an aggregate subgradient method for nonsmooth convex minimization a class of implementable algorithms is described for minimizing any convex not necessarily differentiable functionf of several variables the methods require only the calculation off and one subgradient off at designated points they generalize lemarechal s bundle method more specifically instead of using all previously computed subgradients in search direction finding subproblems that are quadratic programming problems the methods use an aggregate subgradient which is recursively updated as the algorithms proceed each algorithm yields a minimizing sequence of points and iff has any minimizers then this sequence converges to a solution of the problem particular members of this algorithm class terminate whenf is piecewise linear the methods are easy to implement and have flexible storage requirements and computational effort per iteration that can be controlled by a user

a simple label switching algorithm for semisupervised structural svms in structured output learning obtaining labeled data for real world applications is usually costly while unlabeled examples are available in abundance semisupervised structured classification deals with a small number of labeled examples and a large number of unlabeled structured data in this work we consider semisupervised structural support vector machines with domain constraints the optimization problem which in general is not convex contains the loss terms associated with the labeled and unlabeled examples along with the domain constraints we propose a simple optimization approach that alternates between solving a supervised learning problem and a constraint matching problem solving the constraint matching problem is difficult for structured prediction and we propose an efficient and effective label switching method to solve it the alternating optimization is carried out within a deterministic annealing framework which helps in effective constraint matching and avoiding poor local minima which are not very useful the algorithm is simple and easy to implement further it is suitable for any structured output learning problem where exact inference is available experiments on benchmark sequence labeling data sets and a natural language parsing data set show that the proposed approach though simple achieves comparable generalization performance

new variants of bundle methods in this paper we describe a number of new variants of bundle methods for nonsmooth unconstrained and constrained convex optimization convex concave games and variational inequalities we outline the ideas underlying these methods and present rate of convergence estimates

pegasos primal estimated sub gradient solver for svm we describe and analyze a simple and effective stochastic sub gradient descent algorithm for solving the optimization problem cast by support vector machines svm we prove that the number of iterations required to obtain a solution of accuracy epsilon is tilde o 1 epsilon where each iteration operates on a single training example in contrast previous analyses of stochastic gradient descent methods for svms require omega 1 epsilon 2 iterations as in previously devised svm solvers the number of iterations also scales linearly with 1 where is the regularization parameter of svm for a linear kernel the total run time of our method is tilde o d lambda epsilon where d is a bound on the number of non zero features in each example since the run time does not depend directly on the size of the training set the resulting algorithm is especially suited for learning from large datasets our approach also extends to non linear kernels while working solely on the primal objective function though in this case the runtime does depend linearly on the training set size our algorithm is particularly well suited for large text classification problems where we demonstrate an order of magnitude speedup over previous svm learning methods

incorporating the loss function into discriminative clustering of structured outputs clustering using the hilbert schmidt independence criterion cluhsic is a recent clustering algorithm that maximizes the dependence between cluster labels and data observations according to the hilbert schmidt independence criterion hsic it is unique in that structure information on the cluster outputs can be easily utilized in the clustering process however while the choice of the loss function is known to be very important in supervised learning with structured outputs we will show in this paper that cluhsic is implicitly using the often inappropriate zero one loss we propose an extension called cluhsical which stands for clustering using hsic and loss which explicitly considers both the output dependency and loss function its optimization problem has the same form as cluhsic except that its partition matrix is constructed in a different manner experimental results on a number of datasets with structured outputs show that cluhsical often outperforms cluhsic in terms of both structured loss and clustering accuracy

a version of the bundle idea for minimizing a nonsmooth function conceptual idea convergence analysis numerical results during recent years various proposals for the minimization of a nonsmooth functional have been made amongst these the bundle concept turned out to be an especially fruitful idea based on this concept a number of authors have developed codes that can successfully deal with nonsmooth problems the aim of the paper is to show that by adding some features of the trust region philosophy to the bundle concept the end result is a distinguished member of the bundle family with a more stable behaviour than some other bundle versions the reliability and efficiency of this code is demonstrated on the standard academic test examples and on some real life problems

bundle methods for regularized risk minimization a wide variety of machine learning problems can be described as minimizing a regularized risk functional with different algorithms using different notions of risk and different regularizers examples include linear support vector machines svms gaussian processes logistic regression conditional random fields crfs and lasso amongst others this paper describes the theory and implementation of a scalable and modular convex solver which solves all these estimation problems it can be parallelized on a cluster of workstations allows for data locality and can deal with regularizers such as l1 and l2 penalties in addition to the unified framework we present tight convergence bounds which show that our algorithm converges in o 1 e steps to e precision for general convex problems and in o log 1 e steps for continuously differentiable problems we demonstrate the performance of our general purpose solver on a variety of publicly available data sets

faster algorithms for max product message passing maximum a posteriori inference in graphical models is often solved via message passing algorithms such as the junction tree algorithm or loopy belief propagation the exact solution to this problem is well known to be exponential in the size of the maximal cliques of the triangulated model while approximate inference is typically exponential in the size of the model s factors in this paper we take advantage of the fact that many models have maximal cliques that are larger than their constituent factors and also of the fact that many factors consist only of latent variables i e they do not depend on an observation this is a common case in a wide variety of applications that deal with grid tree and ring structured models in such cases we are able to decrease the exponent of complexity for message passing by 0 5 for both exact and approximate inference we demonstrate that message passing operations in such models are equivalent to some variant of matrix multiplication in the tropical semiring for which we offer an o n2 5 expected case solution

efficient algorithms for exact inference in sequence labeling svms the task of structured output prediction deals with learning general functional dependencies between arbitrary input and output spaces in this context two loss sensitive formulations for maximum margin training have been proposed in the literature which are referred to as margin and slack rescaling respectively the latter is believed to be more accurate and easier to handle nevertheless it is not popular due to the lack of known efficient inference algorithms therefore margin rescaling which requires a similar type of inference as normal structured prediction is the most often used approach focusing on the task of label sequence learning we here define a general framework that can handle a large class of inference problems based on hamming like loss functions and the concept of decomposability for the underlying joint feature map in particular we present an efficient generic algorithm that can handle both rescaling approaches and is guaranteed to find an optimal solution in polynomial time

on the algorithmic implementation of multiclass kernel based vector machines in this paper we describe the algorithmic implementation of multiclass kernel based vector machines our starting point is a generalized notion of the margin to multiclass problems using this notion we cast multiclass categorization problems as a constrained optimization problem with a quadratic objective function unlike most of previous approaches which typically decompose a multiclass problem into multiple independent binary classification tasks our notion of margin yields a direct method for training multiclass predictors by using the dual of the optimization problem we are able to incorporate kernels with a compact set of constraints and decompose the dual problem into multiple optimization problems of reduced size we describe an efficient fixed point algorithm for solving the reduced optimization problems and prove its convergence we then discuss technical details that yield significant running time improvements for large datasets finally we describe various experiments with our approach comparing it to previously studied kernel based methods our experiments indicate that for multiclass problems we attain state of the art accuracy

trust region newton method for logistic regression large scale logistic regression arises in many applications such as document classification and natural language processing in this paper we apply a trust region newton method to maximize the log likelihood of the logistic regression model the proposed method uses only approximate newton steps in the beginning but achieves fast convergence in the end experiments show that it is faster than the commonly used quasi newton approach for logistic regression we also extend the proposed method to large scale l2 loss linear support vector machines svm

lattice green function in uniform magnetic fields the green function of a square tight binding model in magnetic fields is expressed by means of continued fractions for rational fields the expression is reduced to a more useful one the green function expressing the out going wave is numerically computed at arbitrary sites for the first time it is found that the absolute value of the green function takes maximum values on the reciprocal lattice of the magnetic brillouin zone

online ranking by projecting we discuss the problem of ranking instances in our framework each instance is associated with a rank or a rating which is an integer in 1 to k our goal is to find a rank prediction rule that assigns each instance a rank that is as close as possible to the instance s true rank we discuss a group of closely related online algorithms analyze their performance in the mistake bound model and prove their correctness we describe two sets of experiments with synthetic data and with the eachmovie data set for collaborative filtering in the experiments we performed our algorithms outperform online algorithms for regression and classification applied to ranking

discrete time local value iteration adaptive dynamic programming admissibility and termination analysis in this paper a novel local value iteration adaptive dynamic programming adp algorithm is developed to solve infinite horizon optimal control problems for discrete time nonlinear systems the focuses of this paper are to study admissibility properties and the termination criteria of discrete time local value iteration adp algorithms in the discrete time local value iteration adp algorithm the iterative value functions and the iterative control laws are both updated in a given subset of the state space in each iteration instead of the whole state space for the first time admissibility properties of iterative control laws are analyzed for the local value iteration adp algorithm new termination criteria are established which terminate the iterative local adp algorithm with an admissible approximate optimal control law finally simulation results are given to illustrate the performance of the developed algorithm

ℵ0 categorical structures smoothly approximated by finite substructures a classification is given of primitive k0 categorical structures which are smoothly approximated by a chain of finite homogeneous substructures the proof uses the classification of finite simple groups and some representation theory the main theorems give information about a class of structures more general than the x0 categorical co stable structures examined by cherlin harrington and lachlan

a bundle newton method for nonsmooth unconstrained minimization an algorithm based on a combination of the polyhedral and quadratic approximation is given for finding stationary points for unconstrained minimization problems with locally lips chitz problem functions that are not necessarily convex or differentiable global convergence of the algorithm is established under additional assumptions it is shown that the algorithm generates newton iterations and that the convergence is superlinear some encouraging numerical experience is reported

proximity control in bundle methods for convex proximal bundle methods for minimizing a convex functionf generate a sequence x k by takingxk 1 to be the minimizer of hat f k x u k x x k 2 2 where hat f k is a sufficiently accurate polyhedral approximation tof andu k 0 the usual choice ofu k 1 may yield very slow convergence a technique is given for choosing u k adaptively that eliminates sensitivity to objective scaling some encouraging numerical experience is reported

efficient exact inference with loss augmented objective in structured learning structural support vector machine svm is an elegant approach for building complex and accurate models with structured outputs however its applicability relies on the availability of efficient inference algorithms the state of the art training algorithms repeatedly perform inference to compute a subgradient or to find the most violating configuration in this paper we propose an exact inference algorithm for maximizing nondecomposable objectives due to special type of a high order potential having a decomposable internal structure as an important application our method covers the loss augmented inference which enables the slack and margin scaling formulations of structural svm with a variety of dissimilarity measures e g hamming loss precision and recall f beta loss intersection over union and many other functions that can be efficiently computed from the contingency table we demonstrate the advantages of our approach in natural language parsing and sequence segmentation applications

ultraconservative online algorithms for multiclass problems in this paper we study a paradigm to generalize online classification algorithms for binary classification problems to multiclass problems the particular hypotheses we investigate maintain one prototype vector per class given an input instance a multiclass hypothesis computes a similarity score between each prototype and the input instance and sets the predicted label to be the index of the prototype achieving the highest similarity to design and analyze the learning algorithms in this paper we introduce the notion of ultraconservativeness ultraconservative algorithms are algorithms that update only the prototypes attaining similarity scores which are higher than the score of the correct label s prototype we start by describing a family of additive ultraconservative algorithms where each algorithm in the family updates its prototypes by finding a feasible solution for a set of linear constraints that depend on the instantaneous similarity scores we then discuss a specific online algorithm that seeks a set of prototypes which have a small norm the resulting algorithm which we term mira for margin infused relaxed algorithm is ultraconservative as well we derive mistake bounds for all the algorithms and provide further analysis of mira using a generalized notion of the margin for multiclass problems we discuss the form the algorithms take in the binary case and show that all the algorithms from the first family reduce to the perceptron algorithm while mira provides a new perceptron like algorithm with a margin dependent learning rate we then return to multiclass problems and describe an analogous multiplicative family of algorithms with corresponding mistake bounds we end the formal part by deriving and analyzing a multiclass version of li and long s romma algorithm we conclude with a discussion of experimental results that demonstrate the merits of our algorithms

on the green function of the almost mathieu operator the square tight binding model in a magnetic field leads to the almost mathieu operator which for rational fields reduces to a q q matrix depending on the components of the wavevector in the magnetic brillouin zone we calculate the corresponding green function without explicit knowledge of eigenvalues and eigenfunctions and obtain analytical expressions for the diagonal and the first off diagonal elements the results which are consistent with the zero magnetic field case can be used to calculate several quantities of physical interest e g the density of states over the entire spectrum impurity levels in a magnetic field

the small index property for infinite dimensional classical groups we show that the countable dimensional analogues of the finite classical groups have the strong small index property that is if g is one of these groups and v is its natural module and if h is a subgroup of g of index less than 2 then there exists a finite dimensional subspace x of v such that h is sandwiched between the pointwise and setwise stabilisers in g of x