Finite Time Identification in Unstable Linear Systems

Identification of the parameters of stable linear dynamical systems is a well-studied problem in the literature, both in the low and high-dimensional settings. However, there are hardly any results for the unstable case, especially regarding finite time bounds. For this setting, classical results on least-squares estimation of the dynamics parameters are not applicable and therefore new concepts and technical approaches need to be developed to address the issue. Unstable linear systems arise in key real applications in control theory, econometrics, and finance. This study establishes finite time bounds for the identification error of the least-squares estimates for a fairly large class of heavy-tailed noise distributions, and transition matrices of such systems. The results relate the time length (samples) required for estimation to a function of the problem dimension and key characteristics of the true underlying transition matrix and the noise distribution. To establish them, appropriate concentration inequalities for random matrices and for sequences of martingale differences are leveraged

Randomized Algorithms for Data-Driven Stabilization of Stochastic Linear Systems

Data-driven control strategies for dynamical systems with unknown parameters are popular in theory and applications. An essential problem is to prevent stochastic linear systems becoming destabilized, due to the uncertainty of the decision-maker about the dynamical parameter. Two randomized algorithms are proposed for this problem, but the performance is not sufficiently investigated. Further, the effect of key parameters of the algorithms such as the magnitude and the frequency of applying the randomizations is not currently available. This work studies the stabilization speed and the failure probability of data-driven procedures. We provide numerical analyses for the performance of two methods: stochastic feedback, and stochastic parameter. The presented results imply that as long as the number of statistically independent randomizations is not too small, fast stabilization is guaranteed

Sample Complexity Lower Bounds for Linear System Identification

This paper establishes problem-specific sample complexity lower bounds for linear system identification problems. The sample complexity is defined in the PAC framework: it corresponds to the time it takes to identify the system parameters with prescribed accuracy and confidence levels. By problem-specific, we mean that the lower bound explicitly depends on the system to be identified (which contrasts with minimax lower bounds), and hence really captures the identification hardness specific to the system. We consider both uncontrolled and controlled systems. For uncontrolled systems, the lower bounds are valid for any linear system, stable or not, and only depend of the system finite-time controllability gramian. A simplified lower bound depending on the spectrum of the system only is also derived. In view of recent finitetime analysis of classical estimation methods (e.g. ordinary least squares), our sample complexity lower bounds are tight for many systems. For controlled systems, our lower bounds are not as explicit as in the case of uncontrolled systems, but could well provide interesting insights into the design of control policy with minimal sample complexity

Finite-time Identification of Stable Linear Systems: Optimality of the Least-Squares Estimator

We present a new finite-time analysis of the estimation error of the Ordinary Least Squares (OLS) estimator for stable linear time-invariant systems. We characterize the number of observed samples (the length of the observed trajectory) sufficient for the OLS estimator to be $(\varepsilon,\delta)$-PAC, i.e., to yield an estimation error less than $\varepsilon$ with probability at least $1-\delta$. We show that this number matches existing sample complexity lower bounds [1,2] up to universal multiplicative factors (independent of ($\varepsilon,\delta)$ and of the system). This paper hence establishes the optimality of the OLS estimator for stable systems, a result conjectured in [1]. Our analysis of the performance of the OLS estimator is simpler, sharper, and easier to interpret than existing analyses. It relies on new concentration results for the covariates matrix

Finite Time Adaptive Stabilization of LQ Systems

Stabilization of linear systems with unknown dynamics is a canonical problem in adaptive control. Since the lack of knowledge of system parameters can cause it to become destabilized, an adaptive stabilization procedure is needed prior to regulation. Therefore, the adaptive stabilization needs to be completed in finite time. In order to achieve this goal, asymptotic approaches are not very helpful. There are only a few existing non-asymptotic results and a full treatment of the problem is not currently available. In this work, leveraging the novel method of random linear feedbacks, we establish high probability guarantees for finite time stabilization. Our results hold for remarkably general settings because we carefully choose a minimal set of assumptions. These include stabilizability of the underlying system and restricting the degree of heaviness of the noise distribution. To derive our results, we also introduce a number of new concepts and technical tools to address regularity and instability of the closed-loop matrix.Comment: arXiv admin note: substantial text overlap with arXiv:1711.0723

Finite Sample Analysis of Stochastic System Identification

In this paper, we analyze the finite sample complexity of stochastic system identification using modern tools from machine learning and statistics. An unknown discrete-time linear system evolves over time under Gaussian noise without external inputs. The objective is to recover the system parameters as well as the Kalman filter gain, given a single trajectory of output measurements over a finite horizon of length $N$. Based on a subspace identification algorithm and a finite number of $N$ output samples, we provide non-asymptotic high-probability upper bounds for the system parameter estimation errors. Our analysis uses recent results from random matrix theory, self-normalized martingales and SVD robustness, in order to show that with high probability the estimation errors decrease with a rate of $1/\sqrt{N}$. Our non-asymptotic bounds not only agree with classical asymptotic results, but are also valid even when the system is marginally stable.Comment: Under revie

Input Perturbations for Adaptive Control and Learning

This paper studies adaptive algorithms for simultaneous regulation (i.e., control) and estimation (i.e., learning) of Multiple Input Multiple Output (MIMO) linear dynamical systems. It proposes practical, easy to implement control policies based on perturbations of input signals. Such policies are shown to achieve a worst-case regret that scales as the square-root of the time horizon, and holds uniformly over time. Further, it discusses specific settings where such greedy policies attain the information theoretic lower bound of logarithmic regret. To establish the results, recent advances on self-normalized martingales together with a novel method of policy decomposition are leveraged

Optimism-Based Adaptive Regulation of Linear-Quadratic Systems

The main challenge for adaptive regulation of linear-quadratic systems is the trade-off between identification and control. An adaptive policy needs to address both the estimation of unknown dynamics parameters (exploration), as well as the regulation of the underlying system (exploitation). To this end, optimism-based methods which bias the identification in favor of optimistic approximations of the true parameter are employed in the literature. A number of asymptotic results have been established, but their finite time counterparts are few, with important restrictions. This study establishes results for the worst-case regret of optimism-based adaptive policies. The presented high probability upper bounds are optimal up to logarithmic factors. The non-asymptotic analysis of this work requires very mild assumptions; (i) stabilizability of the system's dynamics, and (ii) limiting the degree of heaviness of the noise distribution. To establish such bounds, certain novel techniques are developed to comprehensively address the probabilistic behavior of dependent random matrices with heavy-tailed distributions.Comment: 28 page

Certainty Equivalence is Efficient for Linear Quadratic Control

We study the performance of the certainty equivalent controller on Linear Quadratic (LQ) control problems with unknown transition dynamics. We show that for both the fully and partially observed settings, the sub-optimality gap between the cost incurred by playing the certainty equivalent controller on the true system and the cost incurred by using the optimal LQ controller enjoys a fast statistical rate, scaling as the square of the parameter error. To the best of our knowledge, our result is the first sub-optimality guarantee in the partially observed Linear Quadratic Gaussian (LQG) setting. Furthermore, in the fully observed Linear Quadratic Regulator (LQR), our result improves upon recent work by Dean et al. (2017), who present an algorithm achieving a sub-optimality gap linear in the parameter error. A key part of our analysis relies on perturbation bounds for discrete Riccati equations. We provide two new perturbation bounds, one that expands on an existing result from Konstantinov et al. (1993), and another based on a new elementary proof strategy.Comment: In the current version we extended our analysis to the case of partially observable systems, i.e. we provided a suboptimality analysis for the Linear Quadratic Gaussian (LQG) settin

On Adaptive Linear-Quadratic Regulators

Performance of adaptive control policies is assessed through the regret with respect to the optimal regulator, which reflects the increase in the operating cost due to uncertainty about the dynamics parameters. However, available results in the literature do not provide a quantitative characterization of the effect of the unknown parameters on the regret. Further, there are problems regarding the efficient implementation of some of the existing adaptive policies. Finally, results regarding the accuracy with which the system's parameters are identified are scarce and rather incomplete. This study aims to comprehensively address these three issues. First, by introducing a novel decomposition of adaptive policies, we establish a sharp expression for the regret of an arbitrary policy in terms of the deviations from the optimal regulator. Second, we show that adaptive policies based on slight modifications of the Certainty Equivalence scheme are efficient. Specifically, we establish a regret of (nearly) square-root rate for two families of randomized adaptive policies. The presented regret bounds are obtained by using anti-concentration results on the random matrices employed for randomizing the estimates of the unknown parameters. Moreover, we study the minimal additional information on dynamics matrices that using them the regret will become of logarithmic order. Finally, the rates at which the unknown parameters of the system are being identified are presented