on preconditioned iterative methods for burgers equations we study the newton method and a fixed point method for solving the system of nonlinear equations arising from the sinc galerkin discretization of the burgers equations in each step of the newton method or the fixed point method a structured subsystem of linear equations is obtained and needs to be solved numerically in this paper preconditioning techniques are applied to solve such linear subsystems the bounds for eigenvalues of the preconditioned matrices are derived and numerical examples are given to illustrate the effectiveness of the proposed methods we also find that a combination of the newton fixed point iteration with the preconditioned gmres method is quite efficient for the sinc galerkin discretization of the burgers equations

transient flow in bounded randomly heterogeneous domains 1 exact conditional moment equations and recursive approximations in a recent paper we developed exact nonlocal conditional moment equations for transient flow in bounded domains driven by random forcing terms sources initial head and boundary conditions whereas our conditional mean equations took into account the randomness of forcing terms our conditional second moment equations did not we do so in this brief addendum

worst case linear discriminant analysis as scalable semidefinite feasibility problems in this paper we propose an efficient semidefinite programming sdp approach to worst case linear discriminant analysis wlda compared with the traditional lda wlda considers the dimensionality reduction problem from the worst case viewpoint which is in general more robust for classification however the original problem of wlda is non convex and difficult to optimize in this paper we reformulate the optimization problem of wlda into a sequence of semidefinite feasibility problems to efficiently solve the semidefinite feasibility problems we design a new scalable optimization method with a quasi newton method and eigen decomposition being the core components the proposed method is orders of magnitude faster than standard interior point sdp solvers experiments on a variety of classification problems demonstrate that our approach achieves better performance than standard lda our method is also much faster and more scalable than standard interior point sdp solvers based wlda the computational complexity for an sdp with m constraints and matrices of size d by d is roughly reduced from mathcal o m 3 md 3 m 2 d 2 to mathcal o d 3 m d in our case

high dimensional nonconvex stochastic optimization by doubly stochastic successive convex approximation in this paper we consider supervised learning problems over training sets in which the number of training examples and the dimension of feature vectors are both large we focus on the case where the loss function defining the quality of the parameter we wish to estimate may be non convex but also has a convex regularization we propose a doubly stochastic successive convex approximation scheme dssc able to handle non convex regularized expected risk minimization the method operates by decomposing the decision variable into blocks and operating on random subsets of blocks at each step fusing the merits of stochastic approximation with block coordinate methods and then implements successive convex approximation in contrast to many stochastic convex methods whose almost sure behavior is not guaranteed in non convex settings dssc attains almost sure convergence to a stationary solution of the problem moreover we show that the proposed dssc algorithm achieves stationarity at a rate of mathcal o log t t 1 4 numerical experiments on a non convex variant of a lasso regression problem show that dssc performs favorably in this setting we then apply this method to the task of dictionary learning from high dimensional visual data collected from a ground robot and observe reliable convergence behavior for a difficult non convex stochastic program

validation analysis of mirror descent stochastic approximation method the main goal of this paper is to develop accuracy estimates for stochastic programming problems by employing stochastic approximation sa type algorithms to this end we show that while running a mirror descent stochastic approximation procedure one can compute with a small additional effort lower and upper statistical bounds for the optimal objective value we demonstrate that for a certain class of convex stochastic programs these bounds are comparable in quality with similar bounds computed by the sample average approximation method while their computational cost is considerably smaller

the sample average approximation method for stochastic discrete optimization in this paper we study a monte carlo simulation based approach to stochastic discrete optimization problems the basic idea of such methods is that a random sample is generated and the expected value function is approximated by the corresponding sample average function the obtained sample average optimization problem is solved and the procedure is repeated several times until a stopping criterion is satisfied we discuss convergence rates stopping rules and computational complexity of this procedure and present a numerical example for the stochastic knapsack problem

variance based extragradient methods with line search for stochastic variational inequalities in this paper we propose dynamic sampled stochastic approximated ds sa extragradient methods for stochastic variational inequalities svis that are robust with respect to an unknown lipschitz c

minimax and risk averse multistage stochastic programming in this paper we study relations between the minimax risk averse and nested formulations of multistage stochastic programming problems in particular we discuss conditions for time consistency of such formulations of stochastic problems we also describe a connection between law invariant coherent risk measures and the corresponding sets of probability measures in their dual representation finally we discuss a minimax approach with moment constraints to the classical inventory model

bundle level type methods uniformly optimal for smooth and nonsmooth convex optimization the main goal of this paper is to develop uniformly optimal first order methods for convex programming cp by uniform optimality we mean that the first order methods themselves do not require the input of any problem parameters but can still achieve the best possible iteration complexity bounds by incorporating a multi step acceleration scheme into the well known bundle level method we develop an accelerated bundle level method and show that it can achieve the optimal complexity for solving a general class of black box cp problems without requiring the input of any smoothness information such as whether the problem is smooth nonsmooth or weakly smooth as well as the specific values of lipschitz constant and smoothness level we then develop a more practical restricted memory version of this method namely the accelerated prox level apl method we investigate the generalization of the apl method for solving certain composite cp problems and an important class of saddle point problems recently studied by nesterov math program 103 127 152 2005 we present promising numerical results for these new bundle level methods applied to solve certain classes of semidefinite programming and stochastic programming problems

learning in games with continuous action sets and unknown payoff functions this paper examines the convergence of no regret learning in games with continuous action sets for concreteness we focus on learning via dual averaging a widely used class of no regret learning schemes where players take small steps along their individual payoff gradients and then mirror the output back to their action sets in terms of feedback we assume that players can only estimate their payoff gradients up to a zero mean error with bounded variance to study the convergence of the induced sequence of play we introduce the notion of variational stability and we show that stable equilibria are locally attracting with high probability whereas globally stable equilibria are globally attracting with probability 1 we also discuss some applications to mixed strategy learning in finite games and we provide explicit estimates of the method s convergence speed

robust stochastic approximation approach to stochastic programming in this paper we consider optimization problems where the objective function is given in a form of the expectation a basic difficulty of solving such stochastic optimization problems is that the involved multidimensional integrals expectations cannot be computed with high accuracy the aim of this paper is to compare two computational approaches based on monte carlo sampling techniques namely the stochastic approximation sa and the sample average approximation saa methods both approaches the sa and saa methods have a long history current opinion is that the saa method can efficiently use a specific say linear structure of the considered problem while the sa approach is a crude subgradient method which often performs poorly in practice we intend to demonstrate that a properly modified sa approach can be competitive and even significantly outperform the saa method for a certain class of convex stochastic problems we extend the analysis to the case of convex concave stochastic saddle point problems and present in our opinion highly encouraging results of numerical experiments

three dimensional steady state flow to a well in a randomly heterogeneous bounded aquifer 1 we consider flow in a confined aquifer of uniform thickness due to a well of zero radius that fully penetrates the aquifer and discharges at a constant rate if the lateral extent of the aquifer is infinite a steady state flow regime never develops it is however well known that if the aquifer is additionally uniform a quasi steady state region extends from the well out to a cylindrical surface whose radius expands as the square root of time on the expanding surface head is uniform and time invariant inside this surface head at any time is described by a steady state solution a rigorous analysis of the analogous situation in a randomly heterogeneous aquifer would require the solution of a three dimensional transient stochastic flow problem in an aquifer of infinite lateral extent here we take a different approach by developing a three dimensional steady solution for mean flow to a well in a randomly heterogeneous aquifer with a cylindrical prescribed head boundary in analogy to the uniform case we expect our solution to approximate a quasi steady state region whose radius is initially small in comparison to the horizontal correlation scale of log conductivity but grows with time to become eventually much larger we treat log conductivity as a statistically homogeneous random field characterized by a gaussian spatial covariance function that may have different horizontal and vertical correlation scales our solution consists of analytical expressions for the ensemble mean and variance of head in the aquifer to second order in the standard deviation of log conductivity it is based on recursive approximations of exact nonlocal moment equations that are free of distributional assumptions and so apply to both gaussian and non gaussian log conductivity fields the analytical solution is supported by numerical monte carlo simulations it clarifies the manner in which relationships between the horizontal and vertical scales of the quasi steady state region and those of statistical anisotropy impact the statistical moments of drawdown and the equivalent and apparent hydraulic conductivities of the aquifer both conductivities are shown to exhibit a scale effect by growing with distance from the well within a radius of one to two horizontal integral scales from it

a central limit theorem and hypotheses testing for risk averse stochastic programs we study statistical properties of the optimal value and optimal solutions of the sample average approximation of risk averse stochastic problems central limit theorem type results are derived for the optimal value when the stochastic program is expressed in terms of a law invariant coherent risk measure having a discrete kusuoka representation the obtained results are applied to hypotheses testing problems aiming at comparing the optimal values of several risk averse convex stochastic programs on the basis of samples of the underlying random vectors we also consider nonasymptotic tests based on confidence intervals on the optimal values of the stochastic programs obtained using the stochastic mirror descent algorithm numerical simulations show how to use our developments to choose among different distributions and on the considered class of risk averse stochastic programs the asymptotic tests show better results

a note on preconditioned block toeplitz matrices in siam j sci statist comput 13 1992 pp 948 966 ku and kuo proposed and analysed a block circulant preconditioner r mn for solving a family of block toeplitz systems t mn v b for a special class of block matrices called the quadrantally symmetric toeplitz matrices they proved that the eigenvalues of r mn 1 t mn are clustered around one except at most o m n outliers with t mn generated by a two dimensional rational function the superior convergence rate of the preconditioned conjugate gradient pcg method is explained by the clustering property of the spectrum of r mn 1 t mn however in their analysis there is no discussion on the positive definiteness of the matrix t mn and the preconditioner r mn is assumed to be invertible in this paper we give some results on these two aspects under the assumptions in the above referenced paper we prove that if the generating function f x y of t mn is positive then t mn is posit

flow to partially penetrating wells in unconfined heterogeneous aquifers mean head and interpretation of pumping tests 1 a partially penetrating well of length lw and radius rw starts to pump at constant discharge qw at t 0 from an unconfined aquifer of thickness d the aquifer is of random and stationary conductivity characterized by kg geometric mean y2 log conductivity variance and i and iv the horizontal and vertical integral scales the flow problem is solved under a few simplifying assumptions commonly adopted in the literature for homogeneous media rw lw 1 linearization of the free surface condition and constant drainable porosity n additionally it is assumed that rw i 1 and lw iv 1 to simplify the well boundary conditions and that a first order approximation in y2 extended to finite y2 on a conjectural basis is adopted the solution is obtained for the mean head field h r z t and the associated water table equation the main result of the analysis is that the flow domain can be divided into three zones for h 1 the neighborhood of the well r i where h qw lwka h0 r z tkefuv nd with h0 being the zero order solution pertaining to a homogeneous and isotropic aquifer ka being the conductivity arithmetic mean and kefuv being the effective vertical conductivity in mean uniform flow 2 an exterior zone r i in which h qw lwkefuh h0 r z tkefuv nd with kefuh being the horizontal effective conductivity and 3 an intermediate zone in which the solution requires a few numerical quadratures not carried out here the application to pumping tests reveals that identification of the aquifer parameters for homogeneous and anisotropic aquifers by commonly used methods can be applied for the drawdown measured in an observation well of length lowiv to ensure exchange of space and ensemble head averages in the second zone in order to identify kefuh kefuv and n in contrast the use of the drawdown in the well first zone leads to an overestimation of kefuh by the factor ka kefuh

the extended thiem s solution including the impact of heterogeneity 1 in this study we present a formula for the hydraulic head describing the mean drawdown of a three dimensional steady state pumping test in heterogeneous anisotropic porous media effectively by modeling the hydraulic conductivity k x as spatial random function and using the upscaling method coarse graining we succeed in deriving a closed form solution hefw r which we understand as an extension of thiem s formula to heterogeneous media the solution hefw r does not only depend on the radial distance r but accounts also for the statistics of k x namely geometric mean kg variance 2 horizontal correlation length l and anisotropy ratio e we perform a sensitivity analysis on the parameters of hefw r and implement an inverse estimation strategy using numerical pumping tests we show the applicability of hefw r on the interpretation of drawdown data this will be done for both an ensemble of as well as for single pumping tests making use of the inverse estimation method we find excellent agreement of estimated parameters with initial values in particular for the horizontal correlation length

optimality functions in stochastic programming optimality functions define stationarity in nonlinear programming semi infinite optimization and optimal control in some sense in this paper we consider optimality functions for stochastic programs with nonlinear possibly nonconvex expected value objective and constraint functions we show that an optimality function directly relates to the difference in function values at a candidate point and a local minimizer we construct confidence intervals for the value of the optimality function at a candidate point and hence provide a quantitative measure of solution quality based on sample average approximations we develop an algorithm for classes of stochastic programs that include cvar problems and utilize optimality functions to select sample sizes

three dimensional stochastic analysis of macrodispersion in aquifers the dispersive mixing resulting from complex flow in three dimensionally heterogeneous porous media is analyzed using stochastic continuum theory stochastic solutions of the perturbed steady flow and solute transport equations are used to construct the macroscopic dispersive flux and evaluate the resulting macrodispersivity tensor in terms of a three dimensional statistically anisotropic input covariance describing the hydraulic conductivity with a statistically isotropic input covariance the longitudinal macrodispersivity is convectively controlled but the transverse macrodispersivity is proportional to the local dispersivity and is several orders of magnitude smaller than the longitudinal term with an arbitrarily oriented anisotropic conductivity covariance all components of the macrodispersivity tensor are convectively controlled and the ratio of transverse to longitudinal dispersivity is of the order of 10 1 in this case the off diagonal components of the dispersivity tensor are significant being numerically larger than the diagonal transverse terms and the transverse dispersion process can be highly anisotropic dispersivities predicted by the stochastic theory are shown to be consistent with controlled field experiments and monte carlo simulations the theory which treats the asymptotic condition of large displacement indicates that a classical gradient transport fickian relationship is valid for large scale displacements

strong duality in nonconvex quadratic optimization with two quadratic constraints we consider the problem of minimizing an indefinite quadratic function subject to two quadratic inequality constraints when the problem is defined over the complex plane we show that strong duality holds and obtain necessary and sufficient optimality conditions we then develop a connection between the image of the real and complex spaces under a quadratic mapping which together with the results in the complex case lead to a condition that ensures strong duality in the real setting preliminary numerical simulations suggest that for random instances of the extended trust region subproblem the sufficient condition is satisfied with a high probability furthermore we show that the sufficient condition is always satisfied in two classes of nonconvex quadratic problems finally we discuss an application of our results to robust least squares problems

radially convergent flow in heterogeneous porous media we present an analytical solution for apparent effective transmissivity under radially convergent steady state flow conditions produced by constant pumping from a single well of finite radius rw apparent effective transmissivity te is defined as the value that relates the expected values of flow and head gradient at a certain location the domain is two dimensional of annular shape and the size of the pumping well is explicitly taken into account the solution for the steady state heads is obtained by solving the perturbed flow equation and substituting it into darcy s law to obtain a consistent second order expansion for te we show that apparent effective transmissivity is a scalar for any choice of isotropic covariance model with an expression given in integral form our main result is that te in a heterogeneous statistically isotropic random field under radial steady state flow conditions is a monotonie increasing function of r distance from the well that rises from the harmonic mean of the point transmissivity values close to the well and tends asymptotically towards the geometric mean far from the well the asymptotic value is reached at a distance of a few integral scales 1 5 2 for the gaussian model and 3 5 for the exponential one the apparent effective transmissivity versus normalized r curves are in excellent agreement with previously published numerical work carried out using monte carlo method

block triangular and skew hermitian splitting methods for positive definite linear systems by further generalizing the concept of hermitian or normal and skew hermitian splitting for a non hermitian and positive definite matrix we introduce a new splitting called positive definite and skew hermitian splitting pss and then establish a class of pss methods similar to the hermitian or normal and skew hermitian splitting hss or nss method for iteratively solving the positive definite systems of linear equations theoretical analysis shows that the pss method converges unconditionally to the exact solution of the linear system with the upper bound of its convergence factor dependent only on the spectrum of the positive definite splitting matrix and independent of the spectrum of the skew hermitian splitting matrix as well as the eigenvectors of all matrices involved when we specialize the pss to block triangular or triangular and skew hermitian splitting btss or tss the pss method naturally leads to a btss or tss iteration method which may be more practical and efficient than the hss and nss iteration methods applications of the btss method to the linear systems of block two by two structures are discussed in detail numerical experiments further show the effectiveness of our new methods

a note on well boundary condition for flow through heterogeneous formations 1 in modeling flow toward wells it is mathematically convenient to replace the well by a singularity line along its axis in the case of homogeneous aquifers confined flow and fully penetrating wells the strength of the sources is constant and the common condition of constant head on the well is satisfied exactly it also leads to constant flux on the well envelope in the case of aquifers of spatially variable conductivity the constant head boundary condition can be satisfied accurately if the well radius is much smaller than the log conductivity horizontal integral scale by selecting a variable source strength that is proportional to the local conductivity along the well indelman et al 1996 fiori et al 1998 indelman 2003a 2003b in this case the equivalent conductivity tends to the arithmetic mean near the well it changes with distance approaching from above the effective conductivity in uniform horizontal mean flow far from the well selecting inadvertently a constant source strength violates the condition of constant well head the equivalent conductivity tends to the harmonic mean near the well and approaches from below the effective one in uniform mean flow the equivalent conductivity is derived for both types of sources for stratified formations as well as for aquifers of a three dimensional heterogeneous structure the implications on interpretation of head measurements by the two models is illustrated by a few examples

a class of iteration methods based on the hss for toeplitz systems of weakly nonlinear equations for toeplitz systems of weakly nonlinear equations combining the separability and strong dominance between the linear and the nonlinear terms with the hermitian and skew hermitian splitting hss iteration technique we establish two nonlinear composite iteration schemes called picard csss and nonlinear csss like iteration methods which are based on a special case of the hss where the symmetric part h 1 2 a a t is a centrosymmetric matrix and the skew symmetric part h 1 2 a a t is a skew centrosymmetric matrix the advantages of these methods are that they can transfer the linear sub systems involved in inner iteration to two linear systems of half an order besides fast methods are available for computing the two half steps involved in the inner iteration numerical results are provided to further show that both picard csss and nonlinear csss like iteration methods are feasible and effective

introduction to convex optimization in financial markets convexity arises naturally in financial risk management in risk preferences concerning random cash flows convexity corresponds to the fundamental diversification principle convexity is a basic property also of budget constraints both in classical linear models as well as in more realistic models with transaction costs and constraints moreover modern securities markets are based on trading protocols that result in convex trading costs the first part of this paper gives an introduction to certain basic concepts and principles of financial risk management in simple optimization terms the second part reviews some convex optimization techniques used in mathematical and numerical analysis of financial optimization problems

preconditioners for nonsymmetric block toeplitz like plus diagonal linear systems we consider the system of linear equations lu f where l is a nonsymmetric block toeplitz like plus diagonal matrix which arises from the sinc galerkin discretization of differential equations our main contribution is to construct effective preconditioners for this structured coefficient matrix and to derive tight bounds for eigenvalues of the preconditioned matrices moreover we use numerical examples to show that the new preconditioners when applied to the preconditioned gmres method are efficient for solving the system of linear equations

non euclidean restricted memory level method for large scale convex optimization we propose a new subgradient type method for minimizing extremely large scale nonsmooth convex functions over simple domains the characteristic features of the method are a the possibility to adjust the scheme to the geometry of the feasible set thus allowing to get nearly dimension independent and nearly optimal in the large scale case rate of convergence results for minimization of a convex lipschitz continuous function over a euclidean ball a standard simplex and a spectahedron the set of positive semidefinite symmetric matrices of given size with unit trace b flexible handling of accumulated information allowing for tradeoff between the level of utilizing this information and iteration s complexity we present extensions of the scheme for the cases of minimizing non lipschitzian convex objectives finding saddle points of convex concave functions and solving variational inequalities with monotone operators finally we report on encouraging numerical results of experiments with test problems of dimensions up to 66 000

multistep stochastic mirror descent for risk averse convex stochastic programs based on extended polyhedral risk measures we consider risk averse convex stochastic programs expressed in terms of extended polyhedral risk measures we derive computable confidence intervals on the optimal value of such stochastic programs using the robust stochastic approximation and the stochastic mirror descent smd algorithms when the objective functions are uniformly convex we also propose a multistep extension of the stochastic mirror descent algorithm and obtain confidence intervals on both the optimal values and optimal solutions numerical simulations show that our confidence intervals are much less conservative and are quicker to compute than previously obtained confidence intervals for smd and that the multistep stochastic mirror descent algorithm can obtain a good approximate solution much quicker than its nonmultistep counterpart

smoothing techniques for computing nash equilibria of sequential games we develop first order smoothing techniques for saddle point problems that arise in finding a nash equilibrium of sequential games the crux of our work is a construction of suitable prox functions for a certain class of polytopes that encode the sequential nature of the game we also introduce heuristics that significantly speed up the algorithm and decomposed game representations that reduce the memory requirements enabling the application of the techniques to drastically larger games an implementation based on our smoothing techniques computes approximate nash equilibria for games that are more than four orders of magnitude larger than what prior approaches can handle finally we show near linear further speedups from parallelization

asymptotic results of stochastic decomposition for two stage stochastic quadratic programming this paper presents the stochastic decomposition sd algorithms for two classes of stochastic programming problems 1 two stage stochastic quadratic linear programming sqlp in which a quadrati

nonlocal properties of nonuniform averaged flows in heterogeneous media the properties of nonuniform average potential flows in media of stationary random conductivity are studied the mathematical model of average flow is derived as a system of governing equations to be satisfied by mean velocity and mean head the averaged darcy s law determines the effective conductivity as an integral operator of the convolution type relating the mean velocity to the mean head gradient in a nonlocal way in fourier domain the mean velocity is proportional to the mean head gradient the coefficient of proportionality is referred to as the effective conductivity tensor and is derived by perturbation methods in terms of functionals of the conductivity spatial moments it is shown that the effective conductivity cannot be defined uniquely for potential flows however this nonuniqueness does not affect the spatial distribution of the mean head and velocity analytical expressions of the effective conductivity tensor are derived for two and three dimensional flows and for exponential and gaussian correlations of isotropic conductivity the fundamental solution of the governing equations in effective media mean green function is calculated for the same cases two new asymptotic models of the averaged darcy s law are developed to be applicable to large and small scales of heterogeneity several asymptotic expansions of tue two dimensional and three dimensional mean green functions are derived for exponential and gaussian correlations

optimization with reference based robust preference constraints we propose an optimization model using a novel robust preference relationship reference based almost stochastic dominance rsd the concept of rsd addresses the two problems in utility based dec

hybrid deterministic stochastic methods for data fitting many structured data fitting applications require the solution of an optimization problem involving a sum over a potentially large number of measurements incremental gradient algorithms offer inexpensive iterations by sampling a subset of the terms in the sum these methods can make great progress initially but often slow as they approach a solution in contrast full gradient methods achieve steady convergence at the expense of evaluating the full objective and gradient on each iteration we explore hybrid methods that exhibit the benefits of both approaches rate of convergence analysis shows that by controlling the sample size in an incremental gradient algorithm it is possible to maintain the steady convergence rates of full gradient methods we detail a practical quasi newton implementation based on this approach numerical experiments illustrate its potential benefits

a comparison of three methods for selecting values of input variables in the analysis of output from a computer code two types of sampling plans are examined as alternatives to simple random sampling in monte carlo studies these plans are shown to be improvements over simple random sampling with respect to variance for a class of estimators which includes the sample mean and the empirical distribution function

min max generalization for deterministic batch mode reinforcement learning relaxation schemes we study the min max optimization problem introduced in fonteneau et al towards min max reinforcement learning icaart 2010 springer heidelberg 2011 pp 61 77 for computing policies for batch mode reinforcement learning in a deterministic setting with fixed finite time horizon first we show that the min part of this problem is np hard we then provide two relaxation schemes the first relaxation scheme works by dropping some constraints in order to obtain a problem that is solvable in polynomial time the second relaxation scheme based on a lagrangian relaxation where all constraints are dualized can also be solved in polynomial time we also theoretically prove and empirically illustrate that both relaxation schemes provide better results than those given in fonteneau et al 2011 as cited above

band toeplitz preconditioners for block toeplitz systems abstract we consider the solutions of block toeplitz systems with toeplitz blocks by the preconditioned conjugate gradient pcg method here the block toeplitz matrices are generated by nonnegative functions f x y we use band toeplitz matrices as preconditioners the generating functions g x y of the preconditioners are trigonometric polynomials of fixed degree and are determined by minimizing f g f we prove that the condition number of the preconditioned system is o 1 an a priori bound on the number of iterations for convergence is obtained

on sampling rates in simulation based recursions we consider the context of simulation based recursions that is recursions that involve quantities needing to be estimated using a stochastic simulation examples include stochastic adaptations of fixed point and gradient descent recursions obtained by replacing function and derivative values appearing within the recursion by their monte carlo counterparts the primary motivating settings are simulation optimization and stochastic root finding problems where the low point and the zero of a function are sought respectively with only monte carlo estimates of the functions appearing within the problem we ask how much monte carlo sampling needs to be performed within simulation based recursions in order that the resulting iterates remain consistent and more importantly efficient where efficient implies convergence at the fastest possible rate answering these questions involves trading off two types of error inherent in the iterates the deterministic error due to recursion and the stochastic er

transmissivity upscaling in numerical aquifer models of steady well flow unconditional statistics 1 numerical solution of regional scale aquifer flow requires discretizing the transmissivity t typically the numerical element scale r and the transmissivity integral scale i are both of the order of hundreds of meters then the upscaled block transmissivity is random and its statistical moments depend on those of t on flow conditions and on r i modeling y ln t as a two dimensional normally distributed stationary random field we derive unconditional statistics of the upscaled ln accurate to the first order in y2 for strongly nonuniform flow in a circular block of radius r centered on a well of radius rw in an unbounded domain after a proper definition it was found that the ensemble mean is approximately but not exactly equal to ln tg the geometric mean and that the variance ratio 2 y2 which depends on rw i and the shape of the correlation drops slowly from unity for r i 0 to approximately 0 6 for r i 10 hence the variance of the upscaled transmissivity in radial flow is larger than that determined previously for uniform flow additionally defining the equivalent teq as a deterministic value for which the solution of the flow problem renders directly the mean upscaled head drop and specific discharge we find that in radial flow teq th the harmonic mean and grows slowly with increasing r i whereas for mean uniform flow teq tg application of the procedure is illustrated for an example of aquifer with selected values of parameters

correlation structure of flow variables for steady flow toward a well with application to highly anisotropic heterogeneous formations the study a continuation of that of indelman et al 1996 aims at deriving the second order moments of flow variables such as hydraulic head its gradient and the specific discharge for steady flow toward a fully penetrating well in a confined heterogeneous aquifer the log conductivity y ln k is modeled as a three dimensional stationary function of gaussian correlation of anisotropy ratio e by using first order approximations in 2y and e we derive the variance and the vertical integral scale of the piezometric head h of its radial gradient er and of the radial component of the specific discharge qr owing to the nonuniformity of the average flow these quantities are functions of the distance from the well it is shown that the variances of the head 2h and of its gradient 2er as well as the crossvariance e y between er and y vanish at the well whereas the discharge variance 2qr tends to the product between the log conductivity variance 2y and the squared mean discharge qr 2 this behavior pertains to a stratified formation surrounding the well far from the well 75 horizontal y integral scales i the head variance approaches a constant value for r 10i the moments 2er 2qr and ery tend to the corresponding values for uniform flow but with the local mean head gradient replacing the constant one the head vertical integral scale grows indefinitely with r whereas the vertical integral scale of the flux is larger by one log conductivity vertical scale than the one prevailing in uniform flow this latter property is explained by the presence of the source line which increases the correlations in the vertical direction the present results may be used in identifying the log conductivity statistical parameters from flowmeter velocity measurements in piezometers surrounding pumping or injecting wells

cauchy noise loss for stochastic optimization of random matrix models via free deterministic equivalents abstract for random matrix models the parameter estimation based on the traditional likelihood functions is not straightforward in particular when we have only one sample matrix we introduce a new parameter optimization method for random matrix models which works even in such a case the method is based on the spectral distribution instead of the traditional likelihood in the method the cauchy noise has an essential role because the free deterministic equivalent which is a tool in free probability theory allows us to approximate the spectral distribution perturbed by cauchy noises by a smooth and accessible density function moreover we study an asymptotic property of determination gap which has a similar role as generalization gap besides we propose a new dimensionality recovery method for the signal plus noise model and experimentally demonstrate that it recovers the rank of the signal part even if the true rank is not small it is a simultaneous rank selection and parameter estimation procedure

on cscs based iteration methods for toeplitz system of weakly nonlinear equations abstract for toeplitz system of weakly nonlinear equations by using the separability and strong dominance between the linear and the nonlinear terms and using the circulant and skew circulant splitting cscs iteration technique we establish two nonlinear composite iteration schemes called picard cscs and nonlinear cscs like iteration methods respectively the advantage of these methods is that they do not require accurate computation and storage of jacobian matrix and only need to solve linear sub systems of constant coefficient matrices therefore computational workloads and computer storage may be saved in actual implementations theoretical analysis shows that these new iteration methods are local convergent under suitable conditions numerical results show that both picard cscs and nonlinear cscs like iteration methods are feasible and effective for some cases

on the complexity of integer programming a simple proof that integer programming ts in x s given the proof also estabhshes that there s a pseudopolynomial tune algorithm for integer programmmg with any fixed number of constraints

accelerated schemes for a class of variational inequalities we propose a novel stochastic method namely the stochastic accelerated mirror prox samp method for solving a class of monotone stochastic variational inequalities svi the main idea of the proposed algorithm is to incorporate a multi step acceleration scheme into the stochastic mirror prox method the developed samp method computes weak solutions with the optimal iteration complexity for svis in particular if the operator in svi consists of the stochastic gradient of a smooth function the iteration complexity of the samp method can be accelerated in terms of their dependence on the lipschitz constant of the smooth function for svis with bounded feasible sets the bound of the iteration complexity of the samp method depends on the diameter of the feasible set for unbounded svis we adopt the modified gap function introduced by monteiro and svaiter for solving monotone inclusion and show that the iteration complexity of the samp method depends on the distance from the initial point to the set of strong solutions it is worth noting that our study also significantly improves a few existing complexity results for solving deterministic variational inequality problems we demonstrate the advantages of the samp method over some existing algorithms through our preliminary numerical experiments

a stochastic quasi newton method for large scale optimization the question of how to incorporate curvature information into stochastic approximation methods is challenging the direct application of classical quasi newton updating techniques for deterministic optimization leads to noisy curvature estimates that have harmful effects on the robustness of the iteration in this paper we propose a stochastic quasi newton method that is efficient robust and scalable it employs the classical bfgs update formula in its limited memory form and is based on the observation that it is beneficial to collect curvature information pointwise and at spaced intervals one way to do this is through subsampled hessian vector products this technique differs from the classical approach that would compute differences of gradients at every iteration and where controlling the quality of the curvature estimates can be difficult we present numerical results on problems arising in machine learning that suggest that the proposed method shows much promise

transient well type flows in heterogenous formations 1 the problem of averaging transient flows by sources of a given head boundary condition in heterogeneous formations of random conductivity is investigated the study generalizes the recently developed mathematical model of average transient nonuniform flow indelman 1996 tartakovsky and neuman 1998a the latter allows calculating the mean head for sources of flux boundary condition and as such is not applicable to modeling common well flows to account for the head condition the source term in the local flow model is modeled by a random function proportional to the hydraulic conductivity the local flow equations are further averaged in order to determine the mean flow variables it is shown that the effective conductivity cannot be defined for arbitrary initial head distributions this precludes deriving the mathematical model of an average transient flow with the head boundary condition however the average flow equations are derived for the important particular case of a uniformly distributed initial head the effective conductivity tensor is defined for an arbitrary heterogeneous formation and analyzed in detail for isotropic media it is shown that at the initial stage of the transient flow the effective conductivity is larger than the conductivity arithmetic mean the fundamental solution of the average flow equations mean green function corresponding to the mean head distribution due to the instantaneous injection of the unit mean amount of water through a point source is derived at first order in the conductivity variance and is compared with the mean head for the flux boundary condition

primal dual subgradient methods for convex problems in this paper we present a new approach for constructing subgradient schemes for different types of nonsmooth problems with convex structure our methods are primal dual since they are always able to generate a feasible approximation to the optimum of an appropriately formulated dual problem besides other advantages this useful feature provides the methods with a reliable stopping criterion the proposed schemes differ from the classical approaches divergent series methods mirror descent methods by presence of two control sequences the first sequence is responsible for aggregating the support functions in the dual space and the second one establishes a dynamically updated scale between the primal and dual spaces this additional flexibility allows to guarantee a boundedness of the sequence of primal test points even in the case of unbounded feasible set however we always assume the uniform boundedness of subgradients we present the variants of subgradient schemes for nonsmooth convex minimization minimax problems saddle point problems variational inequalities and stochastic optimization in all situations our methods are proved to be optimal from the view point of worst case black box lower complexity bounds

steady flow toward wells in heterogeneous formations mean head and equivalent conductivity we consider steady flow of water in a confined aquifer toward a fully penetrating well of radius rw figure 1 the hydraulic conductivity k is modeled as a three dimensional stationary random space function the two point covariance of y in k kg is of axisymmetric anisotropy with i and i the horizontal and vertical integral scales respectively and kg the geometric mean of k unlike previous studies which assumed constant flux the well boundary condition is of given constant head figure 1 the aim of the study is to derive the mean head h and the mean specific discharge q as functions of the radial coordinate r and of the parameters y2 e i i and rw i an approximate solution is obtained at first order in y2 by replacing the well by a line source of strength proportional to k and by assuming ergodicity i e equivalence between space averages over the vertical and h q ensemble means an equivalent conductivity keq is defined as the fictitious one of a homogeneous aquifer which conveys the same discharge q as the actual one for the given head hw in the well and a given head in a piezometer at distance r from the well this definition corresponds to the transmissivity determined in a pumping test by an observer that measures hw andq the main result of the study is the relationship 19 keq ka 1 kefu where ka is the conductivity arithmetic mean and kefu is the effective conductivity for mean uniform flow in the horizontal direction in the same aquifer the weight coefficient 10 has the simple approximate expression ln r i in r rw near the well 0 and keq ka which is easily understood since for rw i 1 the formation behaves locally like a stratified one in contrast far from the well 1 and keq kefu the flow being slowly varying there since ka kefu our result indicates that the transmissivity is overestimated in a pumping test in a steady state and it decreases with the distance from the well however the difference between ka and kefu is small for highly anisotropic formations for which e 1 a nonlocal effective conductivity which depends only on the heterogeneous structure is derived in appendix a along the lines of indelman and abramovich 1994

bayesian inference with optimal maps we present a new approach to bayesian inference that entirely avoids markov chain simulation by constructing a map that pushes forward the prior measure to the posterior measure existence and uniqueness of a suitable measure preserving map is established by formulating the problem in the context of optimal transport theory we discuss various means of explicitly parameterizing the map and computing it efficiently through solution of an optimization problem exploiting gradient information from the forward model when possible the resulting algorithm overcomes many of the computational bottlenecks associated with markov chain monte carlo advantages of a map based representation of the posterior include analytical expressions for posterior moments and the ability to generate arbitrary numbers of independent posterior samples without additional likelihood evaluations or forward solves the optimization approach also provides clear convergence criteria for posterior approximation and facilitates model selection through automatic evaluation of the marginal likelihood we demonstrate the accuracy and efficiency of the approach on nonlinear inverse problems of varying dimension involving the inference of parameters appearing in ordinary and partial differential equations

a parallel decomposition method for nonconvex stochastic multi agent optimization problems this paper considers the problem of minimizing the expected value of a possibly nonconvex cost function parameterized by a random vector variable when the expectation cannot be computed accurately e g because the statistics of the random variables are unknown and or the computational complexity is prohibitive classical stochastic gradient methods for solving this problem may suffer from slow convergence in this paper we propose a stochastic parallel successive convex approximation based best response algorithm for general nonconvex stochastic sum utility optimization problems which arise naturally in the design of multi agent networks the proposed novel decomposition approach enables all users to update their optimization variables in parallel by solving a sequence of strongly convex subproblems one for each user almost sure convergence to stationary points is proved we then customize the algorithmic framework to solve the stochastic sum rate maximization problem over single input single output siso frequency selective interference channels multiple input multiple output mimo interference channels and mimo multiple access channels numerical results corroborate that the proposed algorithms can converge faster than state of the art stochastic gradient schemes while achieving the same or better sum rates

on preconditioned iterative methods for certain time dependent partial differential equations when the newton method or the fixed point method is employed to solve the systems of nonlinear equations arising in the sinc galerkin discretization of certain time dependent partial differential equations in each iteration step we need to solve a structured subsystem of linear equations iteratively by for example a krylov subspace method such as the preconditioned gmres in this paper based on the tensor and the toeplitz structures of the linear subsystems we construct structured preconditioners for their coefficient matrices and estimate the eigenvalue bounds of the preconditioned matrices under certain assumptions numerical examples are given to illustrate the effectiveness of the proposed preconditioning methods it has been shown that a combination of the newton fixed point iteration with the preconditioned gmres method is efficient and robust for solving the systems of nonlinear equations arising from the sinc galerkin discretization of the time dependent partial differential equations

percentile optimization for markov decision processes with parameter uncertainty markov decision processes are an effective tool in modeling decision making in uncertain dynamic environments because the parameters of these models typically are estimated from data or learned from experience it is not surprising that the actual performance of a chosen strategy often differs significantly from the designer s initial expectations due to unavoidable modeling ambiguity in this paper we present a set of percentile criteria that are conceptually natural and representative of the trade off between optimistic and pessimistic views of the question we study the use of these criteria under different forms of uncertainty for both the rewards and the transitions some forms are shown to be efficiently solvable and others highly intractable in each case we outline solution concepts that take parametric uncertainty into account in the process of decision making

min max feedback model predictive control for constrained linear systems min max feedback formulations of model predictive control are discussed both in the fixed and variable horizon contexts the control schemes the authors discuss introduce in the control optimization the notion that feedback is present in the receding horizon implementation of the control this leads to improved performance compared to standard model predictive control and resolves the feasibility difficulties that arise with the min max techniques that are documented in the literature the stabilizing properties of the methods are discussed as well as some practical implementation details

optimal rates for zero order convex optimization the power of two function evaluations we consider derivative free algorithms for stochastic and nonstochastic convex optimization problems that use only function values rather than gradients focusing on nonasymptotic bounds on convergence rates we show that if pairs of function values are available algorithms for d dimensional optimization that use gradient estimates based on random perturbations suffer a factor of at most sqrt d in convergence rate over traditional stochastic gradient methods we establish such results for both smooth and nonsmooth cases sharpening previous analyses that suggested a worse dimension dependence and extend our results to the case of multiple m ge 2 evaluations we complement our algorithmic development with information theoretic lower bounds on the minimax convergence rate of such problems establishing the sharpness of our achievable results up to constant sometimes logarithmic factors

on stochastic gradient and subgradient methods with adaptive steplength sequences traditionally stochastic approximation sa schemes have been popular choices for solving stochastic optimization problems however the performance of standard sa implementations can vary significantly based on the choice of the steplength sequence and in general little guidance is provided about good choices motivated by this gap we present two adaptive steplength schemes for strongly convex differentiable stochastic optimization problems equipped with convergence theory that aim to overcome some of the reliance on user specific parameters the first scheme referred to as a recursive steplength stochastic approximation rsa scheme optimizes the error bounds to derive a rule that expresses the steplength at a given iteration as a simple function of the steplength at the previous iteration and certain problem parameters the second scheme termed as a cascading steplength stochastic approximation csa scheme maintains the steplength sequence as a piecewise constant decreasing function with the reduction in the steplength occurring when a suitable error threshold is met then we allow for nondifferentiable objectives but with bounded subgradients over a certain domain in such a regime we propose a local smoothing technique based on random local perturbations of the objective function that leads to a differentiable approximation of the function assuming a uniform distribution on the local randomness we establish a lipschitzian property for the gradient of the approximation and prove that the obtained lipschitz bound grows at a modest rate with problem size this facilitates the development of an adaptive steplength stochastic approximation framework which now requires sampling in the product space of the original measure and the artificially introduced distribution

three dimensional numerical analysis of steady state pumping tests in heterogeneous confined aquifers 1 we study three dimensional steady state flow field generated by a fully penetrating well extracting a constant water discharge from a confined aquifer of constant thickness the hydraulic conductivity k is modeled as an axisymmetric stationary random space function and the flow equation is solved by using the seven node finite volume numerical scheme in view of application in aquifer characterization we analyze how the equivalent conductivity keq varies as a function of the distance from the extraction well the numerical simulations showed that the boundary condition at the well exerts a large impact on keq and that the first order solution in y2 obtained by indelman et al 1996 is valid up to y2 0 5 for higher values of y2 the first order solution overestimates keq with differences that for a given y2 reduce progressively as the formation becomes more anisotropic a simple inverse procedure based on keq obtained through cross well interference is finally proposed and tested with the objective of estimating the parameters of the geostatistical model of variability of the hydraulic conductivity a few parametric examples showed that a suitable interpretation of pumping tests can be used to safely estimate parameters such as the geometric mean of the hydraulic conductivity its variance and although with larger uncertainty the horizontal integral scale while the inference of the anisotropy ratio is highly uncertain and error prone

explicit group inverse of an innovative patterned matrix in this paper we present an innovative patterned matrix rfpl toeplitz matrix is neither the extension of toeplitz matrix nor its special case we show that the group inverse of this new patterned matrix can be represented as the sum of products of lower and upper triangular toeplitz matrices first the explicit expression of the group inverse of an rfpl toeplitz matrix is obtained second the decomposition of the group inverse is given finally an example demonstrates availability of the two methods for the group inverse

on projected stochastic gradient descent algorithm with weighted averaging for least squares regression the problem of least squares regression of a d dimensional unknown parameter is considered a stochastic gradient descent based algorithm with weighted iterate averaging that uses a single pass over the data is studied and its convergence rate is analyzed we first consider a bounded constraint set of the unknown parameter under some standard regularity assumptions we provide an explicit o 1 k upper bound on the convergence rate depending on the variance due to the additive noise in the measurements and the size of the constraint set we show that the variance term dominates the error and decreases with rate 1 k while the term that is related to the size of the constraint set decreases with rate log k k 2 we then compare the asymptotic ratio rho between the convergence rate of the proposed scheme and the empirical risk minimizer erm as the number of iterations approaches infinity we show that rho leq 4 for all d geq 1 when the random entries of the sensing vector are uncorrelated and identically distributed we further improve the upper bound by showing that rho leq 4 3 for the case of d 1 and unbounded parameter set when the random sensing entries are equal across time simulation results demonstrate strong performance of the algorithm as compared to existing methods and coincide with rho leq 4 3 even for large d in practice

statistical guarantees for the em algorithm from population to sample based analysis we develop a general framework for proving rigorous guarantees on the performance of the em algorithm and a variant known as gradient em our analysis is divided into two parts a treatment of these algorithms at the population level in the limit of infinite data followed by results that apply to updates based on a finite set of samples first we characterize the domain of attraction of any global maximizer of the population likelihood this characterization is based on a novel view of the em updates as a perturbed form of likelihood ascent or in parallel of the gradient em updates as a perturbed form of standard gradient ascent leveraging this characterization we then provide non asymptotic guarantees on the em and gradient em algorithms when applied to a finite set of samples we develop consequences of our general theory for three canonical examples of incompletedata problems mixture of gaussians mixture of regressions and linear regression with covariates missing completely at random in each case our theory guarantees that with a suitable initialization a relatively small number of em or gradient em steps will yield with high probability an estimate that is within statistical error of the mle we provide simulations to confirm this theoretically predicted behavior

monotonic calibrated interpolated look up tables real world machine learning applications may have requirements beyond accuracy such as fast evaluation times and interpretability in particular guaranteed monotonicity of the learned function with respect to some of the inputs can be critical for user confidence we propose meeting these goals for low dimensional machine learning problems by learning flexible monotonic functions using calibrated interpolated look up tables we extend the structural risk minimization framework of lattice regression to monotonic functions by adding linear inequality constraints in addition we propose jointly learning interpretable calibrations of each feature to normalize continuous features and handle categorical or missing data at the cost of making the objective non convex we address large scale learning through parallelization mini batching and random sampling of additive regularizer terms case studies on real world problems with up to sixteen features and up to hundreds of millions of training samples demonstrate the proposed monotonic functions can achieve state of the art accuracy in practice while providing greater transparency to users

a class of iteration methods based on the generalized preconditioned hermitian and skew hermitian splitting for weakly nonlinear systems abstract for large sparse systems of weakly nonlinear equations based on the separable property and strong dominance between the linear and the nonlinear terms bai and yang studied two nonlinear composite iteration schemes called picard hss and nonlinear hss like methods see z z bai x yang on hss based iteration methods for weakly nonlinear systems appl numer math 59 12 2009 2923 2936 in this paper we generalize these methods and propose a class of generalized nonlinear composite splitting iteration schemes called picard gphss and nonlinear gphss like iteration methods to solve the large sparse systems of weakly nonlinear equations we derive conditions for guaranteeing the local convergence of these iterative methods and derive some special case of iterative methods by choosing different parameters and preconditioned matrices numerical experiments are used to demonstrate the feasibility and effectiveness and an efficient preconditioner is presented for the new methods in actual implementation the efficiency is effectively testified by some comparisons with numerical results

robust inversion dimensionality reduction and randomized sampling we consider a class of inverse problems in which the forward model is the solution operator to linear odes or pdes this class admits several dimensionality reduction techniques based on data averaging or sampling which are especially useful for large scale problems we survey these approaches and their connection to stochastic optimization the data averaging approach is only viable however for a least squares misfit which is sensitive to outliers in the data and artifacts unexplained by the forward model this motivates us to propose a robust formulation based on the student s t distribution of the error we demonstrate how the corresponding penalty function together with the sampling approach can obtain good results for a large scale seismic inverse problem with 50 corrupted data

prediction of steady state flow in nonuniform geologic media by conditional moments exact nonlocal formalism effective conductivities and weak approximation we consider the effect of measuring randomly varying local hydraulic conductivities k x on one s ability to predict steady state flow within a bounded domain driven by random source and boundary functions more precisely we consider the prediction of local hydraulic head h x and darcy flux q x by means of their unbiased ensemble moments h x and q x conditioned on measurements of k x these predictors satisfy a deterministic flow equation in which q x x h x r x where x is a relatively smooth unbiased estimate of k x and r x is a residual flux we derive a compact integral expression for r x which is rigorously valid for a broad class of k x fields including fractals it demonstrates that q x is nonlocal and non darcian so that an effective hydraulic conductivity does not generally exist we show analytically that under uniform mean flow the effective conductivity may be a scalar a symmetric or a nonsymmetric tensor or a set of directional scalars which do not form a tensor we demonstrate numerically that in two dimensional mean radial flow it may increase from the harmonic mean of k x near interior and boundary sources to the geometric mean far from such sources for cases where r x can neither be expressed nor approximated by a local expression we propose a weak integral approximation closure which appears to work well in media with pronounced heterogeneity and improves as the quantity and quality of k x measurements increase the nonlocal deterministic flow equation can be solved numerically by standard methods our theory shows clearly how the scale of grid discretization should relate to the scale quantity and quality of available data after providing explicit approximations for the second moments of head and flux prediction errors we conclude by discussing practical methods to compute x from noisy measurements of k x and to calculate required second moments of the associated estimation errors when k x is lognormal

the subdifferential of measurable composite max integrands and smoothing approximation the subdifferential calculus for the expectation of nonsmooth random integrands involves many fundamental and challenging problems in stochastic optimization it is known that for clarke regular integrands the clarke subdifferential of the expectation equals the expectation of their clarke subdifferential in particular this holds for convex integrands however little is known about the calculation of clarke subgradients for the expectation of non regular integrands the focus of this contribution is to approximate clarke subgradients for the expectation of random integrands by smoothing methods applied to the integrand a framework for how to proceed along this path is developed and then applied to a class of measurable composite max integrands this class contains non regular integrands from stochastic complementarity problems as well as stochastic optimization problems arising in statistical learning

displacement structure theory and applications in this survey paper we describe how strands of work that are important in two different fields matrix theory and complex function theory have come together in some work on fast computational algorithms for matrices with what we call displacement structure in particular a fast triangularization procedure can be developed for such matrices generalizing in a striking way an algorithm presented by schur 1917 j reine angew math 147 1917 pp 205 232 in a paper on checking when a power series is bounded in the unit disc this factorization algorithm has a surprisingly wide range of significant applications going far beyond numerical linear algebra we mention among others inverse scattering analytic and unconstrained rational interpolation theory digital filter design adaptive filtering and state space least squares estimation

on normal and skew hermitian splitting iteration methods for large sparse continuous sylvester equations this paper is concerned with some generalizations of the hermitian and skew hermitian splitting hss iteration for solving continuous sylvester equations the main contents we will introduce are the normal and skew hermitian splitting nss iteration methods for the continuous sylvester equations it is shown that the new schemes can outperform the standard hss method in some situations theoretical analysis shows that the nss methods converge unconditionally to the exact solution of the continuous sylvester equations moreover we derive the upper bound of the contraction factor of the nss iterations numerical experiments further show the effectiveness of our new methods

erratum transient flow in bounded randomly heterogeneous domains 1 exact conditional moment equations and recursive approximations water resources research 1998 34 1 1 12

cyclic displacements and decompositions of inverse matrices for cupl toeplitz matrices abstract in this paper we deal mainly with a class of cupl toeplitz matrices 32 without toeplitz structure which are close to the toeplitz matrices in the sense that their 1 1 cyclic displacements coincide with cyclic displacement of some toeplitz matrices by constructing the corresponding displacement of the matrices we derive the formulas on representation of the inverses of the cupl toeplitz matrices in the form of sums of products of factor row first plus last right circulants furthermore the inverses of the cupl hankel matrices can be also obtained which benefits from the relation between the cupl toeplitz matrices and the cupl hankel matrices

least squares policy iteration we propose a new approach to reinforcement learning for control problems which combines value function approximation with linear architectures and approximate policy iteration this new approach is motivated by the least squares temporal difference learning algorithm lstd for prediction problems which is known for its efficient use of sample experiences compared to pure temporal difference algorithms heretofore lstd has not had a straightforward application to control problems mainly because lstd learns the state value function of a fixed policy which cannot be used for action selection and control without a model of the underlying process our new algorithm least squares policy iteration lspi learns the state action value function which allows for action selection without a model and for incremental policy improvement within a policy iteration framework lspi is a model free off policy method which can use efficiently and reuse in each iteration sample experiences collected in any manner by separating the sample collection method the choice of the linear approximation architecture and the solution method lspi allows for focused attention on the distinct elements that contribute to practical reinforcement learning lspi is tested on the simple task of balancing an inverted pendulum and the harder task of balancing and riding a bicycle to a target location in both cases lspi learns to control the pendulum or the bicycle by merely observing a relatively small number of trials where actions are selected randomly lspi is also compared against q learning both with and without experience replay using the same value function architecture while lspi achieves good performance fairly consistently on the difficult bicycle task q learning variants were rarely able to balance for more than a small fraction of the time needed to reach the target location

acceleration of stochastic approximation by averaging a new recursive algorithm of stochastic approximation type with the averaging of trajectories is investigated convergence with probability one is proved for a variety of classical optimization and identification problems it is also demonstrated for these problems that the proposed algorithm achieves the highest possible rate of convergence

stochastic root finding and efficient estimation of convex risk measures reliable risk measurement is a key problem for financial institutions and regulatory authorities the current industry standard value at risk has several deficiencies improved risk measures have been suggested and analyzed in the recent literature but their computational implementation has largely been neglected so far we propose and investigate stochastic approximation algorithms for the convex risk measure utility based shortfall risk our approach combines stochastic root finding schemes with importance sampling we prove that the resulting shortfall risk estimators are consistent and asymptotically normal and provide formulas for confidence intervals the performance of the proposed algorithms is tested numerically we finally apply our techniques to the normal copula model which is also known as the industry model creditmetrics this provides guidance for future implementations in practice

a new approach to the maximum flow problem all previously known efficient maximum flow algorithms work by finding augmenting paths either one path at a time as in the original ford and fulkerson algorithm or all shortest length augmenting paths at once using the layered network approach of dinic an alternative method based on the preflow concept of karzanov is introduced a preflow is like a flow except that the total amount flowing into a vertex is allowed to exceed the total amount flowing out the method maintains a preflow in the original network and pushes local flow excess toward the sink along what are estimated to be shortest paths the algorithm and its analysis are simple and intuitive yet the algorithm runs as fast as any other known method on dense graphs achieving an o n 3 time bound on an n vertex graph by incorporating the dynamic tree data structure of sleator and tarjan we obtain a version of the algorithm running in o nm log n 2 m time on an n vertex m edge graph this is as fast as any known method for any graph density and faster on graphs of moderate density the algorithm also admits efficient distributed and parallel implementations a parallel implementation running in o n 2 log n time using n processors and o m space is obtained this time bound matches that of the shiloach vishkin algorithm which also uses n processors but requires o n 2 space

manifold identification in dual averaging for regularized stochastic online learning iterative methods that calculate their steps from approximate subgradient directions have proved to be useful for stochastic learning problems over large and streaming data sets when the objective consists of a loss function plus a nonsmooth regularization term the solution often lies on a low dimensional manifold of parameter space along which the regularizer is smooth when an l1 regularizer is used to induce sparsity in the solution for example this manifold is defined by the set of nonzero components of the parameter vector this paper shows that a regularized dual averaging algorithm can identify this manifold with high probability before reaching the solution this observation motivates an algorithmic strategy in which once an iterate is suspected of lying on an optimal or near optimal manifold we switch to a local phase that searches in this manifold thus converging rapidly to a near optimal point computational results are presented to verify the identification property and to illustrate the effectiveness of this approach

optimal affine invariant smooth minimization algorithms we formulate an affine invariant implementation of the accelerated first order algorithm in y nesterov dokl math 27 1983 pp 372 376 its complexity bound is proportional to an affine invariant regularity constant defined with respect to the minkowski gauge of the feasible set we extend these results to more general problems optimizing holder smooth functions using p uniformly convex prox terms and derive an algorithm whose complexity better fits the geometry of the feasible set and adapts to both the best holder smoothness parameter and the best gradient lipschitz constant finally we detail matching complexity lower bounds when the feasible set is an ell p ball in this setting our upper bounds on iteration complexity for the algorithm in y nesterov dokl math 27 1983 pp 372 376 are thus optimal in terms of target precision smoothness and problem dimension

proximity without consensus in online multiagent optimization we consider stochastic optimization problems in multiagent settings where a network of agents aims to learn parameters that are optimal in terms of a global convex objective while giving preference to locally observed streaming information to do so we depart from the canonical decentralized optimization framework where agreement constraints are enforced and instead formulate a problem where each agent minimizes a global convex objective while enforcing network proximity constraints this formulation includes online consensus optimization as a special case but allows for the more general hypothesis that there is data heterogeneity across the network to solve this problem we propose using a stochastic saddle point algorithm inspired by arrow and hurwicz this method yields a decentralized algorithm for processing observations sequentially received at each node of the network using lagrange multipliers to penalize the discrepancy between them only neighboring nodes exchange model information we establish that under a constant step size regime the time average suboptimality and constraint violation are contained in a neighborhood whose radius vanishes with increasing number of iterations as a consequence we prove that the time average primal vectors converge to the optimal objective while satisfying the network proximity constraints we apply this method to the problem of sequentially estimating a correlated random field in a sensor network as well as an online source localization problem both of which demonstrate the empirical validity of the aforementioned convergence results

incremental majorization minimization optimization with application to large scale machine learning majorization minimization algorithms consist of successively minimizing a sequence of upper bounds of the objective function these upper bounds are tight at the current estimate and each iteration monotonically drives the objective function downhill such a simple principle is widely applicable and has been very popular in various scientific fields especially in signal processing and statistics we propose an incremental majorization minimization scheme for minimizing a large sum of continuous functions a problem of utmost importance in machine learning we present convergence guarantees for nonconvex and convex optimization when the upper bounds approximate the objective up to a smooth error we call such upper bounds first order surrogate functions more precisely we study asymptotic stationary point guarantees for nonconvex problems and for convex ones we provide convergence rates for the expected objective function value we apply our scheme to composite optimization and obtain a new incremental proximal gradient algorithm with linear convergence rate for strongly convex functions our experiments show that our method is competitive with the state of the art for solving machine learning problems such as logistic regression when the number of training samples is large enough and we demonstrate its usefulness for sparse estimation with nonconvex penalties

optimal dynamic treatment regimes summary a dynamic treatment regime is a list of decision rules one per time interval for how the level of treatment will be tailored through time to an individual s changing status the goal of this paper is to use experimental or observational data to estimate decision regimes that result in a maximal mean response to explicate our objective and to state the assumptions we use the potential outcomes model the method proposed makes smooth parametric assumptions only on quantities that are directly relevant to the goal of estimating the optimal rules we illustrate the methodology proposed via a small simulation

parallel nonlinear aor method and its convergence abstract we set up a class of parallel nonlinear aor method in the sense of matrix multi splitting for solving the large scale system of nonlinear equations ax x b with a l rn nonsingular b rn and rn rn being continuously diagonal the global as well as the monotone convergence of this method is proved