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

a general framework for population based distributed optimization over networks abstract this paper presents a population based solution for the distributed optimization problem where the overall objective function is defined as an average of local cost functions corresponding to the nodes of a network populations are introduced for the nodes to cooperatively find the global optimum of the overall objective function the main challenge is that each population cannot know the overall objective function and thus cannot directly evaluate the quality of their individuals in each iteration to overcome this difficulty we introduce consensus methods to design a message passing protocol under which the local estimates can converge to the same value we present a general framework that consists of consensus search consensus evaluation population evolution and local stopping steps compared with mathematical methods this framework solves in network distributed optimization problems without requiring a convexity assumption on the objective functions the standard pso and one of its variants called mco are introduced for testing and comparison two simulated examples under different network topologies illustrate the feasibility of our approach

fixed point optimization algorithms for distributed optimization in networked systems this paper considers a networked system with a finite number of users and deals with the problem of minimizing the sum of all users objective functions over the intersection of all users constraint sets onto which the projection cannot be easily implemented the main objective of this paper is to devise distributed optimization algorithms which enable each user to find the solution of the problem without using other users objective functions and constraint sets to reach this goal we first introduce easily implementable nonexpansive mappings of which the intersection of the fixed point sets is equal to the constraint set in the problem we formulate the problem as a convex minimization problem over the intersection of the fixed point sets of the nonexpansive mappings we then present an iterative algorithm based on the conventional incremental subgradient methods which use the projection for solving the problem the algorithm can be implemented by using nonexpansive mappings other than the project

stochastic primal dual hybrid gradient algorithm with arbitrary sampling and imaging applications we propose a stochastic extension of the primal dual hybrid gradient algorithm studied by chambolle and pock in 2011 to solve saddle point problems that are separable in the dual variable the anal

combining ordered subsets and momentum for accelerated x ray ct image reconstruction statistical x ray computed tomography ct reconstruction can improve image quality from reduced dose scans but requires very long computation time ordered subsets os methods have been widely used for research in x ray ct statistical image reconstruction and are used in clinical pet and spect reconstruction in particular os methods based on separable quadratic surrogates os sqs are massively parallelizable and are well suited to modern computing architectures but the number of iterations required for convergence should be reduced for better practical use this paper introduces os sqs momentum algorithms that combine nesterov s momentum techniques with os sqs methods greatly improving convergence speed in early iterations if the number of subsets is too large the os sqs momentum methods can be unstable so we propose diminishing step sizes that stabilize the method while preserving the very fast convergence behavior experiments with simulated and real 3d ct scan data illustrate the performance of the proposed algorithms

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

block iterative algorithms with underrelaxed bregman projections the notion of relaxation is well understood for orthogonal projections onto convex sets for general bregman projections it was considered only for hyperplanes and the question of how to relax bregman projections onto convex sets that are not linear i e not hyperplanes or half spaces has remained open a definition of the underrelaxation of bregman projections onto general convex sets is given here which includes as special cases the underrelaxed orthogonal projections and the underrelaxed bregman projections onto linear sets as given by de pierro and iusem j optim theory appl 51 1986 pp 421 440 with this new definition we construct a block iterative projection algorithmic scheme and prove its convergence to a solution of the convex feasibility problem the practical importance of relaxation parameters in the application of such projection algorithms to real world problems is demonstrated on a problem of image reconstruction from projections

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

a q deformed completely integrable bose gas model the authors construct the hamiltonian of a new quantum integrable q boson lattice model in 1 1 dimensions which has q bosons as dynamical variables and solve it for its energy eigenstates and energy eigenvalues under periodic boundary conditions of finite period this model can be regarded as the q deformation of the integrable lattice bose gas the quantum lattice nonlinear schrodinger quantum lattice nls model in appropriate continuum limits both of these lattice models become the quantum repulsive nls model the bose gas in the lattice case the distinction between the q bosons and ordinary bosons leads to significant new features particularly in the energy spectrum for which the q boson lattice model has an optical branch and bound state spectra as well as the one acoustic branch of the quantum lattice nls model it is argued that this sort of distinction is generic for discrete systems with a finite number of degrees of freedom as opposed to continuum field theories like the bose gas where quantization by quantum group quantization or by canonical quantization leads to the same results various properties of the q bosons are given and used in particular a q boson primakov holstein transformation is given which connects the q boson lattice model with the xxz models of arbitrary spin

the quantum group suq 2 and a q analogue of the boson operators a new realisation of the quantum group suq 2 is constructed by means of a q analogue to the jordan schwinger mapping determining thereby both the complete representation structure and q analogues to the wigner and racah operators to achieve this realisation a new elementary object is defined a q analogue to the harmonic oscillator the uncertainty relation for position and momentum in a q harmonic oscillator is quite unusual

differential calculus on compact matrix pseudogroups quantum groups the paper deals with non commutative differential geometry the general theory of differential calculus on quantum groups is developed bicovariant bimodules as objects analogous to tensor bundles over lie groups are studied tensor algebra and external algebra constructions are described it is shown that any bicovariant first order differential calculus admits a natural lifting to the external algebra so the external derivative of higher order differential forms is well defined and obeys the usual properties the proper form of the cartan maurer formula is found the vector space dual to the space of left invariant differential forms is endowed with a bilinear operation playing the role of the lie bracket commutator generalized antisymmetry relation and jacobi identity are proved

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

a fully well balanced positive and entropy satisfying godunov type method for the shallow water equations this work is devoted to the derivation of a fully well balanced numerical scheme for the well known shallow water model during the last two decades several well balanced strategies have been introduced with a special attention to the exact capture of the stationary states associated with the so called lake at rest by fully well balanced we mean here that the proposed godunov type method is also able to preserve stationary states with non zero velocity the numerical procedure is shown to preserve the positiveness of the water height and satisfies a discrete entropy inequality

a multi well balanced scheme for the shallow water mhd system with topography the shallow water magnetohydrodynamic system involves several families of physically relevant steady states in this paper we design a well balanced numerical scheme for the one dimensional shallow water magnetohydrodynamic system with topography that resolves exactly a large range of steady states two variants are proposed with slightly different families of preserved steady states they are obtained by a generalized hydrostatic reconstruction algorithm involving the magnetic field and with a cutoff parameter to remove singularities the solver is positive in height and semi discrete entropy satisfying which ensures the robustness of the method

incremental subgradient methods for nondifferentiable optimization we consider a class of subgradient methods for minimizing a convex function that consists of the sum of a large number of component functions this type of minimization arises in a dual context from lagrangian relaxation of the coupling constraints of large scale separable problems the idea is to perform the subgradient iteration incrementally by sequentially taking steps along the subgradients of the component functions with intermediate adjustment of the variables after processing each component function this incremental approach has been very successful in solving large differentiable least squares problems such as those arising in the training of neural networks and it has resulted in a much better practical rate of convergence than the steepest descent method in this paper we establish the convergence properties of a number of variants of incremental subgradient methods including some that are stochastic based on the analysis and computational experiments the methods appear very promising and effective for important classes of large problems a particularly interesting discovery is that by randomizing the order of selection of component functions for iteration the convergence rate is substantially improved

a subsonic well balanced reconstruction scheme for shallow water flows we consider the saint venant system for shallow water flows with nonflat bottom in past years efficient well balanced methods have been proposed in order to well resolve solutions close to steady states at rest here we describe a strategy based on a local subsonic steady state reconstruction that allows one to derive a subsonic well balanced scheme preserving exactly all the subsonic steady states it generalizes the now well known hydrostatic solver and like the latter it preserves the nonnegativity of the water height and satisfies a semidiscrete entropy inequality an application to the euler poisson system is proposed

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

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

constrained consensus and optimization in multi agent networks we present distributed algorithms that can be used by multiple agents to align their estimates with a particular value over a network with time varying connectivity our framework is general in that this value can represent a consensus value among multiple agents or an optimal solution of an optimization problem where the global objective function is a combination of local agent objective functions our main focus is on constrained problems where the estimates of each agent are restricted to lie in different convex sets to highlight the effects of constraints we first consider a constrained consensus problem and present a distributed projected consensus algorithm in which agents combine their local averaging operation with projection on their individual constraint sets this algorithm can be viewed as a version of an alternating projection method with weights that are varying over time and across agents we establish convergence and convergence rate results for the projected consensus algorithm we next study a constrained optimization problem for optimizing the sum of local objective functions of the agents subject to the intersection of their local constraint sets we present a distributed projected subgradient algorithm which involves each agent performing a local averaging operation taking a subgradient step to minimize its own objective function and projecting on its constraint set we show that with an appropriately selected stepsize rule the agent estimates generated by this algorithm converge to the same optimal solution for the cases when the weights are constant and equal and when the weights are time varying but all agents have the same constraint set

acceleration method combining broadcast and incremental distributed optimization algorithms this paper considers a networked system consisting of an operator which manages the system and a finite number of subnetworks with all users and studies the problem of minimizing the sum of the operator s and all users objective functions over the intersection of the operator s and all users constraint sets when users in each subnetwork can communicate with each other they can implement an incremental subgradient method that uses the transmitted information from their neighbor users since the operator can communicate with users in the subnetworks it can implement a broadcast distributed algorithm that uses all available information in the subnetworks we present an iterative method combining broadcast and incremental distributed optimization algorithms our method has faster convergence and a wider range of application compared with conventional distributed algorithms we also prove that under certain assumptions our method converges to the solution to the problem in the sense of the strong topol

a randomized incremental subgradient method for distributed optimization in networked systems we present an algorithm that generalizes the randomized incremental subgradient method with fixed stepsize due to nedic and bertsekas siam j optim 12 2001 pp 109 138 our novel algorithm is particularly suitable for distributed implementation and execution and possible applications include distributed optimization e g parameter estimation in networks of tiny wireless sensors the stochastic component in the algorithm is described by a markov chain which can be constructed in a distributed fashion using only local information we provide a detailed convergence analysis of the proposed algorithm and compare it with existing both deterministic and randomized incremental subgradient methods

additive logistic regression a statistical view of boosting boosting is one of the most important recent developments in classification methodology boosting works by sequentially applying a classification algorithm to reweighted versions of the training data and then taking a weighted majority vote of the sequence of classifiers thus produced for many classification algorithms this simple strategy results in dramatic improvements in performance we show that this seemingly mysterious phenomenon can be understood in terms of well known statistical principles namely additive modeling and maximum likelihood for the two class problem boosting can be viewed as an approximation to additive modeling on the logistic scale using maximum bernoulli likelihood as a criterion we develop more direct approximations and show that they exhibit nearly identical results to boosting direct multiclass generalizations based on multinomial likelihood are derived that exhibit performance comparable to other recently proposed multiclass generalizations of boosting in most situations and far superior in some we suggest a minor modification to boosting that can reduce computation often by factors of 10 to 50 finally we apply these insights to produce an alternative formulation of boosting decision trees this approach based on best first truncated tree induction often leads to better performance and can provide interpretable descriptions of the aggregate decision rule it is also much faster computationally making it more suitable to large scale data mining applications

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

convergence rates of distributed nesterov like gradient methods on random networks we consider distributed optimization in random networks where n nodes cooperatively minimize the sum i 1n fi x of their individual convex costs existing literature proposes distributed gradient like methods that are computationally cheap and resilient to link failures but have slow convergence rates in this paper we propose accelerated distributed gradient methods that 1 are resilient to link failures 2 computationally cheap and 3 improve convergence rates over other gradient methods we model the network by a sequence of independent identically distributed random matrices w k drawn from the set of symmetric stochastic matrices with positive diagonals the network is connected on average and the cost functions are convex differentiable with lipschitz continuous and bounded gradients we design two distributed nesterov like gradient methods that modify the d ng and d nc methods that we proposed for static networks we prove their convergence rates in terms of the expected optimality gap at the cost function let k and k be the number of per node gradient evaluations and per node communications respectively then the modified d ng achieves rates o logk k and o logk k and the modified d nc rates o 1 k2 and o 1 k2 where 0 is arbitrarily small for comparison the standard distributed gradient method cannot do better than 1 k2 3 and 1 k2 3 on the same class of cost functions even for static networks simulation examples illustrate our analytical findings

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

study of quommutators of quantum variables and generalized derivatives a general deformation of the heisenberg algebra is introduced with two deformed operators instead of just one this is generalized to many variables and permits the simultaneous existence of coherent states and the transposition of creation operators

on wick algebras with additional twisted commutation relations we study the construction of the c symmetric creation and annihilation operators which form a wick algebra with some additional relations this wick algebra is an algebra of the operators which act on the associative algebra of polynomials with noncommutative variables divided by an ideal generated by quadratic relations between the variables as a result the consistency conditions for such a representation are obtained the creators and annihilator have to preserve ideal where which leads us to consistency conditions

convergent incremental optimization transfer algorithms application to tomography no convergent ordered subsets os type image reconstruction algorithms for transmission tomography have been proposed to date in contrast in emission tomography there are two known families of convergent os algorithms methods that use relaxation parameters and methods based on the incremental expectation maximization em approach this paper generalizes the incremental em approach by introducing a general framework incremental optimization transfer the proposed algorithms accelerate convergence speeds and ensure global convergence without requiring relaxation parameters the general optimization transfer framework allows the use of a very broad family of surrogate functions enabling the development of new algorithms this paper provides the first convergent os type algorithm for nonconcave penalized likelihood pl transmission image reconstruction by using separable paraboloidal surrogates sps which yield closed form maximization steps we found it is very effective to achieve fast convergence rates by starting with an os algorithm with a large number of subsets and switching to the new transmission incremental optimization transfer triot algorithm results show that triot is faster in increasing the pl objective than nonincremental ordinary sps and even os sps yet is convergent

incremental least squares methods and the extended kalman filter in this paper we propose and analyze nonlinear least squares methods which process the data incrementally one data block at a time such methods are well suited for large data sets and real time operation and have received much attention in the context of neural network training problems we focus on the extended kalman filter which may be viewed as an incremental version of the gauss newton method we provide a nonstochastic analysis of its convergence properties and we discuss variants aimed at accelerating its convergence

choosing parameters in block iterative or ordered subset reconstruction algorithms viewed abstractly all the algorithms considered here are designed to provide a nonnegative solution x to the system of linear equations y px where y is a vector with positive entries and p a matrix whose entries are nonnegative and with no purely zero columns the expectation maximization maximum likelihood method as it occurs in emission tomography and the simultaneous multiplicative algebraic reconstruction technique are slow to converge on large data sets accelerating convergence through the use of block iterative or ordered subset versions of these algorithms is a topic of considerable interest these block iterative versions involve relaxation and normalization parameters the correct selection of which may not be obvious to all users the algorithms are not faster merely by virtue of being block iterative the correct choice of the parameters is crucial through a detailed discussion of the theoretical foundations of these methods we come to a better understanding of the precise roles these parameters play

distributed nash equilibrium seeking this paper presents an asynchronous gossip based algorithm for finding a nash equilibrium ne of a game in a distributed multi player network the algorithm is designed in such a way that players make decisions based on estimates of the other players actions obtained from local neighbors using a set of standard assumptions on the cost functions and communication graph the paper proves almost sure convergence to a ne for diminishing step sizes for constant step sizes an error bound on expected distance from a ne is established the effectiveness of the proposed algorithm is demonstrated via simulation for both diminishing and constant step sizes

the iterated kalman smoother as a gauss newton method the kalman smoother is known to be the maximum likelihood estimator when the measurement and transition functions are affine i e a linear function plus a constant a new proof of this result is presented that shows that the kalman smoother decomposes a large least squares problem into a sequence of much smaller problems the iterated kalman smoother is then presented and shown to be a gauss newton method for maximizing the likelihood function in the nonaffine case this method takes advantage of the decomposition obtained with the kalman smoother

convergence analysis of iterative methods for nonsmooth convex optimization over fixed point sets of quasi nonexpansive mappings this paper considers a networked system with a finite number of users and supposes that each user tries to minimize its own private objective function over its own private constraint set it is assumed that each user s constraint set can be expressed as a fixed point set of a certain quasi nonexpansive mapping this enables us to consider the case in which the projection onto the constraint set cannot be computed efficiently this paper proposes two methods for solving the problem of minimizing the sum of their nondifferentiable convex objective functions over the intersection of their fixed point sets of quasi nonexpansive mappings in a real hilbert space one method is a parallel subgradient method that can be implemented under the assumption that each user can communicate with other users the other is an incremental subgradient method that can be implemented under the assumption that each user can communicate with its neighbors investigation of the two methods convergence properties for a constant step size reveals that with a small constant step size they approximate a solution to the problem consideration of the case in which the step size sequence is diminishing demonstrates that the sequence generated by each of the two methods strongly converges to the solution to the problem under certain assumptions convergence rate analysis of the two methods under certain situations is provided to illustrate the two methods efficiency this paper also discusses nonsmooth convex optimization over sublevel sets of convex functions and provides numerical comparisons that demonstrate the effectiveness of the proposed methods

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

on certain quantum deformations of gl n r in this article all deformations of the general linear group subject to certain restrictions which in particular ensure a smooth passage to the lie group limit are obtained representations are given in terms of certains sets of creation and annihilation operators these creation and annihilation operators may belong to a generalization of the q quark type or q hadronic type with q boson or q fermion behavior this leads to a natural definition of q direct sums of q algebras

an adaptive projected subgradient approach to learning in diffusion networks we present an algorithm that minimizes asymptotically a sequence of nonnegative convex functions over diffusion networks in the proposed algorithm at each iteration the nodes in the network have only partial information of the cost function but they are able to achieve consensus on a possible minimizer asymptotically to account for possible node failures position changes and or reachability problems because of moving obstacles jammers etc the algorithm can cope with changing network topologies and cost functions a desirable feature in online algorithms where information arrives sequentially many projection based algorithms can be straightforwardly extended to probabilistic diffusion networks with the proposed scheme the system identification problem in distributed networks is given as one example of a possible application

the ordered subsets mirror descent optimization method with applications to tomography we describe an optimization problem arising in reconstructing three dimensional medical images from positron emission tomography pet a mathematical model of the problem based on the maximum likelihood principle is posed as a problem of minimizing a convex function of several million variables over the standard simplex to solve a problem of these characteristics we develop and implement a new algorithm ordered subsets mirror descent and demonstrate theoretically and computationally that it is well suited for solving the pet reconstruction problem

an incremental gradient projection method with momentum term and adaptive stepsize rule we consider an incremental gradient method with momentum term for minimizing the sum of continuously differentiable functions this method uses a new adaptive stepsize rule that decreases the stepsize whenever sufficient progress is not made we show that if the gradients of the functions are bounded and lipschitz continuous over a certain level set then every cluster point of the iterates generated by the method is a stationary point in addition if the gradient of the functions have a certain growth property then the method is either linearly convergent in some sense or the stepsizes are bounded away from zero the new stepsize rule is much in the spirit of heuristic learning rules used in practice for training neural networks via backpropagation as such the new stepsize rule may suggest improvements on existing learning rules finally extension of the method and the convergence results to constrained minimization is discussed as are some implementation issues and numerical experience

zero gradient sum algorithms for distributed convex optimization the continuous time case this technical note presents a set of continuous time distributed algorithms that solve unconstrained separable convex optimization problems over undirected networks with fixed topologies the algorithms are developed using a lyapunov function candidate that exploits convexity and are called zero gradient sum zgs algorithms as they yield nonlinear networked dynamical systems that evolve invariantly on a zero gradient sum manifold and converge asymptotically to the unknown optimizer we also describe a systematic way to construct zgs algorithms show that a subset of them actually converge exponentially and obtain lower and upper bounds on their convergence rates in terms of the network topologies problem characteristics and algorithm parameters including the algebraic connectivity laplacian spectral radius and function curvatures the findings of this technical note may be regarded as a natural generalization of several well known algorithms and results for distributed consensus to distributed convex optimization

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

example of infinite statistics i give a second quantized example of infinite statistics in which any representation of the symmetric group can occur for identical particles and discuss the properties of the example

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

fast distributed gradient methods we study distributed optimization problems when n nodes minimize the sum of their individual costs subject to a common vector variable the costs are convex have lipschitz continuous gradient with constant l and bounded gradient we propose two fast distributed gradient algorithms based on the centralized nesterov gradient algorithm and establish their convergence rates in terms of the per node communications k and the per node gradient evaluations k our first method distributed nesterov gradient achieves rates o logk k and o logk k our second method distributed nesterov gradient with consensus iterations assumes at all nodes knowledge of l and w the second largest singular value of the n n doubly stochastic weight matrix w it achieves rates o 1 k2 and o 1 k2 0 arbitrarily small further we give for both methods explicit dependence of the convergence constants on n and w simulation examples illustrate our findings

generally deformed oscillator isospectral oscillator system and hermitian phase operator the generally deformed oscillator gdo and its multiphoton realization as well as the coherent and squeezed vacuum states are studied we discuss in particular the gdo depending on a complex parameter q therefore we call it q gdo together with the finite dimensional cyclic representations as a realistic physical system of a gdo the isospectral oscillator system is studied and it is found that its coherent and squeezed vacuum states are closely related to those of the oscillator it is pointed out that starting from the q gdo with q root of unity one can define the hermitian phase operators in quantum optics consistently and algebraically the new creation and annihilation operators of the pegg barnett type phase operator theory are defined by using the cyclic representations and these operators degenerate to those of the ordinary oscillator in the classical limit