mean field game for linear quadratic stochastic recursive systems abstract this paper focuses on linear quadratic lq for short mean field games described by forward backward stochastic differential equations fbsdes for short in which the individual control region is postulated to be convex the decentralized strategies and consistency condition are represented by a kind of coupled mean field fbsdes with projection operators the well posedness of consistency condition system is obtained using the monotonicity condition method the nash equilibrium property is discussed as well

robust mean field linear quadratic gaussian games with unknown l 2 disturbance this paper considers a class of mean field linear quadratic gaussian games with model uncertainty the drift term in the dynamics of the agents contains a common unknown function we take a robust

nonsymmetric values of nonatomic and mixed games this paper presents a new unifying approach to the study of nonsymmetric or quasi values of nonatomic and mixed games a family ofpath values is defined using an appropriate generalization of mertens diagonal formula a path value possesses the following intuitive description consider a function path attaching to each player a distribution function on 0 1 we think of players as arriving randomly and independently to a meeting when the arrival time of a player is distributed according to each player s payoff is defined as his marginal contribution to the coalition of players that have arrived earlier under certain conditions on a path different subspaces of mixed games pna pm bv fl are shown to be in the domain of the path value the family of path values turns out to be very wide we show that onpna pm and their subspaces the path values are essentially the basic construction blocks extreme points of quasi values

linear quadratic differential games saddle point and riccati differential equation zhang siam j control optim 43 2005 pp 2157 2165 recently established the equivalence between the finiteness of the open loop value of a two player zero sum linear quadratic lq game and the finiteness of its open loop lower and upper values in this paper we complete and sharpen the results of zhang for the finiteness of the lower value of the game by providing a set of necessary and sufficient conditions that emphasizes the feasibility condition 0 0 is a solution of the open loop lower value of the game for the zero initial state then we show that under the assumption of an open loop saddle point in the time horizon 0 t for all initial states there is an open loop saddle point in the time horizon s t for all initial times s 0 le s t and all initial states at time s from this we get an optimality principle and adapt the invariant embedding approach to construct the decoupling symmetrical matrix function p s and show that it is an h 1 0 t solution of the matrix riccati differential equation thence an open loop saddle point in 0 t yields closed loop optimal strategies for both players furthermore a necessary and sufficient set of conditions for the existence of an open loop saddle point in 0 t for all initial states is the convexity concavity of the utility function and the existence of an h 1 0 t symmetrical solution to the matrix riccati differential equation as an illustration of the cases where the open loop lower upper value of the game is infty infty we work out two informative examples of solutions to the riccati differential equation with and without blow up time

unsupervised feature selection based on maximum information and minimum redundancy for hyperspectral images unsupervised feature selection plays an important role in hyperspectral image processing it is a very challenge issue to select an effective feature subset with the unavailability of class labels to select the features maximally preserving the information of original features a maximum joint mutual information mjmi criterion is defined since the high order distribution involved in mjmi is hard to calculate a maximum information and minimum redundancy mimr criterion is derived as the low order approximation of mjmi from information theory many classical unsupervised feature selection criteria can also be considered as the low order approximations of mjmi compared with them mimr requires more relaxed approximation condition moreover a new clonal selection algorithm csa in artificial immune system is devised to optimize the selected features with the guidance of mimr experimental results on several hyperspectral datasets demonstrate that the proposed method obtains better feature subsets compared with classical unsupervised feature selection methods a new mimr criterion is proposed for unsupervised feature selection mimr can select more informative and distinctive features many classical criteria and mimr can be unified into the same framework the theoretical advantage of mimr over many classical criteria is given feature selection problem is solved by combinatorial optimization with a new csa

edgeworth approximation of multivariate differential entropy we develop the general multivariate case of the edgeworth approximation of differential entropy and show that it can be more accurate than the nearest neighbor method in the multivariate case and that it scales better with sample size furthermore we introduce mutual information estimation as an application

feature evaluation and selection with cooperative game theory recent years various information theoretic based measurements have been proposed to remove redundant features from high dimensional data set as many as possible however most traditional information theoretic based selectors will ignore some features which have strong discriminatory power as a group but are weak as individuals to cope with this problem this paper introduces a cooperative game theory based framework to evaluate the power of each feature the power can be served as a metric of the importance of each feature according to the intricate and intrinsic interrelation among features then a general filter feature selection scheme is presented based on the introduced framework to handle the feature selection problem to verify the effectiveness of our method experimental comparisons with several other existing feature selection methods on fifteen uci data sets are carried out using four typical classifiers the results show that the proposed algorithm achieves better results than other methods in most cases

discriminant independent component analysis a conventional linear model based on negentropy maximization extracts statistically independent latent variables which may not be optimal to give a discriminant model with good classification performance in this paper a single stage linear semisupervised extraction of discriminative independent features is proposed discriminant independent component analysis dica presents a framework of linearly projecting multivariate data to a lower dimension where the features are maximally discriminant with minimal redundancy the optimization problem is formulated as the maximization of linear summation of negentropy and weighted functional measure of classification motivated by independence among extracted features fisher linear discriminant is used as the functional measure of classification experimental results show improved classification performance when dica features are used for recognition tasks in comparison to unsupervised principal component analysis and ica and supervised feature extraction techniques like linear discriminant analysis lda conditional ica and those based on information theoretic learning approaches dica features also give reduced data reconstruction error in comparison to lda and ica method based on negentropy maximization

non autonomous second order hamiltonian systems abstract we study the existence of periodic solutions for a second order non autonomous dynamical system containing variable kinetic energy terms our assumptions balance the interaction between the kinetic energy and the potential energy with neither one dominating the other we study sublinear problems and the existence of non constant solutions

inferring entropy from structure the thermodynamic definition of entropy can be extended to nonequilibrium systems based on its relation to information to apply this definition in practice requires access to the physical system s microstates which may be prohibitively inefficient to sample or difficult to obtain experimentally it is beneficial therefore to relate the entropy to other integrated properties which are accessible out of equilibrium we focus on the structure factor which describes the spatial correlations of density fluctuations and can be directly measured by scattering the information gained by a given structure factor regarding an otherwise unknown system provides an upper bound for the system s entropy we find that the maximum entropy model corresponds to an equilibrium system with an effective pair interaction approximate closed form relations for the effective pair potential and the resulting entropy in terms of the structure factor are obtained as examples the relations are used to estimate the entropy of an exactly solvable model and two simulated systems out of equilibrium the focus is on low dimensional examples where our method as well as a recently proposed compression based one can be tested against a rigorous direct sampling technique the entropy inferred from the structure factor is found to be consistent with the other methods superior for larger system sizes and accurate in identifying global transitions our approach allows for extensions of the theory to more complex systems and to higher order correlations

value iteration algorithm for mean field games abstract in the literature existence of mean field equilibria has been established for discrete time mean field games under both the discounted cost and the average cost optimality criteria in this paper we provide a value iteration algorithm to compute stationary mean field equilibrium for both the discounted cost and the average cost criteria whose existence proved previously we establish that the value iteration algorithm converges to the fixed point of a mean field equilibrium operator then using this fixed point we construct a stationary mean field equilibrium in our value iteration algorithm we use q functions instead of value functions

existence of periodic solutions to second order hamiltonian systems with potential indefinite in sign we have presented some new results for periodic solutions to a class of hamiltonian systems with potential indefinite in sign these results improve and correct some known results in the literature

distributed computation of nash equilibria in linear quadratic stochastic differential games in this paper a class of n person stochastic linear quadratic differential games under multiple probabilistic modeling is studied with each player acquiring a noisy measurement of the initial state conditions for the existence and uniqueness of the nash equilibrium is obtained and a method is provided for an iterative distributed computation of the solution the distributed algorithm involves learning in the policy space and it does not require that each player knows the others perceptions of the probabilistic model underlying the decision process for the finite horizon problem such an iteration converges whenever the length of the time horizon is sufficiently small and the limit in this case is an affine policy for all players if the underlying distributions are jointly gaussian when the horizon is infinite and a discount factor is used in the cost functionals the iteration converges under conditions depending on the magnitude of the discount factor the limiting policies again being affine in

mean field games for large population multiagent systems with markov jump parameters in this paper distributed games for large population multiagent systems with random time varying parameters are investigated where the agents are coupled via their individual costs and the structure parameters are a family of independent markov chains with identical generators the cost function of each agent is a long run average tracking type functional with an unknown mean field coupling nonlinear term as reference signal to reduce the computational complexity the mean field approach is applied to construct distributed strategies the population statistics effect pse is used to approximate the average effect of all the agents and the distributed strategies are given through solving a markov jump tracking problem here the pse is a deterministic quantity and can be obtained by solving the stackelberg equilibrium of an auxiliary two player game it is shown that the closed loop system is uniformly stable and the distributed strategies are asymptotically optimal in the sense of nash equilibrium

linear quadratic risk sensitive and robust mean field games this paper considers two classes of large population stochastic differential games connected to optimal and robust decentralized control of large scale multiagent systems the first problem p1 is one where each agent minimizes an exponentiated cost function capturing risk sensitive behavior whereas in the second problem p2 each agent minimizes a worst case risk neutral cost function where the worst case stems from the presence of an adversary entering each agent s dynamics characterized by a stochastic differential equation in both problems the individual agents are coupled through the mean field term included in each agent s cost function which captures the average or mass behavior of the agents we solve both p1 and p2 via mean field game theory specifically we first solve a generic risk sensitive optimal control problem and a generic stochastic zero sum differential game where the corresponding optimal controllers are applied by each agent to construct the mean field systems of p1 and p2 we then characterize an approximated mass behavior effect on an individual agent via a fixed point analysis of the mean field system for each problem p1 and p2 we show that the approximated mass behavior is in fact the best estimate of the actual mass behavior in various senses as the population size n goes to infinity moreover we show that for finite n there exist epsilon nash equilibria for both p1 and p2 where the corresponding individual nash strategies are decentralized in terms of local state information and the approximated mass behavior we also show that epsilon can be taken to be arbitrarily small when n is sufficiently large we show that the epsilon nash equilibria of p1 and p2 are partially equivalent in the sense that the individual nash strategies share identical control laws but the approximated mass behaviors for p1 and p2 are different since in p2 the mass behavior is also affected by the associated worst case disturbance finally we prove that the nash equilibria for p1 and p2 both feature robustness and as the parameter characterizing this robustness becomes infinite the two nash equilibria become identical and equivalent to that of the risk neutral case as in the one agent risk sensitive and robust control theory

mean field games with common noise a theory of existence and uniqueness is developed for general stochastic differential mean field games with common noise the concepts of strong and weak solutions are introduced in analogy with the theory of stochastic differential equations and existence of weak solutions for mean field games is shown to hold under very general assumptions examples and counter examples are provided to enlighten the underpinnings of the existence theory finally an analog of the famous result of yamada and watanabe is derived and it is used to prove existence and uniqueness of a strong solution under additional assumptions

linear quadratic mean field games with a major player the multi scale approach this paper considers linear quadratic lq mean field games with a major player and analyzes an asymptotic solvability problem it starts with a large scale system of coupled dynamic programming equations and applies a re scaling technique introduced in huang and zhou 2018 and huang and zhou 2020 to derive a set of riccati equations in lower dimensions the solvability of which determines the necessary and sufficient condition for asymptotic solvability we next derive the mean field limit of the strategies and the value functions finally we show that the two decentralized strategies can be interpreted as the best responses of a major player and a representative minor player embedded in an infinite population which have the property of consistent mean field approximations

a linear discriminant analysis method based on mutual information maximization we present a new linear discriminant analysis method based on information theory where the mutual information between linearly transformed input data and the class labels is maximized first we introduce a kernel based estimate of mutual information with a variable kernel size furthermore we devise a learning algorithm that maximizes the mutual information w r t the linear transformation two experiments are conducted the first one uses a toy problem to visualize and compare the transformation vectors in the original input space the second one evaluates the performance of the method for classification by employing cross validation tests on four datasets from the uci repository various classifiers are investigated our results show that this method can significantly boost class separability over conventional methods especially for nonlinear classification

existence of a solution to an equation arising from the theory of mean field games abstract we construct a small time strong solution to a nonlocal hamilton jacobi equation 1 1 introduced in 48 the so called master equation originating from the theory of mean field games we discover a link between metric viscosity solutions to local hamilton jacobi equations studied in 2 19 20 and solutions to 1 1 as a consequence we recover the existence of solutions to the first order mean field games equations 1 2 first proved in 48 and make a more rigorous connection between the master equation 1 1 and the mean field games equations 1 2

nonlinear filtering theory for mckean vlasov type stochastic differential equations we consider estimation theory for partially observed stochastic dynamical systems whose state equations are given by mckean vlasov type stochastic differential equations and hence contain a measure term corresponding to the distribution of the solution of the state process nonlinear filtering equations are derived in this framework based upon the classification that the measure term is either stochastic or deterministic and that either the state or the measure term is estimated when the measure term is deterministic the standard theory holds further when the measure is random the induced functions in the dynamics of the state become random and a similar recursion for the optimal filter is obtained the joint estimation of state and the measure term is next considered the extended state in this setup is shown to be a banach space valued stochastic process with random functions in its state dynamics and a nonlinear filtering equation for this setup is provided the first step of such an analysis requ

large population lqg games involving a major player the nash certainty equivalence principle we consider linear quadratic gaussian lqg games with a major player and a large number of minor players the major player has a significant influence on others the minor players individually have negligible impact but they collectively contribute mean field coupling terms in the individual dynamics and costs to overcome the dimensionality difficulty and obtain decentralized strategies the so called nash certainty equivalence methodology is applied the control synthesis is preceded by a state space augmentation via a set of aggregate quantities giving the mean field approximation subsequently within the population limit the lqg game is decomposed into a family of limiting two player games as each is locally seen by a representative minor player next when solving these limiting two player games we impose certain interaction consistency conditions such that the aggregate quantities initially assumed coincide with the ones replicated by the closed loop of a large number of minor players this procedure leads to a set of decentralized strategies for the original lqg game which is an varepsilon nash equilibrium

mean field linear quadratic control uniform stabilization and social optimality abstract this paper is concerned with uniform stabilization and social optimality for general mean field linear quadratic control systems where subsystems are coupled via individual dynamics and costs and the state weight is not assumed with the definiteness condition for the finite horizon problem we first obtain a set of forward backward stochastic differential equations fbsdes from variational analysis and construct a feedback type control by decoupling the fbsdes for the infinite horizon problem by using solutions to two riccati equations we design a set of decentralized control laws which is further proved to be asymptotically social optimal some equivalent conditions are given for uniform stabilization of the systems in different cases respectively finally the proposed decentralized controls are compared to the asymptotic optimal strategies in previous works

mean field linear quadratic gaussian lqg games for stochastic integral systems in this technical note we formulate and investigate a class of mean field linear quadratic gaussian lqg games for stochastic integral systems unlike other literature on mean field games where the individual states follow the controlled stochastic differential equations sdes the individual states in our large population system are characterized by a class of stochastic volterra type integral equations we obtain the nash certainty equivalence nce equation and hence derive the set of associated decentralized strategies the epsilon nash equilibrium properties are also verified due to the intrinsic integral structure the techniques and estimates applied here are significantly different from those existing results in mean field lqg games for stochastic differential systems for example some fredholm equation in the mean field setup is introduced for the first time as for applications two types of stochastic delayed systems are formulated as the special cases of our stochastic integral system and relevant mean field lqg games are discussed

feature selection using localized generalization error for supervised classification problems using rbfnn a pattern classification problem usually involves using high dimensional features that make the classifier very complex and difficult to train with no feature reduction both training accuracy and generalization capability will suffer this paper proposes a novel hybrid filter wrapper type feature subset selection methodology using a localized generalization error model the localized generalization error model for a radial basis function neural network bounds from above the generalization error for unseen samples located within a neighborhood of the training samples iteratively the feature making the smallest contribution to the generalization error bound is removed moreover the novel feature selection method is independent of the sample size and is computationally fast the experimental results show that the proposed method consistently removes large percentages of features with statistically insignificant loss of testing accuracy for unseen samples in the experiments for two of the datasets the classifiers built using feature subsets with 90 of features removed by our proposed approach yield average testing accuracies higher than those trained using the full set of features finally we corroborate the efficacy of the model by using it to predict corporate bankruptcies in the us

estimating optimal feature subsets using efficient estimation of high dimensional mutual information a novel feature selection method using the concept of mutual information mi is proposed in this paper in all mi based feature selection methods effective and efficient estimation of high dimensional mi is crucial in this paper a pruned parzen window estimator and the quadratic mutual information qmi are combined to address this problem the results show that the proposed approach can estimate the mi in an effective and efficient way with this contribution a novel feature selection method is developed to identify the salient features one by one also the appropriate feature subsets for classification can be reliably estimated the proposed methodology is thoroughly tested in four different classification applications in which the number of features ranged from less than 10 to over 15000 the presented results are very promising and corroborate the contribution of the proposed feature selection methodology

periodic second order superlinear hamiltonian systems abstract we consider periodic solutions for superlinear second order non autonomous dynamical systems including both kinetic and potential terms we study the existence of nontrivial and ground state solutions

homoclinic orbits for second order hamiltonian systems possessing superquadratic potentials received by the editors november 27 1990 1991 mathematics subject classification primary 34c99 58e99 58f99 the first author was supported by ministero p i gruppo 40 calcolo delle variazioni the second author s research was sponsored in part by the u s army research office under contract daal03 87 k 0043 the national science foundation under grant mcs 8110556 and the office of naval research under grant n00014 88 ko134 any reproduction for the purpose of the united states govemment is permitted

invariant optimal feature selection a distance discriminant and feature ranking based solution the goal of feature selection is to find the optimal subset consisting of m features chosen from the total n features one critical problem for many feature selection methods is that an exhaustive search strategy has to be applied to seek the best subset among all the possible nm feature subsets which usually results in a considerably high computational complexity the alternative suboptimal feature selection methods provide more practical solutions in terms of computational complexity but they cannot promise that the finally selected feature subset is globally optimal we propose a new feature selection algorithm based on a distance discriminant fsdd which not only solves the problem of the high computational costs but also overcomes the drawbacks of the suboptimal methods the proposed method is able to find the optimal feature subset without exhaustive search or branch and bound algorithm the most difficult problem for optimal feature selection the search problem is converted into a feature ranking problem following rigorous theoretical proof such that the computational complexity can be greatly reduced the proposed method is invariant to the linear transformation of data when a diagonal transformation matrix is applied fsdd was compared with relieff and mrmrmid based on mutual information on 8 data sets the experiment results show that fsdd outperforms the other two methods and is highly efficient

the reduced parzen classifier the parzen density estimate is known to be an effective tool for estimating the bayes error given a set of training samples from the class distributions an algorithm is developed to select a given number of representative samples whose parzen density estimate closely matches that of the entire sample set using this reduced representative set a piecewise quadratic classifier which provides nearly optimal performance is designed

feature subset selection using separability index matrix effective feature subset selection fss is an important step when designing engineering systems that classify complex data in real time the electromyographic emg signal based walking assistance system is a typical system that requires an efficient computational architecture for classification the performance of such a system depends largely on a criterion function that assesses the quality of selected feature subsets however many well known conventional criterion functions use less relevant features for classification or they have a high computational cost here we propose a new criterion function that provides more effective fss the proposed criterion function known as a separability index matrix sim provides features pertinent to the classification task and a very low computational cost this new function produces to a simple feature selection algorithm when combined with the forward search paradigm we performed extensive experimental comparisons in terms of classification accuracy and computational costs to confirm that the proposed algorithm outperformed other filter type feature selection methods that are based on various distance measures including inter intra euclidean mahalanobis and bhattacharyya distances we then applied the proposed method to a gait phase recognition problem in our emg signal based walking assistance system we demonstrated that the proposed method performed competitively when compared with other wrapper type feature selection methods in terms of class separability and recognition rate

relative performance of mutual information estimation methods for quantifying the dependence among short and noisy data commonly used dependence measures such as linear correlation cross correlogram or kendall s tau cannot capture the complete dependence structure in data unless the structure is restricted to linear periodic or monotonic mutual information mi has been frequently utilized for capturing the complete dependence structure including nonlinear dependence recently several methods have been proposed for the mi estimation such as kernel density estimators kde k nearest neighbors knn edgeworth approximation of differential entropy and adaptive partitioning of the xy plane however outstanding gaps in the current literature have precluded the ability to effectively automate these methods which in turn have caused limited adoptions by the application communities this study attempts to address a key gap in the literature specifically the evaluation of the above methods to choose the best method particularly in terms of their robustness for short and noisy data based on comparisons with the theoretical mi estimates which can be computed analytically as well with linear correlation and kendall s tau here we consider smaller data sizes such as 50 100 and 1 000 where this study considers 50 and 100 data points as very short and 1 000 as short we consider a broader class of functions specifically linear quadratic periodicmore and chaotic contaminated with artificial noise with varying noise to signal ratios the case studies presented here are motivated by domain consideration in the earth sciences where the data are short and noisy our results indicate kde as the best choice for very short data at relatively high noise to signal levels whereas the performance of knn is the best for short data at relatively low noise levels as well as for short data consistently across noise levels in addition the optimal smoothing parameter of a gaussian kernel appears to be the best choice for kde while three nearest neighbors appear optimal for knn thus in situations where the approximate data sizes are known in advance and exploratory data analysis and or domain knowledge can be used to provide a priori insights on the noise to signal ratios the results in the paper point to a way forward for automating the process of mi estimation less

robust mean field games for coupled markov jump linear systems abstractwe consider robust stochastic large population games for coupled markov jump linear systems mjlss the n agents individual mjlss are governed by different infinitesimal generators and are affected not only by the control input but also by an individual disturbance or adversarial input the mean field term representing the average behaviour of n agents is included in the individual worst case cost function to capture coupling effects among agents to circumvent the computational complexity and analyse the worst case effect of the disturbance we use robust mean field game theory to design low complexity robust decentralised controllers and to characterise the associated worst case disturbance we show that with the individual robust decentralised controller and the corresponding worst case disturbance which constitute a saddle point solution to a generic stochastic differential game for mjlss the actual mean field behaviour can be approximated by a deterministic function which is a fixed

input feature selection by mutual information based on parzen window mutual information is a good indicator of relevance between variables and have been used as a measure in several feature selection algorithms however calculating the mutual information is difficult and the performance of a feature selection algorithm depends on the accuracy of the mutual information in this paper we propose a new method of calculating mutual information between input and class variables based on the parzen window and we apply this to a feature selection algorithm for classification problems

convex analysis for lqg systems with applications to major minor lqg mean field game systems abstract we develop a convex analysis approach for solving lqg optimal control problems and apply it to major minor mm lqg mean field game mfg systems the approach retrieves the best response strategies for the major agent and all minor agents that attain an nash equilibrium an important and distinctive advantage to this approach is that unlike the classical approach in the literature we are able to avoid imposing assumptions on the evolution of the mean field in particular this provides a tool for dealing with complex and non standard systems

supervised feature selection by clustering using conditional mutual information based distances in this paper a supervised feature selection approach is presented which is based on metric applied on continuous and discrete data representations this method builds a dissimilarity space using information theoretic measures in particular conditional mutual information between features with respect to a relevant variable that represents the class labels applying a hierarchical clustering the algorithm searches for a compression of the information contained in the original set of features the proposed technique is compared with other state of art methods also based on information measures eventually several experiments are presented to show the effectiveness of the features selected from the point of view of classification accuracy

ϵ nash mean field games for general linear quadratic systems with applications this paper is concerned with general mean field mf linear quadratic lq games of stochastic large population system where the individual diffusion coefficient can depend on both the state and the control of the agent moreover the control weight in the cost functional could be indefinite the asymptotic suboptimality property namely nash equilibrium of the decentralized strategies for the lq games is derived through the consistency condition the impact of the population s collective behaviors and the consistency of the mean field estimation are illustrated by the numerical results a pricing problem is also studied for which the decentralized suboptimal price is obtained

correntropy based feature selection using binary projection most feature selection algorithms based on information theoretic learning itl adopt ranking process or greedy search as their searching strategies the former selects features individually so that it ignores feature interaction and dependencies the latter heavily relies on the search paths as only one path will be explored with no possible back track in addition both strategies typically lead to heuristic algorithms to cope with these problems this article proposes a novel feature selection framework based on correntropy in itl namely correntropy based feature selection using binary projection bpfs our framework selects features by projecting the original high dimensional data to a low dimensional space through a special binary projection matrix the formulated objective function aims at maximizing the correntropy between selected features and class labels and this function can be efficiently optimized via standard mathematical tools we apply the half quadratic method to optimize the objective function in an iterative manner where each iteration reduces to an assignment subproblem which can be highly efficiently solved with some off the shelf toolboxes comparative experiments on six real world datasets indicate that our framework is effective and efficient

linear quadratic mean field games asymptotic solvability and relation to the fixed point approach mean field game theory has been developed largely following two routes one of them called the direct approach starts by solving a large scale game and next derives a set of limiting equations as the population size tends to infinity the second route is to apply mean field approximations and formalize a fixed point problem by analyzing the best response of a representative player this paper addresses the connection and difference of the two approaches in a linear quadratic lq setting we first introduce an asymptotic solvability notion for the direct approach which means for all sufficiently large population sizes the corresponding game has a set of feedback nash strategies in addition to a mild regularity requirement we provide a necessary and sufficient condition for asymptotic solvability and show that in this case the solution converges to a mean field limit this is accomplished by developing a re scaling method to derive a low dimensional ordinary differential equation ode system where a non symmetric riccati ode has a central role we next compare with the fixed point approach which determines a two point boundary value tpbv problem and show that asymptotic solvability implies feasibility of the fixed point approach but the converse is not true we further address non uniqueness in the fixed point approach and examine the long time behavior of the non symmetric riccati ode in the asymptotic solvability problem

on the feature selection criterion based on an approximation of multidimensional mutual information we derive the feature selection criterion presented in 1 and 2 from the multidimensional mutual information between features and the class our derivation 1 specifies and validates the lower order dependency assumptions of the criterion and 2 mathematically justifies the utility of the criterion by relating it to bayes classification error

analytic solution for a class of linear quadratic open loop nash games a method for solving the asymmetric coupled riccati type matrix differential equations for open loop nash strategy in linear quadratic games is presented the class of games studied here is one in which the state weighting matrices in player s cost functionals are proportional to each other by writing in a special order the necessary conditions for open loop nash strategy a matrix with specific properties is derived these properties are then exploited to solve the two point boundary value problem some special cases are discussed and a simple example is given to illustrate the solution procedure

social optima in mean field lqg control centralized and decentralized strategies we study a class of linear quadratic gaussian lqg control problems with n decision makers where the basic objective is to minimize a social cost as the sum of n individual costs containing mean field coupling the exact socially optimal solution determining a particular pareto optimum requires centralized information for each agent and has high implementational complexity as an alternative we subsequently exploit a mean field structure in the centralized optimal control problem to develop decentralized cooperative optimization so that each agent only uses its own state and a function which may be computed offline the resulting set of strategies asymptotically achieves the social optimum as n a key feature in this scheme is to let each agent optimize a new cost as the sum of its own cost and another component capturing its social impact on all other agents we also discuss the relationship between the decentralized cooperative solution and the so called nash certainty equivalence based solution presented in previous work on mean field lqg games

image covariance based subspace method for face recognition this paper proposes a new subspace method that is based on image covariance obtained from windowed features of images a windowed input feature consists of a number of pixels and the dimension of input space is determined by the number of windowed features each element of an image covariance matrix can be obtained from the inner product of two windowed features the 2d pca and 2d lda methods are then obtained from principal component analysis and linear discriminant analysis respectively using the image covariance matrix in the case of 2d lda there is no need for pca preprocessing and the dimension of subspace can be greater than the number of classes because the within class and between class image covariance matrices have full ranks comparative experiments are performed using the feret cmu and orl databases of facial images the experimental results show that the proposed 2d lda provides the best recognition rate among several subspace methods in all of the tests

a leader follower stochastic linear quadratic differential game a leader follower stochastic differential game is considered with the state equation being a linear ito type stochastic differential equation and the cost functionals being quadratic we allow that the coefficients of the system and those of the cost functionals are random the controls enter the diffusion of the state equation and the weight matrices for the controls in the cost functionals are not necessarily positive definite the so called open loop strategies are considered only thus the follower first solves a stochastic linear quadratic lq optimal control problem with the aid of a stochastic riccati equation then the leader turns to solve a stochastic lq problem for a forward backward stochastic differential equation if such an lq problem is solvable one obtains an open loop solution to the two person leader follower stochastic differential game moreover it is shown that the open loop solution admits a state feedback representation if a new stochastic riccati equation is solvable

ϵ nash mean field games for linear quadratic systems with random jumps and applications this paper studies linear quadratic gaussian lqg games of stochastic large population system with jump diffusion processes the most distinguishing feature compared with the well studied mean fi

mean field production output control with sticky prices nash and social solutions this paper presents an application of mean field control to dynamic production optimization with sticky prices and adjustment costs both noncooperative and cooperative solutions are considered by solving auxiliary limiting optimal control problems subject to consistent mean field approximations two sets of decentralized strategies are obtained and further shown to asymptotically attain nash equilibria and social optima respectively a numerical example is given to compare market prices firms outputs and costs under the two solution frameworks

periodic solutions of a discrete hamiltonian system with a change of sign in the potential in this paper an existence theorem is obtained for periodic solutions of a second order discrete hamiltonian system with a change of sign in the potential by the minimax methods in the critical point theory

periodic solutions of second order discrete hamiltonian systems with potential indefinite in sign abstract in this paper an existence theorem is obtained for the periodic solutions of second order discrete hamiltonian systems with potential indefinite in sign by linking theorem in the critical point theory

existence result for periodic solutions of a class of hamiltonian systems with super quadratic potential abstract we have presented a variant existence result for periodic solutions to a class of hamiltonian systems with a potential which is both super quadratic and sign indefinite

weighted parzen windows for pattern classification this paper introduces the weighted parzen window classifier the proposed technique uses a clustering procedure to find a set of reference vectors and weights which are used to approximate the parzen window kernel estimator classifier the weighted parzen window classifier requires less computation and storage than the full parzen window classifier experimental results showed that significant savings could be achieved with only minimal if any error rate degradation for synthetic and real data sets

backward mean field linear quadratic gaussian lqg games full and partial information this paper introduces the backward mean field mf linear quadratic gaussian lqg games for short bmflqg of weakly coupled stochastic large population system in contrast to the well studied forward mean field lqg games the individual state in our large population system follows the backward stochastic differential equation bsde whose terminal instead initial condition should be prescribed two classes of bmflqg games are discussed here and their decentralized strategies are derived through the consistency condition in the first class the individual agents of large population system are weakly coupled in their state dynamics and the full information can be accessible to all agents in the second class the coupling structure lies in the cost functional with only partial information structure in both classes the asymptotic near optimality property namely epsilon nash equilibrium of decentralized strategies are verified to this end some estimates to bsde are presented in the large population setting

epsilon nash equilibria for partially observed lqg mean field games with a major player huang 2010 and nguyen and huang 2012 solved the linear quadratic mean field systems and control problem in the case where there is a major agent i e non asymptotically vanishing as the population size goes to infinity together with a population of minor agents i e individually asymptotically negligible the new feature in this case is that the mean field becomes stochastic and then by minor agent state extension the existence of epsilon nash equilibria together with the individual agents control laws that yield the equilibria may be established this paper presents results initially announced by caines and kizilkale 2013 2014 where it is shown that if the major agent s state is partially observed by the minor agents and if the major agent completely observes its own state all agents can recursively generate estimates in general individually distinct of the major agent s state and the mean field and thence generate feedback controls yielding epsilon nash equilibria

weighted majority games have asymptotic value the asymptotic value of a game v with a continuum set of players i is defined whenever all the sequences of the shapley values of finite games that approximate v have the same limit a weighted majority game is a game of the form f where is a positive measure and fx 1 if x q and fx 0 otherwise and q is a real number 0 q i in this paper we prove that all weighted majority games have asymptotic values this result is then used further to show that if v is of the form v f where is a probability measure and f is a function of bounded variation on 0 1 that is continuous at 0 and at 1 then v has an asymptotic value this bad previously been known only when f is absolutely continuous or when has at most finitely many atoms or when is purely atomic thus the essential novelty is that even when has countably many atoms and a nonatomic part f has an asymptotic value we also show that f does not necessarily have an asymptotic value when is a signed measure

mean field stackelberg games aggregation of delayed instructions in this paper we consider an n player interacting strategic game in the presence of a endogenous dominating player who gives direct influence on individual agents through its impact on their control in the sense of stackelberg game and then on the whole community each individual agent is subject to a delay effect on collecting information specifically at a delay time from the dominating player the size of his delay is completely known by the agent while to others including the dominating player his delay plays as a hidden random variable coming from a common fixed distribution by invoking a noncanonical fixed point property we show that for a general class of finite n player games each of them converges to the mean field counterpart which may possess an optimal solution that can serve as an epsilon nash equilibrium for the corresponding finite n player game second we provide with explicit solutions a comprehensive study on the corresponding linear quadratic mean field games of

a stochastic maximum principle for delayed mean field stochastic differential equations and its applications in this technical note we discuss the stochastic optimal control problems of mean field stochastic differential delayed equations mfsddes which arise naturally from various backgrounds including economics finance engineering and physics etc to this end some new estimates are used to handle the complex structure of our controlled system due to the presence of both delay and mean field characters as the main result a stochastic maximum principle for the mean field stochastic optimal control with delay mfsocd is derived in terms of necessary and sufficient conditions in particular applying the convex variation and duality relation we obtain the necessary condition for optimality see theorem 1 in addition the sufficient condition of the optimality is also obtained under some convex condition see theorem 2 based on our maximum principle the related mean field linear quadratic delayed mflqd optimal control problems are also investigated the optimal control is derived and its existence is also verified refer theorem 3 as illustration an example is also proposed and its explicit optimal control is derived

the evolutionary game of pressure or interference resistance and collaboration in this paper we extend the framework of the evolutionary inspection game put forward recently by the author and coworkers to a large class of conflict interactions to address the pressure executed by the major player or principal on the large group of small players who can resist this pressure or collaborate with the major player we prove rigorous results on the convergence of various markov decision models of interacting small agents including evolutionary growth i e pairwise in groups and by coalition formation to a deterministic evolution on the distributions of the state spaces of small players paying main attention to situations with an infinite state space of small players we supply precise rates of convergence the theoretical results of the paper are applied to the analysis of the processes of inspection corruption cyber security counter terrorism banks and firms merging strategically enhanced preferential attachment and many other

linear quadratic gaussian mixed games with continuum parametrized minor players we consider a mean field linear quadratic gaussian game with a major player and a large number of minor players parametrized by a continuum set the mean field generated by the minor players is approximated by a random process depending only on the initial state and the brownian motion of the major player and this leads to two limiting optimal control problems with random coefficients which are solved subject to a consistency requirement on the mean field approximation the set of decentralized strategies constructed from the limiting control problems has an varepsilon nash equilibrium property when applied to the large but finite population model

stochastic maximum principle for mean field type optimal control under partial information this technical note is concerned with a partially observed optimal control problem whose novel feature is that the cost functional is of mean field type hence determining the optimal control is time inconsistent in the sense that bellman s dynamic programming principle does not hold a maximum principle is established using girsanov s theorem and convex variation some nonlinear filtering results for backward stochastic differential equations bsdes are developed by expressing the solutions of the bsdes as some ito s processes an illustrative example is demonstrated in terms of the maximum principle and the filtering

lq nash games with random entrance an infinite horizon major player and minor players of finite horizons we study dynamic games with randomly entering players staying in the game for different lengths of time particularly a class of discrete time linear quadratic lq games involving a major player who has an infinite time horizon and a random number of minor players is considered the number of the new minor players entering at some instant of time is random and it is described by a markov chain the problem of the characterization of a nash equilibrium consisting of linear feedback strategies is reformulated as a set of coupled finite and infinite horizon lq optimal control problems for markov jump linear systems mjls sufficient conditions characterizing nash equilibrium are then derived the problem of games involving a large number of minor players is then addressed using a mean field mf approach and asymptotic varepsilon nash equilibrium results are derived sufficient conditions for the existence of a mf nash equilibrium are finally stated

a new maximum relevance minimum multicollinearity mrmmc method for feature selection and ranking a new maximum relevanceminimum multicollinearity mrmmc method is proposed the proposed mrmrc algorithm was applied to a number of real life datasets experimental results are reported and compared with several state of the art methods numerical analysis results confirmed the promising performance of the proposed method a substantial amount of datasets stored for various applications are often high dimensional with redundant and irrelevant features processing and analysing data under such circumstances is time consuming and makes it difficult to obtain efficient predictive models there is a strong need to carry out analyses for high dimensional data in some lower dimensions and one approach to achieve this is through feature selection this paper presents a new relevancy redundancy approach called the maximum relevanceminimum multicollinearity mrmmc method for feature selection and ranking which can overcome some shortcomings of existing criteria in the proposed method relevant features are measured by correlation characteristics based on conditional variance while redundancy elimination is achieved according to multiple correlation assessment using an orthogonal projection scheme a series of experiments were conducted on eight datasets from the uci machine learning repository and results show that the proposed method performed reasonably well for feature subset selection

social optima in mean field linear quadratic gaussian models with markov jump parameters this paper investigates social optima of mean field linear quadratic gaussian lqg control models with markov jump parameters the common objective of the agents is to minimize a social cost the cost average of the whole society in the cost functions there are coupled mean field terms first we consider the centralized case and get a parameterized equation of mean field effect then we design a set of distributed strategies by solving a limiting optimal control problem in an augmented state space subject to the consistency requirement for mean field approximation it is shown that the set of distributed strategies is asymptotically team optimal and the asymptotically optimal social cost value can be obtained explicitly the optimal social average cost is compared with the optimal individual cost in mean field games by virtue of the explicit expressions and the difference is further illustrated by a numerical example

large population cost coupled lqg problems with nonuniform agents individual mass behavior and decentralized varepsilon nash equilibria we consider linear quadratic gaussian lqg games in large population systems where the agents evolve according to nonuniform dynamics and are coupled via their individual costs a state aggregation technique is developed to obtain a set of decentralized control laws for the individuals which possesses an epsiv nash equilibrium property a stability property of the mass behavior is established and the effect of inaccurate population statistics on an isolated agent is also analyzed by variational techniques