to either conservative uncertainty sets or various approximations schemes, and to our best knowledge, the current literature lacks an exact, yet tractable, solution. We propose a class of novel uncertainty sets for the system matrices of the linear system. We show that the resulting robust linear quadratic regulator problem enjoys a closed-form solution described through a generalized algebraic Riccati equation arising from dynamic game theory.

In this chapter we discuss differentiable structures on topological manifolds. In particular, we discuss transversality, tubular neighbourhoods, index theory, the degree and the theorems by Poincaré and Hopf and Bobylev and Krasnosel’skiĭ.