Non-stationary Stochastic Optimization

We consider a non-stationary variant of a sequential stochastic optimization problem, in which the underlying cost functions may change along the horizon. We propose a measure, termed variation budget, that controls the extent of said change, and study how restrictions on this budget impact achievable performance. We identify sharp conditions under which it is possible to achieve long-run-average optimality and more refined performance measures such as rate optimality that fully characterize the complexity of such problems. In doing so, we also establish a strong connection between two rather disparate strands of literature: adversarial online convex optimization; and the more traditional stochastic approximation paradigm (couched in a non-stationary setting). This connection is the key to deriving well performing policies in the latter, by leveraging structure of optimal policies in the former. Finally, tight bounds on the minimax regret allow us to quantify the "price of non-stationarity," which mathematically captures the added complexity embedded in a temporally changing environment versus a stationary one

Coding of non-stationary sources as a foundation for detecting change points and outliers in binary time-series

An interesting scheme for estimating and adapting distributions in real-time for non-stationary data has recently been the focus of study for several different tasks relating to time series and data mining, namely change point detection, outlier detection and online compression/sequence prediction. An appealing feature is that unlike more sophisticated procedures, it is as fast as the related stationary procedures which are simply modified through discounting or windowing. The discount scheme makes older observations lose their influence on new predictions. The authors of this article recently used a discount scheme for introducing an adaptive version of the Context Tree Weighting compression algorithm. The mentioned change point and outlier detection methods rely on the changing compression ratio of an online compression algorithm. Here we are beginning to provide theoretical foundations for the use of these adaptive estimation procedures that have already shown practical promise

Non-stationary wave relaxation methods for general linear systems of Volterra equations: convergence and parallel GPU implementation

Producción CientíficaIn the present paper, a parallel-in-time discretization of linear systems of Volterra equations of type u¯(t)=u¯0+∫t0K(t−s)u¯(s) d s+f¯(t),0<t≤T, is addressed. Related to the analytical solution, a general enough functional setting is firstly stated. Related to the numerical solution, a parallel numerical scheme based on the Non-Stationary Wave Relaxation (NSWR) method for the time discretization is proposed, and its convergence is studied as well. A CUDA parallel implementation of the method is carried out in order to exploit Graphics Processing Units (GPUs), which are nowadays widely employed for reducing the computational time of several general purpose applications. The performance of these methods is compared to some sequential implementation. It is revealed throughout several experiments of special interest in practical applications the good performance of the parallel approach.Ministerio de Universidades e Investigación de Italia (MUR), a través del proyecto PRIN 2017 (No. 2017JYCLSF) “Aproximación preservadora de estructuras de problemas evolutivos”Publicación en abierto financiada por el Consorcio de Bibliotecas Universitarias de Castilla y León (BUCLE), con cargo al Programa Operativo 2014ES16RFOP009 FEDER 2014-2020 DE CASTILLA Y LEÓN, Actuación:20007-CL - Apoyo Consorcio BUCL

Non-stationary log-periodogram regression

We study asymptotic properties of the log-periodogram semiparametric estimate of the memory parameter d for non-stationary (d>=1/2) time series with Gaussian increments, extending the results of Robinson (1995) for stationary and invertible Gaussian processes. We generalize the definition of the memory parameter d for non-stationary processes in terms of the (successively) differentiated series. We obtain that the log-periodogram estimate is asymptotically normal for dE[1/2, 3/4) and still consistent for dE[1/2, 1). We show that with adequate data tapers, a modified estimate is consistent and asymptotically normal distributed for any d, including both non-stationary and non-invertible processes. The estimates are invariant to the presence of certain deterministic trends, without any need of estimation.Publicad

Non-stationary de Sitter cosmological models

In this note it is proposed a class of non-stationary de Sitter, rotating and non-rotating, solutions of Einstein's field equations with a cosmological term of variable function.Comment: 11 pages, Latex. International Journal of Modern Physics D (accepted for publication