A new stochastic approximation procedure using quantile curves

Summary Suppose that for each real number x we can sample at will from a population having unknown distribution function F _x (·), that p is a fixed number in (0,1), that a is a fixed real number, and that there is a unique unknown value of x (call it ) such that is the p-th quantile of F .We propose scheme for sequentially approximating , present a theorem giving conditions under which these approximations converge to with probability one, discuss the conditions of the theorem, give some examples, and mention some variations.The research of this author was supported by National Science Foundation Grant No. MCS 77-00841The research of this author was supported by the Research Foundation of the State University of New York     

Strong representation of an adaptive stochastic approximation procedure

We consider a rather general one-dimensional stochastic approximation algorithm where the steplengths might be random. Without assuming a martingale property of the random noise we obtain a strong representation by weighted averages of the error terms. We are able to apply the representation to an adaptive process in the case where the random noise is a martingale difference sequence as well as in the case where the random noise is weakly dependent and some moment conditions are statisfied.     

Near-optimal stochastic approximation for online principal component estimation

Principal component analysis (PCA) has been a prominent tool for high-dimensional data analysis. Online algorithms that estimate the principal component by processing streaming data are of tremendous practical and theoretical interests. Despite its rich applications, theoretical convergence analysis remains largely open. In this paper, we cast online PCA into a stochastic nonconvex optimization problem, and we analyze the online PCA algorithm as a stochastic approximation iteration. The stochastic approximation iteration processes data points incrementally and maintains a running estimate of the principal component. We prove for the first time a nearly optimal finite-sample error bound for the online PCA algorithm. Under the subgaussian assumption, we show that the finite-sample error bound closely matches the minimax information lower bound.     

Continuous procedure of stochastic approximation in a semi-Markov medium

Using the Lyapunov function for an averaged system, we establish conditions for the convergence of the procedure of stochastic approximation

Nearly optimal stochastic approximation for online principal subspace estimation

Principal component analysis(PCA) has been widely used in analyzing high-dimensional data. It converts a set of observed data points of possibly correlated variables into a set of linearly uncorrelated variables via an orthogonal transformation. To handle streaming data and reduce the complexities of PCA,(subspace)online PCA iterations were proposed to iteratively update the orthogonal transformation by taking one observed data point at a time. Existing works on the convergence of(subspace) onli...     

Bayesian multiple change-point estimation with annealing stochastic approximation Monte Carlo

Bayesian multiple change-point models are built with data from normal, exponential, binomial and Poisson distributions with a truncated Poisson prior for the number of change-points and conjugate prior for the distributional parameters. We applied Annealing Stochastic Approximation Monte Carlo (ASAMC) for posterior probability calculations for the possible set of change-points. The proposed methods are studied in simulation and applied to temperature and the number of respiratory deaths in Seoul, South Korea.     

Invariantly recursive credibility estimation

Necessary and sufficient conditions are given for recursivity of credibility estimators to be invariant against changes in term-length. A stationary invariant risk process is analysed, and a concept of asymptotic efficiency is developed.     

Event-driven stochastic approximation

We consider a Robbins-Monro type iteration wherein noisy measurements are event-driven and therefore arrive asynchronously. We propose a modification of step-sizes that ensures desired asymptotic behaviour regardless of this aspect. This generalizes earlier results on asynchronous stochastic approximation wherein the asynchronous behaviour is across different components, but not along the same component of the vector iteration, as is the case considered here.     

Asymptotic normality of a discrete procedure of stochastic approximation in a semi-Markov medium

We obtain sufficient conditions for the asymptotic normality of a jump procedure of stochastic approximation in a semi-Markov medium using a compensating operator of an extended Markov renewal process. The asymptotic representation of the compensating operator guarantees the construction of the generator of a limit diffusion process of the Ornstein-Uhlenbeck type. __________ Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 58, No. 10, pp. 1425–1433, October, 2006.     

Parallel and bootstrapped stochastic approximation

Parallelization of stochastic approximation procedures can reduce computation and total observation time of a system. Concerning the number of all observations used by the pure sequential and the suggested parallel method a weak invariance principle implies the asymptotic equivalence of both methods. A loglog invariance principle and a rate of a.s. convergence result describe the pathwise properties. Due to the parallel design asymptotic confidence regions can readily be constructed either by computing the bootstrap distribution or the Gaussian limit distribution determined by the empirical covariance     

Adaptive stochastic approximation algorithm

In this paper, stochastic approximation (SA) algorithm with a new adaptive step size scheme is proposed. New adaptive step size scheme uses a fixed number of previous noisy function values to adjust steps at every iteration. The algorithm is formulated for a general descent direction and almost sure convergence is established. The case when negative gradient is chosen as a search direction is also considered. The algorithm is tested on a set of standard test problems. Numerical results show good performance and verify efficiency of the algorithm compared to some of existing algorithms with adaptive step sizes.     

Parallel Lagrangean approximation procedure

We investigate the potential of the parallelised Lagrangean approximation procedure (PLAP) for certain combinatorial optimisation problems in manufacturing systems. The framework of a PLAP is proposed for some combinatorial manufacturing problems which are decomposable into well-structured subproblems. The synchronous PLAP for the multistage dynamic lot-sizing problem is implemented on a parallel computer (Alliant FX/4) and its computational experience is reported as a promising application of vector-concurrent computing.     

Measurement scheduling for recursive team estimation

We consider a decentralized LQG measurement scheduling problem in which every measurement is costly, no communication between observers is permitted, and the observers' estimation errors are coupled quadratically. This setup, motivated by considerations from organization theory, models measurement scheduling problems in which cost, bandwidth, or security constraints necessitate that estimates be decentralized, although their errors are coupled. We show that, unlike the centralized case, in the decentralized case the problem of optimizing the time integral of the measurement cost and the quadratic estimation error is fundamentally stochastic, and we characterize the -optimal open-loop schedules as chattering solutions of a deterministic Lagrange optimal control problem. Using a numerical example, we describe also how this deterministic optimal control problem can be solved by nonlinear programming.This research was supported in part by ARPA Grant N00174-91-C-0116 and NSF Grant NCR-92-04419.     

Geometric structure in stochastic approximation

1. IntroductionLet f: Re -- R be a differelltiable fUnction. f reaChes its extremes on the setJ = {x E R"lfx(x) = 0}, (1.1)where,x(X) = (V,..., V)". (1.2)If jx can be observed exactly at any x e R", then there are various numerical methods toconstruct {xh}, xk E Re such that the distance d(xk, J) between uk and J tends to zero ask -- co. However, in many application problems jx can only be observed with noise, i.e.,the observation at time k 1 isYk 1 = fi(~k) (k 1, (1'3)where xk is …     

Weak approximation of stochastic equations

We approximate the solution of nonlinear stochastic equations driven by a Gaussian white noise by solutions of similar equations, where the Gaussian noise is replaced by a weighted Poissonian point process.     

On stochastic estimation

We consider a local random searching method to approximate a root of a specified equation. If such roots, which can be regarded as estimators for the Euclidean parameter of a statistical experiment, have some asymptotic optimality properties, the local random searching method leads to asymptotically optimal estimators in such cases. Application to simple first order autoregressive processes and some simulation results for such models are also included.     

Spatial approximation of stochastic convolutions

We study linear stochastic evolution partial differential equations driven by additive noise. We present a general and flexible framework for representing the infinite dimensional Wiener process, which drives the equation. Since the eigenfunctions and eigenvalues of the covariance operator of the process are usually not available for computations, we propose an expansion in an arbitrary frame. We show how to obtain error estimates when the truncated expansion is used in the equation. For the stochastic heat and wave equations, we combine the truncated expansion with a standard finite element method and derive a priori bounds for the mean square error. Specializing the frame to biorthogonal wavelets in one variable, we show how the hierarchical structure, support and cancelation properties of the primal and dual bases lead to near sparsity and can be used to simplify the simulation of the noise and its update when new terms are added to the expansion.     

Bounds for a class of stochastic recursive equations

In this note we develop a framework for computing upper and lower bounds of an exponential form for a class of stochastic recursive equations with uniformly recurrent Markov modulated inputs. These bounds generalize Kingman's bounds for queues with renewal inputs.     

Preferences estimation without approximation

We devise an estimation methodology which allows preferences estimation and comparative statics analysis without a reliance on Taylor’s approximations and the indirect utility function.