incremental stochastic subgradient algorithms for convex optimization this paper studies the effect of stochastic errors on two constrained incremental subgradient algorithms the incremental subgradient algorithms are viewed as decentralized network optimization algorithms as applied to minimize a sum of functions when each component function is known only to a particular agent of a distributed network first the standard cyclic incremental subgradient algorithm is studied in this the agents form a ring structure and pass the iterate in a cycle when there are stochastic errors in the subgradient evaluations sufficient conditions on the moments of the stochastic errors are obtained that guarantee almost sure convergence when a diminishing step size is used in addition almost sure bounds on the algorithm s performance with a constant step size are also obtained next the markov randomized incremental subgradient method is studied this is a noncyclic version of the incremental algorithm where the sequence of computing agents is modeled as a time nonhomogeneous markov chain such a model is appropriate for mobile networks as the network topology changes across time in these networks convergence results and error bounds for the markov randomized method in the presence of stochastic errors for diminishing and constant step sizes are obtained

superiority of superposition multiaccess with single user decoding over tdma in the low snr regime this paper studies the gaussian multiaccess channel with multiantenna basestation in the low signal to noise ratio snr regime we compare the spectral efficiencies of the optimal superposition channel sharing scheme and two simple alternatives the time division multiaccess tdma scheme and superposition multiaccess with single user decoding ssd due to the fact that ssd but not tdma exploits the multiuser multiplexing gain the relative spectral efficiency of ssd over tdma grows drastically as the number of antennas at the basestation increases the results suggest that in the low snr regime with multiple antenna basestation tdma s suboptimality can no longer be offset by its simplicity since ssd can achieve much higher spectral efficiency while the simplicities of the two channel sharing schemes are similar

modular invariant representations of infinite dimensional lie algebras and superalgebras abstract in this paper we launch a program to describe and classify modular invariant representations of infinite dimensional lie algebras and superalgebras we prove a character formula for a large class of highest weight representations l of a kac moody algebra unk with a symmetrizable cartan matrix generalizing the weyl kac character formula kac v g 1974 funct anal appl 8 68 70 in the case of an affine unk this class includes modular invariant representations of arbitrary rational level m t u where t unk z and u unk n are relatively prime and m g g u g is the dual coxeter number we write the characters of these representations in terms of theta functions and calculate their asymptotics generalizing the results of kac and peterson kac v g peterson d h 1984 adv math 53 125 264 and of kac and wakimoto kac v g wakimoto m 1988 adv math 70 156 234 for the u 1 integrable case we work out in detail the case unk a1 1 in particular classifying all its modular invariant representations furthermore we show that the modular invariant representations of the virasoro algebra vir are precisely the minimal series of belavin et al belavin a a polyakov a m zamolodchikov a b 1984 nucl phys b 241 333 380 using the character formulas of feigin and fuchs feigin b l fuchs d b 1984 lect notes math 1060 230 245 we show that tensoring the basic representation and modular invariant representations of a1 1 produces all modular invariant representations of vir generalizing the results of goddard et al goddard p kent a olive d 1986 commun math phys 103 105 119 and of kac and wakimoto kac v g wakimoto m 1986 lect notes phys 261 345 371 in the unitary case we study the general branching functions as well all these results are generalized to the kac moody superalgebras introduced by kac kac v g 1978 adv math 30 85 136 and to n 1 super virasoro algebras we work out in detail the case of the superalgebra b 0 1 1 showing in particular that restricting to its even part produces again all modular invariant representations of vir these results lead to general conjectures about asymptotic behavior of positive energy representations and classification of modular invariant representations

quantum reduction for affine superalgebras we extend the homological method of quantization of generalized drinfeld sokolov reductions to affine superalgebras this leads in particular to a unified representation theory of superconformal algebras

an optimal method for stochastic composite optimization this paper considers an important class of convex programming cp problems namely the stochastic composite optimization sco whose objective function is given by the summation of general nonsmooth and smooth stochastic components since sco covers non smooth smooth and stochastic cp as certain special cases a valid lower bound on the rate of convergence for solving these problems is known from the classic complexity theory of convex programming note however that the optimization algorithms that can achieve this lower bound had never been developed in this paper we show that the simple mirror descent stochastic approximation method exhibits the best known rate of convergence for solving these problems our major contribution is to introduce the accelerated stochastic approximation ac sa algorithm based on nesterov s optimal method for smooth cp nesterov in doklady an sssr 269 543 547 1983 nesterov in math program 103 127 152 2005 and show that the ac sa algorithm can achieve the aforementioned lower bound on the rate of convergence for sco to the best of our knowledge it is also the first universally optimal algorithm in the literature for solving non smooth smooth and stochastic cp problems we illustrate the significant advantages of the ac sa algorithm over existing methods in the context of solving a special but broad class of stochastic programming problems

dirty paper coding versus tdma for mimo broadcast channels we compare the capacity of dirty paper coding dpc to that of time division multiple access tdma for a multiple antenna multiple input multiple output mimo gaussian broadcast channel bc we find that the sum rate capacity achievable using dpc of the multiple antenna bc is at most min m k times the largest single user capacity i e the tdma sum rate in the system where m is the number of transmit antennas and k is the number of receivers this result is independent of the number of receive antennas and the channel gain matrix and is valid at all signal to noise ratios snrs we investigate the tightness of this bound in a time varying channel assuming perfect channel knowledge at receivers and transmitters where the channel experiences uncorrelated rayleigh fading and in some situations we find that the dirty paper gain is upper bounded by the ratio of transmit to receive antennas we also show that min m k upper bounds the sum rate gain of successive decoding over tdma for the uplink channel where m is the number of receive antennas at the base station and k is the number of transmitters

convergence of line search methods for unconstrained optimization line search methods are traditional and successful methods for solving unconstrained optimization problems its convergence has attracted more attention in recent years in this paper we analyze the general results on convergence of line search methods with seven line search rules it is clarified that the search direction plays a main role in these methods and that step size guarantees the global convergence in some cases it is also proved that many line search methods have same convergence property these convergence results can enable us to design powerful effective and stable algorithms in practice finally a class of special line search methods is investigated

distributed nonconvex constrained optimization over time varying digraphs this paper considers nonconvex distributed constrained optimization over networks modeled as directed possibly time varying graphs we introduce the first algorithmic framework for the minimization of the sum of a smooth nonconvex nonseparable function the agent s sum utility plus a difference of convex function with nonsmooth convex part this general formulation arises in many applications from statistical machine learning to engineering the proposed distributed method combines successive convex approximation techniques with a judiciously designed perturbed push sum consensus mechanism that aims to track locally the gradient of the smooth part of the sum utility sublinear convergence rate is proved when a fixed step size possibly different among the agents is employed whereas asymptotic convergence to stationary solutions is proved using a diminishing step size numerical results show that our algorithms compare favorably with current schemes on both convex and nonconvex problems

the capacity region of the gaussian multiple input multiple output broadcast channel the gaussian multiple input multiple output mimo broadcast channel bc is considered the dirty paper coding dpc rate region is shown to coincide with the capacity region to that end a new notion of an enhanced broadcast channel is introduced and is used jointly with the entropy power inequality to show that a superposition of gaussian codes is optimal for the degraded vector broadcast channel and that dpc is optimal for the nondegraded case furthermore the capacity region is characterized under a wide range of input constraints accounting as special cases for the total power and the per antenna power constraints

convergence of approximate and incremental subgradient methods for convex optimization we present a unified convergence framework for approximate subgradient methods that covers various stepsize rules including both diminishing and nonvanishing stepsizes convergence in objective values and convergence to a neighborhood of the optimal set we discuss ways of ensuring the boundedness of the iterates and give efficiency estimates our results are extended to incremental subgradient methods for minimizing a sum of convex functions which have recently been shown to be promising for various large scale problems including those arising from lagrangian relaxation

brief paper an adaptive optimization scheme with satisfactory transient performance adaptive optimization ao schemes based on stochastic approximation principles such as the random directions kiefer wolfowitz rdkw the simultaneous perturbation stochastic approximation spsa and the adaptive fine tuning aft algorithms possess the serious disadvantage of not guaranteeing satisfactory transient behavior due to their requirement for using random or random like perturbations of the parameter vector the use of random or random like perturbations may lead to particularly large values of the objective function which may result to severe poor performance or stability problems when these methods are applied to closed loop controller optimization applications in this paper we introduce and analyze a new algorithm for alleviating this problem mathematical analysis establishes satisfactory transient performance and convergence of the proposed scheme under a general set of assumptions application of the proposed scheme to the adaptive optimization of a large scale complex control system demonstrates the efficiency of the proposed scheme

a policy gradient method for semi markov decision processes with application to call admission control solving a semi markov decision process smdp using value or policy iteration requires precise knowledge of the probabilistic model and suffers from the curse of dimensionality to overcome these limitations we present a reinforcement learning approach where one optimizes the smdp performance criterion with respect to a family of parameterised policies we propose an online algorithm that simultaneously estimates the gradient of the performance criterion and optimises it using stochastic approximation we apply our algorithm to call admission control our proposed policy gradient smdp algorithm and its application to admission control is novel

a new class of incremental gradient methods for least squares problems the least mean squares lms method for linear least squares problems differs from the steepest descent method in that it processes data blocks one by one with intermediate adjustment of the parameter vector under optimization this mode of operation often leads to faster convergence when far from the eventual limit and to slower sublinear convergence when close to the optimal solution we embed both lms and steepest descent as well as other intermediate methods within a one parameter class of algorithms and we propose a hybrid class of methods that combine the faster early convergence rate of lms with the faster ultimate linear convergence rate of steepest descent these methods are well suited for neural network training problems with large data sets furthermore these methods allow the effective use of scaling based for example on diagonal or other approximations of the hessian matrix

general results on the convergence of stochastic algorithms a deterministic approach is proposed for proving the convergence of stochastic algorithms of the most general form under necessary conditions on the input noise and reasonable conditions on the nonnecessarily continuous mean field emphasis is placed on the case where more than one stationary point exists we also use this approach to prove the convergence of a stochastic algorithm with markovian dynamics

decentralized delay optimal control for interference networks with limited renewable energy storage in this paper we consider delay minimization for interference networks with renewable energy source where the transmission power of a node comes from both the conventional utility power ac power and the renewable energy source we assume the transmission power of each node is a function of the local channel state local data queue state and local energy queue state only we consider two delay optimization formulations namely the decentralized partially observable markov decision process dec pomdp and noncooperative partially observable stochastic game posg in dec pomdp formulation we derive a decentralized online learning algorithm to determine the control actions and lagrangian multipliers lms simultaneously based on the policy gradient approach under some mild technical conditions the proposed decentralized policy gradient algorithm converges almost surely to a local optimal solution in the noncooperative posg formulation the transmitter nodes are noncooperative we extend the decentralized policy gradient solution and establish the technical proof for almost sure convergence of the learning algorithms in both cases the solutions are very robust to model variations

on the nonergodic convergence rate of an inexact augmented lagrangian framework for composite convex programming in this paper we consider the linearly constrained composite convex optimization problem whose objective is a sum of a smooth function and a possibly nonsmooth function we propose an inexact aug

an incremental decomposition method for unconstrained optimization abstract in this work we consider the problem of minimizing a sum of continuously differentiable functions the vector of variables is partitioned into two blocks and we assume that the objective function is convex with respect to a block component problems with this structure arise for instance in machine learning in order to advantageously exploit the structure of the objective function and to take into account that the number of terms of the objective function may be huge we propose a decomposition algorithm combined with a gradient incremental strategy global convergence of the proposed algorithm is proved the results of computational experiments performed on large scale real problems show the effectiveness of the proposed approach with respect to existing algorithms

power control with imperfect exchanges and applications to spectrum sharing in various applications the effect of errors in gradient based iterations is of particular importance when seeking saddle points of the lagrangian function associated with constrained convex optimization problems of particular interest here are problems arising in power control applications where network utility is maximized subject to minimum signal to interference plus noise ratio sinr constraints maximum interference constraints maximum received power constraints or simultaneous minimum and maximum sinr constraints especially when the gradient iterations are executed in a distributed fashion imperfect exchanges among the link nodes may result in erroneous gradient vectors in order to assess and cope with such errors two running averages ergodic sequences are formed from the iterates generated by the perturbed saddle point method each with complementary strengths under the assumptions of problem convexity and error boundedness bounds on the constraint violation and the suboptimality per iteration index are derived the two types of running averages are tested on a spectrum sharing problem with minimum and maximum sinr constraints as well as maximum interference constraints

structure of some categories of representations of infinite dimensional lie algebras the highest weight representations of a finite dimensional complex semisimple lie algebra 9 have been studied extensively and a number of interesting results have been obtained in the last few years an extensive bibliography is given in 7 in 2 3 a category cy of representations of g was introduced which turns out to be the proper setting for considering various questions regarding highest weight representations the aim of this paper is to extend the study of this notion to a kac moody algebra or more generally to a contragredient lie algebra g a corresponding to a square matrix a over c the study of such algebras was begun independently in 8 14 and subsequently a number of interesting results were proved these results show that many important properties from the finite dimensional setup i e from the context of 9 can be proved in the general setup involving g a in particular the notion of category was extended in 9 to the case of kac moody algebras and used in the proof of combinatorial identities

optimality of beamforming in fading mimo multiple access channels we consider the sum capacity of a multi input multi output mimo multiple access channel mac where the receiver has the perfect channel state information csi while the transmitters have either no or partial csi when the transmitters have partial csi it is in the form of either the covariance matrix of the channel or the mean matrix of the channel for the covariance feedback case we mainly consider physical models that result in single sided correlation structures for the mean feedback case we consider physical models that result in in phase received signals under these assumptions we analyze the mimo mac from three different viewpoints first we consider a finite sized system we show that the optimum transmit directions of each user are the eigenvectors of its own channel covariance and mean feedback matrices in the covariance and mean feedback models respectively also we find the conditions under which beamforming is optimal for all users second in the covariance feedback case we prove that the region where beamforming is optimal for all users gets larger with the addition of new users into the system in the mean feedback case we show through simulations that this is not necessarily true third we consider the asymptotic case where the number of users is large we show that in both no and partial csi cases beamforming is asymptotically optimal in particular in the case of no csi we show that a simple form of beamforming which may be characterized as an arbitrary antenna selection scheme achieves the sum capacity in the case of partial csi we show that beamforming in the direction of the strongest eigenvector of the channel feedback matrix achieves the sum capacity finally we generalize our covariance feedback results to double sided correlation structures in the appendix

random algorithms for convex minimization problems this paper deals with iterative gradient and subgradient methods with random feasibility steps for solving constrained convex minimization problems where the constraint set is specified as the intersection of possibly infinitely many constraint sets each constraint set is assumed to be given as a level set of a convex but not necessarily differentiable function the proposed algorithms are applicable to the situation where the whole constraint set of the problem is not known in advance but it is rather learned in time through observations also the algorithms are of interest for constrained optimization problems where the constraints are known but the number of constraints is either large or not finite we analyze the proposed algorithm for the case when the objective function is differentiable with lipschitz gradients and the case when the objective function is not necessarily differentiable the behavior of the algorithm is investigated both for diminishing and non diminishing stepsize values the almost sure convergence to an optimal solution is established for diminishing stepsize for non diminishing stepsize the error bounds are established for the expected distances of the weighted averages of the iterates from the constraint set as well as for the expected sub optimality of the function values along the weighted averages

convergence property of gradient type methods with non monotone line search in the presence of perturbations in this paper two new kinds of methods which are called gradient type method and hybrid projection method with perturbations are proposed and non monotone line search technique is employed at the same time global convergence of these methods is proved only in the case where the gradient function is uniformly continuous on an open convex set containing the iteration sequence numerical examples are given at the end of this paper

gradient convergence in gradient methods with errors we consider the gradient method x t 1 x t g t s t w t where s t is a descent direction of a function f rn to re and w t is a deterministic or stochastic error we assume that gr f is lipschitz continuous that the stepsize g t diminishes to 0 and that s t and w t satisfy standard conditions we show that either f x t to infty or f x t converges to a finite value and gr f x t to0 with probability 1 in the stochastic case and in doing so we remove various boundedness conditions that are assumed in existing results such as boundedness from below of f boundedness of gr f x t or boundedness of xt

on the convergence of the lms algorithm with adaptive learning rate for linear feedforward networks we consider the problem of training a linear feedforward neural network by using a gradient descent like lms learning algorithm the objective is to find a weight matrix for the network by repeatedly presenting to it a finite set of examples so that the sum of the squares of the errors is minimized kohonen showed that with a small but fixed learning rate or stepsize some subsequences of the weight matrices generated by the algorithm will converge to certain matrices close to the optimal weight matrix in this paper we show that by dynamically decreasing the learning rate during each training cycle the sequence of matrices generated by the algorithm will converge to the optimal weight matrix we also show that for any given 0 the lms algorithm with decreasing learning rates will generate an optimal weight matrix i e a matrix of distance at most away from the optimal matrix after o 1 training cycles this is in contrast to 1 log 1 training cycles needed to generate an optima

a stochastic successive minimization method for nonsmooth nonconvex optimization with applications to transceiver design in wireless communication networks consider the problem of minimizing the expected value of a cost function parameterized by a random variable the classical sample average approximation method for solving this problem requires minimization of an ensemble average of the objective at each step which can be expensive in this paper we propose a stochastic successive upper bound minimization method ssum which minimizes an approximate ensemble average at each iteration to ensure convergence and to facilitate computation we require the approximate ensemble average to be a locally tight upper bound of the expected cost function and be easily optimized the main contributions of this work include the development and analysis of the ssum method as well as its applications in linear transceiver design for wireless communication networks and online dictionary learning moreover using the ssum framework we extend the classical stochastic sub gradient method to the case of minimizing a nonsmooth nonconvex objective function and establish its convergence

maximizing sum rate and minimizing mse on multiuser downlink optimality fast algorithms and equivalence via max min sinr maximizing the minimum weighted signal to interference and noise ratio sinr minimizing the weighted sum mean square error mse and maximizing the weighted sum rate in a multiuser downlink system are three important performance objectives in nonconvex joint transceiver and power optimization where all the users have a total power constraint we show that through connections with the nonlinear perron frobenius theory jointly optimizing power and beamformers in the max min weighted sinr problem can be solved optimally in a distributed fashion then connecting these three performance objectives through the arithmetic geometric mean inequality and nonnegative matrix theory we solve the weighted sum mse minimization and the weighted sum rate maximization in the weak interference regimes using fast algorithms in the general case we first establish optimality conditions to the weighted sum mse minimization and the weighted sum rate maximization problems and provide their further connection to the max min weighted sinr problem we then propose a distributed weighted proportional sinr algorithm that leverages our fast max min weighted sinr algorithm to solve for local optimal solution of the two nonconvex problems and give conditions under which global optimality is achieved numerical results are provided to complement the analysis

a stochastic gradient type algorithm for closed loop problems we focus on the numerical solution of closed loop stochastic problems and propose a perturbed gradient algorithm to achieve this goal the main hurdle in such problems is the fact that the control variables are infinite dimensional due to e g the information constraints alternatively said control variables are feedbacks i e functions such controls have hence to be represented in a finite way in order to solve the problem numerically in the same way the gradient of the criterion is itself an infinite dimensional object our algorithm replaces this exact and unknown gradient by a perturbed one which consists of the product of the true gradient evaluated at a random point and a kernel function which extends this gradient to the neighbourhood of the random point proceeding this way we explore the whole space iteration after iteration through random points since each kernel function is perfectly known by a small number of parameters say n the control at iteration k is perfectly known as an infinite dimensional object by at most n k parameters the main strength of this method is that it avoids any discretization of the underlying space provided that we can sample as many points as needed in this space moreover our algorithm can take into account the possible measurability constraints of the problem in a new way finally the randomized strategy implemented by the algorithm causes the most probable parts of the space to be the most explored ones which is a priori an interesting feature in this paper we first prove two convergence results of this algorithm in the strongly convex and convex cases and then give some numerical examples showing the interest of this method for practical stochastic optimization problems

an online gradient based parameter identification algorithm for the neuro fuzzy systems abstract online gradient descent method has been widely applied for parameter learning in neuro fuzzy systems the success of the application relies on the convergence of the learning procedure however there barely have been convergence analyses on the online learning procedure for neuro fuzzy systems in this paper an online gradient learning algorithm with adaptive learning rate is proposed to identify the parameters of the neuro fuzzy systems representing the mamdani fuzzy model with gaussian fuzzy sets we take the reciprocals of the variances of the gaussian membership functions rather than the variances themselves as independent variables when computing the gradient with respect to the variance parameters subsequently oscillation of the gradient value in the learning process can be avoided furthermore some convergence results for this online learning scheme are studied finally three numerical examples are provided to illustrate the performance of the proposed algorithm

communication over fading channels with delay constraints we consider a user communicating over a fading channel with perfect channel state information data are assumed to arrive from some higher layer application and are stored in a buffer until transmitted we study adapting the user s transmission rate and power based on the channel state information as well as the buffer occupancy the objectives are to regulate both the long term average transmission power and the average buffer delay incurred by the traffic two models for this situation are discussed one corresponding to fixed length variable rate codewords and one corresponding to variable length codewords the tradeoff between the average delay and the average transmission power required for reliable communication is analyzed a dynamic programming formulation is given to find all pareto optimal power delay operating points we then quantify the behavior of this tradeoff in the regime of asymptotically large delay in this regime we characterize simple buffer control policies which exhibit optimal characteristics connections to the delay limited capacity and the expected capacity of fading channels are also discussed

steered sequential projections for the inconsistent convex feasibility problem we study a steered sequential gradient algorithm which minimizes the sum of convex functions by proceeding cyclically in the directions of the negative gradients of the functions and using steered step sizes this algorithm is applied to the convex feasibility problem by minimizing a proximity function which measures the sum of the bregman distances to the members of the family of convex sets the resulting algorithm is a new steered sequential bregman projection method

asynchronous stochastic approximations the asymptotic behavior of a distributed asynchronous stochastic approximation scheme is analyzed in terms of a limiting nonautonomous differential equation the relation between the latter and the relative values of suitably rescaled relative frequencies of updates of different components is underscored

spectral efficiency of cdma with random spreading the cdma channel with randomly and independently chosen spreading sequences accurately models the situation where pseudonoise sequences span many symbol periods furthermore its analysis provides a comparison baseline for cdma channels with deterministic signature waveforms spanning one symbol period we analyze the spectral efficiency total capacity per chip as a function of the number of users spreading gain and signal to noise ratio and we quantify the loss in efficiency relative to an optimally chosen set of signature sequences and relative to multiaccess with no spreading white gaussian background noise and equal power synchronous users are assumed the following receivers are analyzed a optimal joint processing b single user matched filtering c decorrelation and d mmse linear processing

spectral efficiency of multicarrier cdma we analyze the spectral efficiency sum rate per subcarrier of randomly spread synchronous multicarrier code division multiple access mc cdma subject to frequency selective fading in the asymptotic regime of number of users and bandwidth going to infinity with a constant ratio both uplink and downlink are considered either conditioned on the subcarrier fading coefficients for nonergodic channels or unconditioned thereon for ergodic channels the following receivers are analyzed a jointly optimum receiver b linear minimum mean square error mmse receiver c decorrelator and d single user matched filter

a convergent incremental gradient method with a constant step size an incremental aggregated gradient method for minimizing a sum of continuously differentiable functions is presented the method requires a single gradient evaluation per iteration and uses a constant step size for the case that the gradient is bounded and lipschitz continuous we show that the method visits infinitely often regions in which the gradient is small under certain unimodality assumptions global convergence is established in the quadratic case a global linear rate of convergence is shown the method is applied to distributed optimization problems arising in wireless sensor networks and numerical experiments compare the new method with other incremental gradient methods

distributed optimization in an energy constrained network analog versus digital communication schemes we consider a distributed optimization problem whereby a network of n nodes sl l 1 n wishes to minimize a common strongly convex function f x x x1 xn t under the constraint that node sl controls variable xl only the nodes locally update their respective variables and periodically exchange their values with their neighbors over a set of predefined communication channels previous studies of this problem have focused mainly on the convergence issue and the analysis of convergence rate in this study we consider noisy communication channels and study the impact of communication energy on convergence in particular we study the minimum amount of communication energy required for nodes to obtain an e minimizer of f x in the mean square sense for linear analog communication schemes we prove that the communication energy to obtain an e minimizer of f x must grow at least at the rate of 1 e and this bound is tight when f is convex quadratic furthermore we show that the same energy requirement can be reduced to o log21 e if a suitable digital communication scheme is used

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

when does online bp training converge the backpropogation bp neural networks have been widely applied in scientific research and engineering the success of the application however relies upon the convergence of the training procedure involved in the neural network learning we settle down the convergence analysis issue through proving two fundamental theorems on the convergence of the online bp training procedure one theorem claims that under mild conditions the gradient sequence of the error function will converge to zero the weak convergence and another theorem concludes the convergence of the weight sequence defined by the procedure to a fixed value at which the error function attains its minimum the strong convergence the weak convergence theorem sharpens and generalizes the existing convergence analysis conducted before while the strong convergence theorem provides new analysis results on convergence of the online bp training procedure the results obtained reveal that with any analytic sigmoid activation function the online bp training procedure is always convergent which then underlies successful application of the bp neural networks

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

dgm a deep learning algorithm for solving partial differential equations abstract high dimensional pdes have been a longstanding computational challenge we propose to solve high dimensional pdes by approximating the solution with a deep neural network which is trained to satisfy the differential operator initial condition and boundary conditions our algorithm is meshfree which is key since meshes become infeasible in higher dimensions instead of forming a mesh the neural network is trained on batches of randomly sampled time and space points the algorithm is tested on a class of high dimensional free boundary pdes which we are able to accurately solve in up to 200 dimensions the algorithm is also tested on a high dimensional hamilton jacobi bellman pde and burgers equation the deep learning algorithm approximates the general solution to the burgers equation for a continuum of different boundary conditions and physical conditions which can be viewed as a high dimensional space we call the algorithm a deep galerkin method dgm since it is similar in spirit to galerkin methods with the solution approximated by a neural network instead of a linear combination of basis functions in addition we prove a theorem regarding the approximation power of neural networks for a class of quasilinear parabolic pdes

characters and fusion rules for w algebras via quantized drinfeld sokolov reduction using the cohomological approach tow algebras we calculate characters and fusion coefficients for their representations obtained from modular invariant representations of affine algebras by the quantized drinfeld sokolov reduction

optimal constellations for the low snr noncoherent mimo block rayleigh fading channel reliable communication over the discrete input continuous output noncoherent multiple input multiple output mimo rayleigh block fading channel is considered when the signal to noise ratio snr per degree of freedom is low two key problems are posed and solved to obtain the optimum discrete input in both problems the average and peak power per space time slot of the input constellation are constrained in the first one the peak power to average power ratio ppapr of the input constellation is held fixed while in the second problem the peak power is fixed independently of the average power in the first ppapr constrained problem the mutual information which grows as o snr2 is maximized up to second order in snr in the second peak constrained problem where the mutual information behaves as o snr the structure of constellations that are optimal up to first order or equivalently that minimize energy per bit are explicitly characterized furthermore among constellations that are first order optimal those that maximize the mutual information up to second order or equivalently the wideband slope are characterized in both ppapr constrained and peak constrained problems the optimal constellations are obtained in closed form as solutions to nonconvex optimizations and interestingly they are found to be identical due to its special structure the common solution is referred to as space time orthogonal rank one modulation or storm in both problems it is seen that storm provides a sharp characterization of the behavior of noncoherent mimo capacity

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

simulation based optimal sensor scheduling with application to observer trajectory planning the sensor scheduling problem can be formulated as a controlled hidden markov model and this paper solves the problem when the state observation and action spaces are continuous this general case is important as it is the natural framework for many applications the aim is to minimise the variance of the estimation error of the hidden state w r t the action sequence we present a novel simulation based method that uses a stochastic gradient algorithm to find optimal actions

incremental proximal methods for large scale convex optimization we consider the minimization of a sum sum i 1 mf i x consisting of a large number of convex component functions f i for this problem incremental methods consisting of gradient or subgradient iterations applied to single components have proved very effective we propose new incremental methods consisting of proximal iterations applied to single components as well as combinations of gradient subgradient and proximal iterations we provide a convergence and rate of convergence analysis of a variety of such methods including some that involve randomization in the selection of components we also discuss applications in a few contexts including signal processing and inference machine learning

from structured data to evolution linear partial differential equations abstract this paper is devoted to the derivation of computational methods for constructing partial differential equations from data following some recent works 7 14 15 20 we propose a methodology based on symbolic calculus 8 9 13 pseudospectral methods 2 3 and stochastic processes 6 in order to determine non constant coefficients of linear evolution partial differential equations pdes from a set of structured data constituted by solutions at given times and positions of an unknown linear pde

generalized kac moody algebras on etudie une classe d algebres de lie qui ont une forme bilineaire contravariante qui est presque definie positive ces algebres generalisent celles de kac moody et peuvent etre considerees comme des algebres de kac moody avec des racines simples imaginaires

decomposition by partial linearization parallel optimization of multi agent systems we propose a novel decomposition framework for the distributed optimization of general nonconvex sum utility functions arising naturally in the system design of wireless multi user interfering systems our main contributions are i the development of the first class of inexact jacobi best response algorithms with provable convergence where all the users simultaneously and iteratively solve a suitably convexified version of the original sum utility optimization problem ii the derivation of a general dynamic pricing mechanism that provides a unified view of existing pricing schemes that are based instead on heuristics and iii a framework that can be easily particularized to well known applications giving rise to very efficient practical jacobi or gauss seidel algorithms that outperform existing ad hoc methods proposed for very specific problems interestingly our framework contains as special cases well known gradient algorithms for nonconvex sum utility problems and many block coordinate descent schemes for convex functions

communication on the grassmann manifold a geometric approach to the noncoherent multiple antenna channel we study the capacity of multiple antenna fading channels we focus on the scenario where the fading coefficients vary quickly thus an accurate estimation of the coefficients is generally not available to either the transmitter or the receiver we use a noncoherent block fading model proposed by marzetta and hochwald see ibid vol 45 p 139 57 1999 the model does not assume any channel side information at the receiver or at the transmitter but assumes that the coefficients remain constant for a coherence interval of length t symbol periods we compute the asymptotic capacity of this channel at high signal to noise ratio snr in terms of the coherence time t the number of transmit antennas m and the number of receive antennas n while the capacity gain of the coherent multiple antenna channel is min m n bits per second per hertz for every 3 db increase in snr the corresponding gain for the noncoherent channel turns out to be m 1 m t bits per second per hertz where m min m n t 2 the capacity expression has a geometric interpretation as sphere packing in the grassmann manifold

weyl kac type character formula for admissible representations of borcherds kac moody lie superalgebras we prove that borcherds kac moody lie superalgebras yield weyl kac type character formulas for some specific types of admissible representations our results generalize the results of kac and wakimoto proc natl acad sci usa 85 4956 4960 1988 by extending the weyl kac character formula for integrable irreducible highest weight modules over kac moody algebras to the case of admissible irreducible highest weight modules over borcherds kac moody lie superalgebras