Universal MMSE Filtering With Logarithmic Adaptive Regret

We consider the problem of online estimation of a real-valued signal corrupted by oblivious zero-mean noise using linear estimators. The estimator is required to iteratively predict the underlying signal based on the current and several last noisy observations, and its performance is measured by the mean-square-error. We describe and analyze an algorithm for this task which: 1. Achieves logarithmic adaptive regret against the best linear filter in hindsight. This bound is assyptotically tight, and resolves the question of Moon and Weissman [1]. 2. Runs in linear time in terms of the number of filter coefficients. Previous constructions required at least quadratic time.Comment: 14 page

DR9.3 Final report of the JRRM and ASM activities

Deliverable del projecte europeu NEWCOM++This deliverable provides the final report with the summary of the activities carried out in NEWCOM++ WPR9, with a particular focus on those obtained during the last year. They address on the one hand RRM and JRRM strategies in heterogeneous scenarios and, on the other hand, spectrum management and opportunistic spectrum access to achieve an efficient spectrum usage. Main outcomes of the workpackage as well as integration indicators are also summarised.Postprint (published version

Statistical Learning Theory for Control: A Finite Sample Perspective

This tutorial survey provides an overview of recent non-asymptotic advances in statistical learning theory as relevant to control and system identification. While there has been substantial progress across all areas of control, the theory is most well-developed when it comes to linear system identification and learning for the linear quadratic regulator, which are the focus of this manuscript. From a theoretical perspective, much of the labor underlying these advances has been in adapting tools from modern high-dimensional statistics and learning theory. While highly relevant to control theorists interested in integrating tools from machine learning, the foundational material has not always been easily accessible. To remedy this, we provide a self-contained presentation of the relevant material, outlining all the key ideas and the technical machinery that underpin recent results. We also present a number of open problems and future directions.Comment: Survey Paper, Submitted to Control Systems Magazine. Second version contains additional motivation for finite sample statistics and more detailed comparison with classical literatur

A Comprehensive Approach to Universal Piecewise Nonlinear Regression Based on Trees

Cataloged from PDF version of article.In this paper, we investigate adaptive nonlinear regression and introduce tree based piecewise linear regression algorithms that are highly efficient and provide significantly improved performance with guaranteed upper bounds in an individual sequence manner. We use a tree notion in order to partition the space of regressors in a nested structure. The introduced algorithms adapt not only their regression functions but also the complete tree structure while achieving the performance of the “best” linear mixture of a doubly exponential number of partitions, with a computational complexity only polynomial in the number of nodes of the tree. While constructing these algorithms, we also avoid using any artificial “weighting” of models (with highly data dependent parameters) and, instead, directly minimize the final regression error, which is the ultimate performance goal. The introduced methods are generic such that they can readily incorporate different tree construction methods such as random trees in their framework and can use different regressor or partitioning functions as demonstrated in the paper

Highly efficient hierarchical online nonlinear regression using second order methods

We introduce highly efficient online nonlinear regression algorithms that are suitable for real life applications. We process the data in a truly online manner such that no storage is needed, i.e., the data is discarded after being used. For nonlinear modeling we use a hierarchical piecewise linear approach based on the notion of decision trees where the space of the regressor vectors is adaptively partitioned based on the performance. As the first time in the literature, we learn both the piecewise linear partitioning of the regressor space as well as the linear models in each region using highly effective second order methods, i.e., Newton–Raphson Methods. Hence, we avoid the well known over fitting issues by using piecewise linear models, however, since both the region boundaries as well as the linear models in each region are trained using the second order methods, we achieve substantial performance compared to the state of the art. We demonstrate our gains over the well known benchmark data sets and provide performance results in an individual sequence manner guaranteed to hold without any statistical assumptions. Hence, the introduced algorithms address computational complexity issues widely encountered in real life applications while providing superior guaranteed performance in a strong deterministic sense. © 2017 Elsevier B.V

Efficient implementation of Newton-raphson methods for sequential data prediction

We investigate the problem of sequential linear data prediction for real life big data applications. The second order algorithms, i.e., Newton-Raphson Methods, asymptotically achieve the performance of the 'best' possible linear data predictor much faster compared to the first order algorithms, e.g., Online Gradient Descent. However, implementation of these second order methods results in a computational complexity in the order of $O(M2)$ for an $M$ dimensional feature vector, where the first order methods offer complexity in the order of $O(M)$. Because of this extremely high computational need, their usage in real life big data applications is prohibited. To this end, in order to enjoy the outstanding performance of the second order methods, we introduce a highly efficient implementation where the computational complexity of these methods is reduced from $O(M2)$ to $O(M)$. The presented algorithm provides the well-known merits of the second order methods while offering a computational complexity similar to the first order methods. We do not rely on any statistical assumptions, hence, both regular and fast implementations achieve the same performance in terms of mean square error. We demonstrate the efficiency of our algorithm on several sequential big datasets. We also illustrate the numerical stability of the presented algorithm. © 1989-2012 IEEE

Efficient implementation of Newton-raphson methods for sequential data prediction

We investigate the problem of sequential linear data prediction for real life big data applications. The second order algorithms, i.e., Newton-Raphson Methods, asymptotically achieve the performance of the 'best' possible linear data predictor much faster compared to the first order algorithms, e.g., Online Gradient Descent. However, implementation of these second order methods results in a computational complexity in the order of $O(M2)$ for an $M$ dimensional feature vector, where the first order methods offer complexity in the order of $O(M)$. Because of this extremely high computational need, their usage in real life big data applications is prohibited. To this end, in order to enjoy the outstanding performance of the second order methods, we introduce a highly efficient implementation where the computational complexity of these methods is reduced from $O(M2)$ to $O(M)$. The presented algorithm provides the well-known merits of the second order methods while offering a computational complexity similar to the first order methods. We do not rely on any statistical assumptions, hence, both regular and fast implementations achieve the same performance in terms of mean square error. We demonstrate the efficiency of our algorithm on several sequential big datasets. We also illustrate the numerical stability of the presented algorithm. © 1989-2012 IEEE

Statistical Learning For System Identification, Estimation, And Control

Despite the recent widespread success of machine learning, we still do not fully understand its fundamental limitations. Going forward, it is crucial to better understand learning complexity, especially in critical decision making applications, where a wrong decision can lead to catastrophic consequences. In this thesis, we focus on the statistical complexity of learning unknown linear dynamical systems, with focus on the tasks of system identification, prediction, and control. We are interested in sample complexity, i.e. the minimum number of samples required to achieve satisfactory learning performance. Our goal is to provide finite-sample learning guarantees, explicitly highlighting how the learning objective depends on the number of samples. A fundamental question we are trying to answer is how system theoretic properties of the underlying process can affect sample complexity. Using recent advances in statistical learning, high-dimensional statistics, and control theoretic tools, we provide finite-sample guarantees in the following settings. i) System Identification. We provide the first finite-sample guarantees for identifying a stochastic partially-observed system; this problem is also known as the stochastic system identification problem. ii) Prediction. We provide the first end-to-end guarantees for learning the Kalman Filter, i.e. for learning to predict, in an offline learning architecture. We also provide the first logarithmic regret guarantees for the problem of learning the Kalman Filter in an online learning architecture, where the data are revealed sequentially. iii) Difficulty of System Identification and Control. Focusing on fully-observed systems, we investigate when learning linear systems is statistically easy or hard, in the finite sample regime. Statistically easy to learn linear system classes have sample complexity that is polynomial with the system dimension. Statistically hard to learn linear system classes have worst-case sample complexity that is at least exponential with the system dimension. We show that there actually exist classes of linear systems, which are hard to learn. Such classes include indirectly excited systems with large degree of indirect excitation. Similar conclusions hold for both the problem of system identification and the problem of learning to control