A Variational Perspective on Accelerated Methods in Optimization

Accelerated gradient methods play a central role in optimization, achieving optimal rates in many settings. While many generalizations and extensions of Nesterov's original acceleration method have been proposed, it is not yet clear what is the natural scope of the acceleration concept. In this paper, we study accelerated methods from a continuous-time perspective. We show that there is a Lagrangian functional that we call the \emph{Bregman Lagrangian} which generates a large class of accelerated methods in continuous time, including (but not limited to) accelerated gradient descent, its non-Euclidean extension, and accelerated higher-order gradient methods. We show that the continuous-time limit of all of these methods correspond to traveling the same curve in spacetime at different speeds. From this perspective, Nesterov's technique and many of its generalizations can be viewed as a systematic way to go from the continuous-time curves generated by the Bregman Lagrangian to a family of discrete-time accelerated algorithms.Comment: 38 pages. Subsumes an earlier working draft arXiv:1509.0361

Imprecise continuous-time Markov chains : efficient computational methods with guaranteed error bounds

Imprecise continuous-time Markov chains are a robust type of continuous-time Markov chains that allow for partially specified time-dependent parameters. Computing inferences for them requires the solution of a non-linear differential equation. As there is no general analytical expression for this solution, efficient numerical approximation methods are essential to the applicability of this model. We here improve the uniform approximation method of Krak et al. (2016) in two ways and propose a novel and more efficient adaptive approximation method. For ergodic chains, we also provide a method that allows us to approximate stationary distributions up to any desired maximal error

On the policy function in continuos time economic models

In this paper, I consider a general class of continuous-time economic models with unbounded horizon. I study the sets of conditions under which the policy function is continuous, Lipschitz continuous, and Cl differentiable. 1 also single out certain postulates which may prevent higher-order differentiability. The analysis provides, therefore, a fmn foundation to the use of dynamic programming methods in continuous time models with unbounded horizo

Power Utility Maximization in Discrete-Time and Continuous-Time Exponential Levy Models

Consider power utility maximization of terminal wealth in a 1-dimensional continuous-time exponential Levy model with finite time horizon. We discretize the model by restricting portfolio adjustments to an equidistant discrete time grid. Under minimal assumptions we prove convergence of the optimal discrete-time strategies to the continuous-time counterpart. In addition, we provide and compare qualitative properties of the discrete-time and continuous-time optimizers.Comment: 18 pages, to appear in Mathematical Methods of Operations Research. The final publication is available at springerlink.co

A temporal logic for the specification and verification of real-time systems

The development of a product typically starts with the specification of the user’s requirements and ends with the design of a system to meet those requirements. Traditional approaches to the specification and analysis of a system abstracted away from any notion of time at the specification level. However, for a real-time system the specification may include timing requirements. Many specification and verification methods for real-time systems are based on the assumption that time is discrete because the verification methods using it are significantly simpler than those using continuous time. Yet real-time systems operate in ‘real’ continuous time and their requirements are often specified using a continuous time model. In this thesis we develop a temporal logic and proof methods for the specifica­tion and verification of a real-time system which can be applied irrespective of whether time is discrete, continuous or dense. The logic is based on the defini­tion of the next operator as the next time point a change in state occurs or if no state change occurs then it is the time point obtained by incrementing the current time by one. We show that this definition of the next operator leads to a logic which is expressive enough for specifying real-time systems where continuous time is required, and in which the verification and proof methods developed for discrete time can be used. To demonstrate the applicability of the logic several varied examples including communication protocols and digital circuits are specified and their real-time properties proved. A compositional proof system which supports hierarchical development of programs is also developed for a real-time extension of a CSP-like language