Square-root Rate Research Articles

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.

Constant-Order Policies for Lost-Sales Inventory Models with Random Supply Functions: Asymptotics and Heuristic

We consider an infinite-horizon lost-sales inventory model where the supply takes positive lead times and is a random function of the order quantity (e.g., random yield/capacity). The optimal policy for this model is computationally intractable; and no heuristic has been proposed in the literature. In this paper, we focus on a simple class of constant-order policies (COP) that place the same order in every period, regardless of the system state. Under some assumptions on the random supply function, we prove that the best COP is asymptotically optimal with large lead times and the optimality gap converges to zero exponentially fast in the lead time. We also prove that, if the mean supply capacity is less than the mean demand, then the best COP is also asymptotically optimal with large penalty costs; otherwise, the long-run average cost of the best COP asymptotically increases at the rate of square-root of the penalty cost. Further, we construct a simple heuristic COP and show that it performs very close to the best COP. Finally, we provide a numerical study to derive further insights into the performance of the best COP.

Efficient Parameter Estimation for Multivariate Jump-Diffusions

This paper develops estimators of the transition density, filters, and parameters of multivariate jump-diffusions with latent components. The drift, volatility, jump intensity, and jump magnitude are allowed to be general functions of the state. Our density and filter estimators converge at the canonical square-root rate, implying computational efficiency. Our parameter estimators have the same asymptotic properties as true maximum likelihood estimators, implying statistical efficiency. Numerical experiments highlight the superior performance of our estimators.

Asymptotic normality and optimalities in estimation of large Gaussian graphical models

The Gaussian graphical model, a popular paradigm for studying relationship among variables in a wide range of applications, has attracted great attention in recent years. This paper considers a fundamental question: When is it possible to estimate low-dimensional parameters at parametric square-root rate in a large Gaussian graphical model? A novel regression approach is proposed to obtain asymptotically efficient estimation of each entry of a precision matrix under a sparseness condition relative to the sample size. When the precision matrix is not sufficiently sparse, or equivalently the sample size is not sufficiently large, a lower bound is established to show that it is no longer possible to achieve the parametric rate in the estimation of each entry. This lower bound result, which provides an answer to the delicate sample size question, is established with a novel construction of a subset of sparse precision matrices in an application of Le Cam's lemma. Moreover, the proposed estimator is proven to have optimal convergence rate when the parametric rate cannot be achieved, under a minimal sample requirement. The proposed estimator is applied to test the presence of an edge in the Gaussian graphical model or to recover the support of the entire model, to obtain adaptive rate-optimal estimation of the entire precision matrix as measured by the matrix $\ell_q$ operator norm and to make inference in latent variables in the graphical model. All of this is achieved under a sparsity condition on the precision matrix and a side condition on the range of its spectrum. This significantly relaxes the commonly imposed uniform signal strength condition on the precision matrix, irrepresentability condition on the Hessian tensor operator of the covariance matrix or the $\ell_1$ constraint on the precision matrix. Numerical results confirm our theoretical findings. The ROC curve of the proposed algorithm, Asymptotic Normal Thresholding (ANT), for support recovery significantly outperforms that of the popular GLasso algorithm.

Intrinsic means on the circle: uniqueness, locus and asymptotics

This paper gives a comprehensive treatment of local uniqueness, asymptotics and numerics for intrinsic sample means on the circle. It turns out that local uniqueness as well as rates of convergence are governed by the distribution near the antipode. If the distribution is locally less than uniform there, we have local uniqueness and asymptotic normality with a square-root rate. With increased proximity to the uniform distribution the rate can be arbitrarily slow, and in the limit, local uniqueness is lost. Further, we give general distributional conditions, e.g., unimodality, that ensure global uniqueness. Along the way, we discover that sample means can occur only at the vertices of a regular polygon which allows to compute intrinsic sample means in linear time from sorted data. This algorithm is finally applied in a simulation study demonstrating the dependence of the convergence rates on the behavior of the density at the antipode.

Kinetics of pyrite oxidation in sodium hydroxide solutions

The kinetics of pyrite oxidation in sodium hydroxide solution were investigated in a stirred reactor, under temperatures ranging from 50 °C to 85 °C, oxygen partial pressures of up to 1 atm, particle size fractions from -150 + 106 to -38 + 10µm (-100 + 150 mesh to -400 mesh + 10 µ), and pH values of up to 12.5. The surface reaction is represented by the rate equation:-dN/dt = Sbk″pO0.5 2[oH- 0.25/(1 +k‴ pO2 0.5) where N represents moles of pyrite, S is the surface area of the solid particles,k″ andk″ are constants,b is a stoichiometric factor, pO2 is the oxygen partial pressure, and [OH-] is the hydroxyl ion concentration. The corresponding fractional conversion (X) vs time behavior follows the shrinking particle model for chemical reaction control: 1 - (1 -X)1/3 =k ct The rate increases with the reciprocal of particle size and has an activation energy of 55.6 kJ/mol (13.6 kcal/mol). The relationship between reaction rate and oxygen partial pressure resembles a Langmuir-type equation and thus suggests that the reaction involves adsorption or desorption of oxygen at the interface. The square-root rate law may be due to the adsorption of a dissociated oxygen molecule. The observed apparent reaction order with respect to the hydroxyl ion concentration is a result of a complex combination of processes involving the oxidation and nydrolysis of iron, oxidation and hydrolysis of sulfur, and the oxygen reduction.

Excess Mass Estimates and Tests for Multimodality

Abstract We propose and study a method for analyzing the modality of a distribution. The method is based on the excess mass functional. This functional measures excessive empirical mass in comparison with multiples of uniform distribution. By the excess mass approach we separate the investigation about the number of modes from questions concerning their location. For distributions on the line, the excess mass functional can be estimated at a square-root rate. The asymptotic behavior of estimators is studied, and tests for multimodality based on the excess mass are derived.

Kinetics of silicide formation by thin films of V on Si and SiO2 substrates

The reaction rate of vacuum-evaporated films of V of the order of 1000 Å thick is investigated by MeV He backscattering spectrometry. On substrates of single-crystal Si and for anneal times up to several hours in the temperature range 570–650°C, VSi2 is formed at a linear rate in time. The activation energy of the process is 1.7±0.2 eV. The presence of oxygen in amounts of 10% can significantly decelerate the reaction. On substrates of SiO2 in the temperature range 730–820°C and anneal times of several hours or less, V3Si is formed at a square-root rate in time. The activation energy of this process is 2.0±0.2 eV.