DOCSLIB.ORG

Learning Output Kernels with Block Coordinate Descent

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Learning Output Kernels with Block Coordinate Descent Learning Output Kernels with Block Coordinate Descent Francesco Dinuzzo [email protected] Max Planck Institute for Intelligent Systems, 72076 T¨ubingen,Germany Cheng Soon Ong [email protected] Department of Computer Science, ETH Z¨urich, 8092 Z¨urich, Switzerland Peter Gehler [email protected] Max Planck Institute for Informatics, 66123 Saarbr¨ucken, Germany Gianluigi Pillonetto [email protected] Department of Information Engineering, University of Padova, 35131 Padova, Italy Abstract the outputs. In this paper, we introduce a method that We propose a method to learn simultane- simultaneously learns a vector-valued function and the ously a vector-valued function and a kernel kernel between the components of the output vector. between its components. The obtained ker- The problem is formulated within the framework of nel can be used both to improve learning per- regularization in RKH spaces. We assume that the formance and to reveal structures in the out- matrix-valued kernel can be decomposed as the prod- put space which may be important in their uct of a scalar kernel and a positive semidefinite kernel own right. Our method is based on the so- matrix that represents the similarity between the out- lution of a suitable regularization problem put components, a structure that has been considered over a reproducing kernel Hilbert space of in a variety of works, (Evgeniou et al., 2005; Capon- vector-valued functions. Although the regu- netto et al., 2008; Baldassarre et al., 2010). larized risk functional is non-convex, we show In practice, an important issue is the choice of the ker- that it is invex, implying that all local min- nel matrix between the outputs. In some cases, one imizers are global minimizers. We derive a may be able to fix it by using prior knowledge about block-wise coordinate descent method that the relationship between the components. However, efficiently exploits the structure of the objec- such prior knowledge is not available in the vast ma- tive functional. Then, we empirically demon- jority of the applications, therefore it is important to strate that the proposed method can improve develop data-driven tools to choose the kernel auto- classification accuracy. Finally, we provide matically. The learned kernel matrix can be also used a visual interpretation of the learned kernel for visualization purposes, revealing interesting struc- matrix for some well known datasets. tures in the output space. For instance, when applying the method to a multiclass classification problem, the learned kernel matrix can be used to cluster the classes 1. Introduction into homogeneous groups. Previous works, such as Classical learning tasks such as binary classification (Zien & Ong, 2007; Lampert & Blaschko, 2008), have and regression can be solved by synthesizing a real- shown that learning convex combinations of kernels on valued function from the data. More generally, prob- the output space can improve performance. lems such as multiple output regression, multitask Our method searches over the whole cone of positive learning, multiclass and multilabel classification can semidefinite matrices. Although the resulting opti- be addressed by first learning a vector-valued func- mization problem is non-convex, we prove that the ob- tion, and then applying a suitable transformation to jective is an invex function (Mishra & Giorgi, 2008), Appearing in Proceedings of the 28 th International Con- namely it has the property that stationary points ference on Machine Learning, Bellevue, WA, USA, 2011. are global minimizers. Then, we propose an efficient Copyright 2011 by the author(s)/owner(s). block-wise coordinate descent algorithm to compute Learning Output Kernels with Block Coordinate Descent an optimal solution. The algorithm alternates between It turns out that every RKHS of Y-valued functions H the solution of a so-called discrete time Sylvester equa- can be associated with a unique positive semidefinite tion and a quadratic optimization problem over the Y-kernel H, called the reproducing kernel. Conversely, cone of positive semidefinite matrices. Exploiting the given a positive semidefinite Y-kernel H on X , there specific structure of the problem, we propose an effi- exists a unique RKHS of Y-valued functions defined cient solution for both steps. over X whose reproducing kernel is H. The standard definition of positive semidefinite (scalar) kernel and The four contributions of this paper are as follows. RKHS (of real-valued functions) can be recovered by First, we introduce a new problem of learning simul- letting Y = . taneously a vector-valued function and the similarity R between its components. Second, we show that our For the rest of the paper, we assume Y = Rm. In non-convex objective is in fact invex, and hence local such a case, L(Y) is just the space of square matrices minimizers are global minimizers. Third, we derive of order m. By fixing a basis fbigi2T for the output an efficient block-wise coordinate descent algorithm to space, where T = f1; : : : ; mg, one can uniquely define solve our problem. Finally, we show empirical evidence an associated (scalar-valued) kernel R over X ×T such that our method can improve performances, and also that discover meaningful structures in the output space. hbi;H(x1; x2)bjiY = R((x1; i); (x2; j)); 2. Spaces of vector-valued functions so that an Y-kernel can be seen as a function that maps We review some facts used to model learning tasks two inputs into the space of square matrices of order m. such as multiple output regression, multiclass and mul- Similarly, by fixing any function g : X!Y, one can tilabel classification. The key idea is to learn functions uniquely define an associated function h : X × T ! R with values in a vector space. such that X g(x) = h(x; i)b : 2.1. Reproducing Kernel Hilbert Spaces i i2T We provide some background here on reproducing ker- Consequently, a space of Y-valued functions over X is nel Hilbert spaces (RKHSs) of vector-valued functions, isomorphic to a standard space of scalar-valued func- i.e. where the outputs are vectors instead of scalars. tions defined over the input set X × T . In conclusion, Let Y denote an Hilbert space with inner product by fixing a basis for the output space, the problem of h·; ·i , and L(Y) the space of bounded linear operators Y learning a vector-valued function can be equivalently from Y into itself. The following definitions introduce regarded as a problem of learning a scalar function de- positive semidefinite Y-kernel and RKHS of Y-valued fined over an enlarged input set, a modeling approach functions, that extend the classical concepts of positive that is also used in the structured output learning set- semidefinite kernel and RKHS. See Micchelli & Pontil ting (Bakir et al., 2007). (2005); Carmeli et al. (2006) for further details. Definition 2.1 (Positive semidefinite Y-kernel). Let 2.2. Learning Tasks X denote a non-empty set and Y an Hilbert space. A symmetric function H : X × X ! L(Y) is called As mentioned in the introduction, many learning tasks positive semidefinite Y-kernel on X if, for any finite can be solved by first learning a function taking val- integer `, the following holds ues in Rm. We list here brief descriptions of some ` ` commonly used models. First of all, in multiple out- X X put regression, each component of the vector directly hyi;H(xi; xj)yjiY ≥ 0; 8(xi; yi) 2 (X ; Y) : i=1 j=1 corresponds to an output. For multiclass classifica- tion, output data are modeled as binary vectors, with We now introduce Hilbert spaces of vector-valued func- +1 in the position corresponding to the class. By tions that can be associated to positive semidefinite interpreting the components of the learned vector- Y-kernels. valued function g as “confidence scores" for the differ- Definition 2.2 (RKHS of Y-valued functions). A Re- ent classes, one can consider a classification rule of the producing Kernel Hilbert Space (RKHS) of Y-valued form yb(x) = arg maxi2T gi(x): For binary multilabel functions g : X!Y is an Hilbert space H such that, classification, outputs are vectors with ±1 at the dif- ferent components. The final classification rule is given for all x 2 X , there exists Cx 2 R such that by the component-wise application of a sign function: kg(x)k ≤ C kgk ; 8g 2 H: Y x H ybi(x) = sign(gi(x)): Learning Output Kernels with Block Coordinate Descent 3. Learning an output kernel In this section, we introduce and study an optimization problem that can be used to learn simultaneously a 3 vector-valued function and a kernel on the outputs. 2 Q 3.1. Matrix notation 1 m m m 0 Let S++ ⊆ S+ ⊆ M denote the open cone of positive definite matrices, the closed cone of positive semidefi- 1 nite matrices, and the space of square matrices of or- 2 m der m, respectively. For any A; B 2 M , denote the 2.0 T 1.5 Frobenius inner product hA; BiF := tr(A B), and the 1.0 p 0.5 C induced norm kAkF := hA; AiF . We denote the 0.2 T 0.4 0.0 identity matrix as I and, for any matrix A, let A de- 0.6 0.5 y L 0.8 note its transpose, and A its Moore-Penrose pseudo- 1.0 1.0 inverse. The Kronecker product is denoted by ⊗, and the vectorization operator by vec(·). In particular, for Figure 1. The objective function for scalar K and L. Al- matrices A; B; C of suitable size, we use the identity though the level sets are non-convex, stationary points are tr(AT BAC) = vec(A)T (CT ⊗ B)vec(A).
Load more

Recommended publications
  • CORE View Metadata, Citation and Similar Papers at Core.Ac.Uk
    View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Bulgarian Digital Mathematics Library at IMI-BAS Serdica Math. J. 27 (2001), 203-218 FIRST ORDER CHARACTERIZATIONS OF PSEUDOCONVEX FUNCTIONS Vsevolod Ivanov Ivanov Communicated by A. L. Dontchev Abstract. First order characterizations of pseudoconvex functions are investigated in terms of generalized directional derivatives. A connection with the invexity is analysed. Well-known first order characterizations of the solution sets of pseudolinear programs are generalized to the case of pseudoconvex programs. The concepts of pseudoconvexity and invexity do not depend on a single definition of the generalized directional derivative. 1. Introduction. Three characterizations of pseudoconvex functions are considered in this paper. The first is new. It is well-known that each pseudo- convex function is invex. Then the following question arises: what is the type of 2000 Mathematics Subject Classification: 26B25, 90C26, 26E15. Key words: Generalized convexity, nonsmooth function, generalized directional derivative, pseudoconvex function, quasiconvex function, invex function, nonsmooth optimization, solution sets, pseudomonotone generalized directional derivative. 204 Vsevolod Ivanov Ivanov the function η from the definition of invexity, when the invex function is pseudo- convex. This question is considered in Section 3, and a first order necessary and sufficient condition for pseudoconvexity of a function is given there. It is shown that the class of strongly pseudoconvex functions, considered by Weir [25], coin- cides with pseudoconvex ones. The main result of Section 3 is applied to characterize the solution set of a nonlinear programming problem in Section 4. The base results of Jeyakumar and Yang in the paper [13] are generalized there to the case, when the function is pseudoconvex.
    [Show full text]
  • Raport De Autoevaluare - 2012
    RAPORT DE AUTOEVALUARE - 2012 - 1. Date de identificare institut/centru : 1.1. Denumire: INSTITUTUL DE MATEMATICA OCTAV MAYER 1.2. Statut juridic: INSTITUTIE PUBLICA 1.3. Act de infiintare: Hotarare nr. 498 privind trecere Institutului de Matematica din Iasi la Academia Romana, din 22.02.1990, Guvernul Romaniei. 1.4. Numar de inregistrare in Registrul Potentialilor Contractori: 1807 1.5. Director general/Director: Prof. Dr. Catalin-George Lefter 1.6. Adresa: Blvd. Carol I, nr. 8, 700505-Iasi, Romania, 1.7. Telefon, fax, pagina web, e-mail: tel :0232-211150 http://www.iit.tuiasi.ro/Institute/institut.php?cod_ic=13. e-mail: [email protected] 2. Domeniu de specialitate : Mathematical foundations 2.1. Conform clasificarii UNESCO: 12 2.2. Conform clasificarii CAEN: CAEN 7310 (PE1) 3. Stare institut/centru 3.1. Misiunea institutului/centrului, directiile de cercetare, dezvoltare, inovare. Rezultate de excelenta in indeplinirea misiunii (maximum 2000 de caractere): Infiintarea institutului, in 1948, a reprezentat un moment esential pentru dezvoltarea, in continuare, a matematicii la Iasi. Cercetarile in prezent se desfasoara in urmatoarele directii: Ecuatii cu derivate partiale (ecuatii stochastice cu derivate partiale si aplicatii in studiul unor probleme neliniare, probleme de viabilitate si invarianta pentru ecuatii si incluziuni diferentiale si aplicatii in teoria controlului optimal, stabilizarea si controlabilitatea ecuatiilor dinamicii fluidelor, a sistemelor de tip reactie-difuzie, etc.), Geometrie (geometria sistemelor mecanice, geometria lagrangienilor pe fibrate vectoriale, structuri geometrice pe varietati riemanniene, geometria foliatiilor pe varietati semiriemanniene, spatii Hamilton etc), Analiza 1 matematica (analiza convexa, optimizare, operatori neliniari in spatii uniforme, etc.), Mecanica (elasticitate, termoelasticitate si modele generalizate in mecanica mediilor continue).
    [Show full text]
  • First Order Characterizations of Pseudoconvex Functions
    Serdica Math. J. 27 (2001), 203-218 FIRST ORDER CHARACTERIZATIONS OF PSEUDOCONVEX FUNCTIONS Vsevolod Ivanov Ivanov Communicated by A. L. Dontchev Abstract. First order characterizations of pseudoconvex functions are investigated in terms of generalized directional derivatives. A connection with the invexity is analysed. Well-known first order characterizations of the solution sets of pseudolinear programs are generalized to the case of pseudoconvex programs. The concepts of pseudoconvexity and invexity do not depend on a single definition of the generalized directional derivative. 1. Introduction. Three characterizations of pseudoconvex functions are considered in this paper. The first is new. It is well-known that each pseudo- convex function is invex. Then the following question arises: what is the type of 2000 Mathematics Subject Classification: 26B25, 90C26, 26E15. Key words: Generalized convexity, nonsmooth function, generalized directional derivative, pseudoconvex function, quasiconvex function, invex function, nonsmooth optimization, solution sets, pseudomonotone generalized directional derivative. 204 Vsevolod Ivanov Ivanov the function η from the definition of invexity, when the invex function is pseudo- convex. This question is considered in Section 3, and a first order necessary and sufficient condition for pseudoconvexity of a function is given there. It is shown that the class of strongly pseudoconvex functions, considered by Weir [25], coin- cides with pseudoconvex ones. The main result of Section 3 is applied to characterize the solution set of a nonlinear programming problem in Section 4. The base results of Jeyakumar and Yang in the paper [13] are generalized there to the case, when the function is pseudoconvex. The second and third characterizations are considered in Sections 5, 6.
    [Show full text]
  • Invex Sets and Functions
    View metadata, citation and similar papers at core.ac.uk brought to you by CORE Journal of Mathematical Analysis and Applications 263, 355–379 (2001) provided by Elsevier - Publisher Connector doi:10.1006/jmaa.2001.7574, available online at http://www.idealibrary.com on p r -Invex Sets and Functions Tadeusz Antczak Faculty of Mathematics, University of Łod´ ´z, Banacha 22, 90-238 Łod´ ´z, Poland E-mail: [email protected] Submitted by William F. Ames Received January 22, 2001 Notions of invexity of a function and of a set are generalized.The notion of an invex function with respect to η can be further extended with the aid of p-invex sets.Slight generalization of the notion of p-invex sets with respect to η leads to a new class of functions.A family of real functions called, in general, p r -pre- invex functions with respect to η (without differentiability) or p r -invex functions with respect to η (in the differentiable case) is introduced.Some (geometric) prop- erties of these classes of functions are derived.Sufficient optimality conditions are obtained for a nonlinear programming problem involving p r -invex functions with respect to η. 2001 Academic Press Key Words: p r -invex set with respect to η; r-invex set with respect to η; p r -pre-invex function with respect to ηp r -invex function with respect to η. 1.INTRODUCTION Convexity plays a vital role in many aspects of mathematical program- ming including, for example, sufficient optimality conditions and duality theorems.An invex function is one of the generalized convex functions
    [Show full text]
  • Arxiv:1608.04636V4 [Cs.LG]
    Linear Convergence of Gradient and Proximal-Gradient Methods Under the Polyak-Lojasiewicz Condition Hamed Karimi, Julie Nutini, and Mark Schmidt Department of Computer Science, University of British Columbia Vancouver, British Columbia, Canada {[email protected]},{jnutini,schmidtm}@cs.ubc.ca Abstract. In 1963, Polyak proposed a simple condition that is sufficient to show a global linear convergence rate for gradient descent. This condition is a special case of theLojasiewicz inequality proposed in the same year, and it does not require strong convexity (or even convexity). In this work, we show that this much-older Polyak-Lojasiewicz (PL) inequality is actually weaker than the main conditions that have been explored to show linear convergence rates without strong convexity over the last 25 years. We also use the PL inequality to give new analyses of randomized and greedy coordinate descent methods, sign-based gradient descent methods, and stochastic gradient methods in the classic setting (with decreasing or constant step-sizes) as well as the variance- reduced setting. We further propose a generalization that applies to proximal-gradient methods for non-smooth optimization, leading to simple proofs of linear convergence of these methods. Along the way, we give simple convergence results for a wide variety of problems in machine learning: least squares, logistic regression, boosting, resilient backpropagation, L1-regularization, support vector machines, stochastic dual coordinate ascent, and stochastic variance-reduced gradient methods. 1 Introduction Fitting most machine learning models involves solving some sort of optimization problem. Gradient descent, and variants of it like coordinate descent and stochastic gradient, are the workhorse tools used by the field to solve very large instances of these problems.
    [Show full text]
  • Glimpses Upon Quasiconvex Analysis Jean-Paul Penot
    Glimpses upon quasiconvex analysis Jean-Paul Penot To cite this version: Jean-Paul Penot. Glimpses upon quasiconvex analysis. 2007. hal-00175200 HAL Id: hal-00175200 https://hal.archives-ouvertes.fr/hal-00175200 Preprint submitted on 27 Sep 2007 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. ESAIM: PROCEEDINGS, Vol. ?, 2007, 1-10 Editors: Will be set by the publisher DOI: (will be inserted later) GLIMPSES UPON QUASICONVEX ANALYSIS Jean-Paul Penot Abstract. We review various sorts of generalized convexity and we raise some questions about them. We stress the importance of some special subclasses of quasiconvex functions. Dedicated to Marc Att´eia R´esum´e. Nous passons en revue quelques notions de convexit´eg´en´eralis´ee.Nous tentons de les relier et nous soulevons quelques questions. Nous soulignons l’importance de quelques classes particuli`eres de fonctions quasiconvexes. 1. Introduction Empires usually are well structured entities, with unified, strong rules (for instance, the length of axles of carts in the Chinese Empire and the Roman Empire, a crucial rule when building a road network). On the contrary, associated kingdoms may have diverging rules and uses.
    [Show full text]
  • Greed Is Good Greedy Optimization Methods for Large-Scale Structured Problems
    Greed is Good Greedy Optimization Methods for Large-Scale Structured Problems by Julie Nutini B.Sc., The University of British Columbia (Okanagan), 2010 M.Sc., The University of British Columbia (Okanagan), 2012 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in The Faculty of Graduate and Postdoctoral Studies (Computer Science) THE UNIVERSITY OF BRITISH COLUMBIA (Vancouver) May 2018 c Julie Nutini 2018 The following individuals certify that they have read, and recommend to the Faculty of Graduate and Postdoctoral Studies for acceptance, the dissertation entitled: Greed is Good: Greedy Optimization Methods for Large-Scale Structured Problems submitted by Julie Nutini in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Computer Science Examining Committee: Mark Schmidt, Computer Science Supervisor Chen Greif, Computer Science Supervisory Committee Member Will Evans, Computer Science Supervisory Committee Member Bruce Shepherd, Computer Science University Examiner Ozgur Yilmaz, Mathematics University Examiner ii Abstract This work looks at large-scale machine learning, with a particular focus on greedy methods. A recent trend caused by big datasets is to use optimization methods that have a cheap iteration cost. In this category are (block) coordinate descent and Kaczmarz methods, as the updates of these methods only rely on a reduced subspace of the problem at each iteration. Prior to our work, the literature cast greedy variations of these methods as computationally expensive with comparable convergence rates to randomized versions. In this dissertation, we show that greed is good. Specifically, we show that greedy coordinate descent and Kaczmarz methods have efficient implementations and can be faster than their randomized counterparts for certain common problem structures in machine learning.
    [Show full text]
  • Global Optimisation for Energy Systems
    View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by The Australian National University Global Optimisation for Energy Systems Ksenia Bestuzheva October 2018 A thesis submitted for the degree of Doctor of Philosophy of The Australian National University © Copyright by Ksenia Bestuzheva 2018 All Rights Reserved Except where otherwise stated, this thesis is my own original work. Ksenia Bestuzheva October 2018 1 Abstract The goal of global optimisation is to find globally optimal solutions, avoiding local optima and other stationary points. The aim of this thesis is to provide more efficient global optimisation tools for energy systems planning and operation. Due to the ongoing increasing of complex- ity and decentralisation of power systems, the use of advanced mathematical techniques that produce reliable solutions becomes necessary. The task of developing such methods is com- plicated by the fact that most energy-related problems are nonconvex due to the nonlinear Alternating Current Power Flow equations and the existence of discrete elements. In some cases, the computational challenges arising from the presence of non-convexities can be tackled by relaxing the definition of convexity and identifying classes of problems that can be solved to global optimality by polynomial time algorithms. One such property is known as invexity and is defined by every stationary point of a problem being a global optimum. This thesis investigates how the relation between the objective function and the structure of the feasible set is connected to invexity and presents necessary conditions for invexity in the general case and necessary and sufficient conditions for problems with two degrees of freedom.
    [Show full text]
  • International Journal of Pure and Applied Mathematics ————————————————————————– Volume 15 No
    International Journal of Pure and Applied Mathematics ————————————————————————– Volume 15 No. 2 2004, 137-250 FUNDAMENTALS OF MIXED QUASI VARIATIONAL INEQUALITIES Muhammad Aslam Noor Etisalat College of Engineering P.O. Box 980, Sharjah, UNITED ARAB EMIRATES e-mail: [email protected] Abstract: Mixed quasi variational inequalities are very important and sig- nificant generalizations of variational inequalities involving the nonlinear bi- function. It is well-known that a large class of problems arising in various branches of pure and applied sciences can be studied in the general framework of mixed quasi variational inequalities. Due to the presence of the bifunction in the formulation of variational inequalities, projection method and its variant forms cannot be used to suggest and analyze iterative methods for mixed quasi variational inequalities. In this paper, we suggest and analyze various itera- tive schemes for solving these variational inequalities using resolvent methods, resolvent equations and auxiliary principle techniques. We discuss the sensitiv- ity analysis, stability of the dynamical systems and well-posedness of the mixed quasi variational inequalities. We also consider and investigate various classes of mixed quasi variational inequalities in the setting of invexity, g-convexity and uniformly prox-regular convexity. The concepts of invexity, g-convexity and uniformly prox-regular convexity are generalizations of the classical convexity in different directions. Some classes of equilibrium problems are introduced and studied. We also suggest several open problems with sufficient information and references. AMS Subject Classification: 49J40, 90C30, 35A15, 47H17 Key Words: mixed quasi variational inequalities, auxiliary principle, fixed- points, resolvent equations, iterative methods, convergence, dynamical systems, global convergence, stability Received: March 11, 2004 c 2004, Academic Publications Ltd.
    [Show full text]
  • Local Cone Approximations in Optimization
    Control and Cybernetics vol. 36 (2007) No. 3 Local cone approximations in optimization by M. Castellani1 and M. Pappalardo2 1 Department of “Sistemi ed Istituzioni per l’Economia” University of L’Aquila 2 Department of “Matematica Applicata” University of Pisa Abstract: We show how to use intensively local cone approxi- mations to obtain results in some fields of optimization theory such as optimality conditions, constraint qualifications, mean value theo- rems and error bound. Keywords: nonsmooth optimization, cone approximations, generalized directional epiderivatives, optimality conditions, con- straint qualifications, mean value theorem, error bound. 1. Introduction Many approaches have been developed in literature for obtaining results in op- timization theory. In this paper we propose a systematic approach to many problems of optimization theory (i.e. necessary and sufficient optimality con- ditions, constraint qualifications, mean value theorems and error bound) which uses local cone approximations of sets. This concept, introduced by Elster and Thierfelder (1988) and studied also in Castellani and Pappalardo (1995), appears to be very useful when used together with separation theorems and generalized derivatives. When intensively applied, we show that it is very flexible in our scope of interest. The first step of this scheme consists in approximating a set with a cone; in particular, when the set is the epigraph of a function, the cone represents the epigraph of a positively homogeneous function, which will be called generalized derivative. The subsequent step consists in considering the case in which the generalized derivative is the difference between two sublinear functions; finally, the most general situation, is the one in which the generalized derivative is a minimum of sublinear functions.
    [Show full text]
  • Learning Output Kernels with Block Coordinate Descent
    Learning output kernels with block coordinate descent Francesco Dinuzzo [email protected] Max Planck Institute for Intelligent Systems, 72076 T¨ubingen,Germany Cheng Soon Ong [email protected] Department of Computer Science, ETH Z¨urich, 8092 Z¨urich, Switzerland Peter Gehler [email protected] Max Planck Institute for Informatics, 66123 Saarbr¨ucken, Germany Gianluigi Pillonetto [email protected] Department of Information Engineering, University of Padova, 35131 Padova, Italy Abstract the outputs. In this paper, we introduce a method that We propose a method to learn simultane- simultaneously learns a vector-valued function and the ously a vector-valued function and a kernel kernel between the components of the output vector. between its components. The obtained ker- The problem is formulated within the framework of nel can be used both to improve learning per- regularization in RKH spaces. We assume that the formance and to reveal structures in the out- matrix-valued kernel can be decomposed as the prod- put space which may be important in their uct of a scalar kernel and a positive semidefinite kernel own right. Our method is based on the so- matrix that represents the similarity between the out- lution of a suitable regularization problem put components, a structure that has been considered over a reproducing kernel Hilbert space of in a variety of works, (Evgeniou et al., 2005; Capon- vector-valued functions. Although the regu- netto et al., 2008; Baldassarre et al., 2010). larized risk functional is non-convex, we show In practice, an important issue is the choice of the ker- that it is invex, implying that all local min- nel matrix between the outputs.
    [Show full text]
  • Higher Order Strongly Exponentially Preinvex Functions
    J. Appl. Math. & Informatics Vol. 39(2021), No. 3 - 4, pp. 469 - 485 https://doi.org/10.14317/jami.2021.469 HIGHER ORDER STRONGLY EXPONENTIALLY PREINVEX FUNCTIONS MUHAMMAD ASLAM NOOR∗ AND KHALIDA INAYAT NOOR Abstract. In this paper, some new classes of the higher order strongly exponentially preinvex functions are introduced. New relationships among various concepts of higher order strongly exponentially preinvex functions are established. It is shown that the optimality conditions of differentiable higher order strongly exponentially preinvex functions can be character- ized by exponentially variational-like inequalities. Parallelogram laws for Banach spaces are obtained as an application. As special cases, one can obtain various new and known results from our results. Results obtained in this paper can be viewed as refinement and improvement of previously known results. AMS Mathematics Subject Classification : 26D15, 49J40, 90C33. Key words and phrases : Convex functions, exponentially preinvex func- tions, monotone operators, Wrght exponentially preinvex functions, variational- like inequalities. 1. Introduction Recently, several extensions and generalizations have been considered for classical convexity. Strongly convex functions were introduced and studied by Polyak [51], which play an important part in the optimization theory and related areas. Karmardian [18] used the strongly convex functions to discuss the unique existence of a solution of the nonlinear complementarity problems. Strongly convex functions are used to investigate the convergence analysis for solving variational inequalities and equilibrium problems, see Zu and Marcotte [59]. See also Nikodem and Pales [19] and Qu and Li [49]. Awan et al[7, 8, 9] have derived Hermite-Hadamard type inequalities, which provide upper and lower estimate for the integrand.
    [Show full text]