Convex Risk

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Perturbation of convex risk minimization and its application in differential private learning algorithms

Perturbation of convex risk minimization and its application in differential private learning algorithms

The first one is the perturbation results for general convex risk minimization algorithms. We studied two cases of the general algorithms. The second one is applied in the following analysis, as it leads to a sharper upper bound of the error between two outputs differ in  sample point. However, the first one is more relaxed, without Lipschitz continuity of the loss function. Based on such perturbation results we obtain a choice for the random terms of the differential private algorithms, i.e., Proposition . This gives us a theoretical and practical construction of differential private algorithms.
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On Robustness Properties of Convex Risk Minimization Methods for Pattern Recognition

On Robustness Properties of Convex Risk Minimization Methods for Pattern Recognition

The paper brings together methods from two disciplines: machine learning theory and robust statis- tics. We argue that robustness is an important aspect and we show that many existing machine learning methods based on the convex risk minimization principle have − besides other good prop- erties − also the advantage of being robust. Robustness properties of machine learning methods based on convex risk minimization are investigated for the problem of pattern recognition. As- sumptions are given for the existence of the influence function of the classifiers and for bounds on the influence function. Kernel logistic regression, support vector machines, least squares and the AdaBoost loss function are treated as special cases. Some results on the robustness of such methods are also obtained for the sensitivity curve and the maxbias, which are two other robustness criteria. A sensitivity analysis of the support vector machine is given.
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Real-valued conditional convex risk measures in
            Lp(ℱ, R)

Real-valued conditional convex risk measures in Lp(ℱ, R)

function satisfies the coerciveness property of Definition 2.8. In particular, this property of coerciveness char- acterizes real-valued conditional convex risk measures in the spaces L p (F , R) in terms of the corresponding penalty functions. This fact, together with the invariance property of Proposition 5.1 will allow us to character- ize conditional convex risk measures defined in a space L ∞ (F, R) which can be extended to a space L p (F, R) and assign values in the space L 1 (G, R); see Remark 5.2.

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Svindland, Gregor
  

(2009):


	Convex Risk Measures Beyond Bounded Risks.


Dissertation, LMU München: Fakultät für Mathematik, Informatik und Statistik

Svindland, Gregor (2009): Convex Risk Measures Beyond Bounded Risks. Dissertation, LMU München: Fakultät für Mathematik, Informatik und Statistik

over all allocations (Y 1 , . . . , Y n ). By cash-invariance an optimal allocation ( Y e 1 , . . . , Y e n ) does not only minimise the group risk (0.1), but by rebalancing the cash we may assume that ρ( Y e i ) ≤ ρ(X i ) too. Thus, we obtain a reduction of group and individual risk. In particular optimal allocations are optimal in Pareto’s sense. The optimal allocation problem has been studied by several authors, see e.g. [5, 23, 16, 7, 1, 29]. We devote chapter 3 to it. Here it is proved that if our model space is L p and if the convex risk measures ρ i are in addition law-invariant, then there always exists an optimal allocation
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To split or not to split: capital allocation with convex risk measures

To split or not to split: capital allocation with convex risk measures

A weaker requirement on risk measures than positive homogeneity / subadditivity is convexity, proposed by Deprez and Gerber (1985), who in- troduce convex risk measures and study them in the context of optimal risk exchanges. While convexity still acknowledges diversification, risk capital is no more scale-independent – in fact it is increasing per unit of exposure. Moreover, it is possible that for some portfolios pooling increases aggregate risk. Convex risk measures were introduced in the mathematical finance literature by F¨ ollmer and Schied (2002) and Frittelli and Rosazza Gianin (2002) and spurred a lively research area including dynamic generalisations (Detlefsen and Scandolo, 2005).
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To split or not to split: Capital allocation with convex risk measures

To split or not to split: Capital allocation with convex risk measures

In Section 4 conditions are examined under which the Aumann-Shapley allocation produces incentives for the splitting of portfolios. If such split- ting turns out to be beneficial in the sense of savings in aggregate capital, the question arises as to how the portfolio should be optimally restructured. By transferring the results in Barrieu and El Karoui (2005) to the present context, it is demonstrated that using a (non-homogenous) convex risk mea- sure for capital allocation produces an incentive for infinite fragmentation of portfolios. This clarifies some of the difficulties for using convex risk measures in a risk management context. It is argued that the rather uncom- fortable issue of portfolio fragmentation can be addressed by posing some cost-induced constraints to the extent that portfolios can be split.
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Classification Methods with Reject Option Based on Convex Risk Minimization

Classification Methods with Reject Option Based on Convex Risk Minimization

We have focused thus far on the case where misclassifying from one class to the other, either g(X ) = 1 while Y = − 1 or g(X) = − 1 while Y = 1, is assigned the same loss. In many applications, however, one type of misclassification may incur a heavier loss than the other. Such situations naturally arise in risk management or medical diagonsis. To this end, the following loss function can be adopted in place of ℓ:

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Nonlinear and evolutionary phenomena in deterministic growing economies

Nonlinear and evolutionary phenomena in deterministic growing economies

Our first proposal deals with an economy populated by representative agents seeking to maximize consumption utility by taking optimal consumption and in- vestment decisions. Agents face convex risk premium on bonds and investment adjustment costs in their budget constraint and accumulate productive capital lin- early. This economic setup is correctly described by an aggregate intertemporal maximization problem, where the budget constraint describes the national account identity for an open economy. The optimal solution to the dynamic optimization problem is given by a nonlinear three-dimensional autonomous dynamical system. This framework is an optimal candidate to test evolutionary economic ideas in an modern orthodox decision setup. The low dimensional dynamics approach to eco- nomic phenomena is an old tradition in evolutionary economics. Early proposals on the coexistence of cycles and growth by Michal Kalecky, Nicholas Kaldor and Richard Goodwin, for example, modelled economies as two-dimensional vector fields. Our proposal goes a step further to deliver a system with multiple equilibria and several bifurcation phenomena, such as the fold-hopf bifurcation. This result is of particu- lar interest. First, fold-hopf bifurcations have the potential to unleash a cascade of complex nonlinear phenomena. Second, to our knowledge, this proposal is the first in the field of economic growth applications to show the existence of this bifurca- tion. The fold-hopf bifurcation and associated nonlinear global phenomena has been gaining attention in several fields of applied mathematics. This is not yet the case in economic dynamics literature, despite the potential of this bifurcation scenario to explain empirically observed macroeconomic phenomena, such as structural change dynamics, as a result of complex global dynamics. We focus our analysis on this and other global meaningful conjectures, and give a thorough description of local dynamics and the global organization of the phase space for this economy. We also portray the existence of nonlinear phenomena arising from the complex organization of the system’s invariant manifolds, and discuss the challenges posed to policy in this environment.
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Robustness Regions for Measures of Risk Aggregation

Robustness Regions for Measures of Risk Aggregation

Thus, neither convex risk measures such as ES nor VaR, are robust on L 1 . VaR requires strictly increasing distribution functions. Convex risk measures like ES place requirements on the tail of the underlying dis- tribution functions via the uniform integrability condition, see Theorem 3.5. A comparative assessment of those two risk measures thus relies on whether strict increasingness or uniform integrability is a more real- istic constraint on the set of distributions on which the risk measure is to be evaluated. This depends on the context of the application. For example, in reinsurance problems where distributions with constant parts can occur, uniform integrability may be a more suitable assumption. On the other hand, when dealing with an asset return with an approximately bell-shaped density but arbitrarily heavy tails, strict increasingness of the distribution appears to be a more appropriate condition.
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Risk Measures and Nonlinear Expectations

Risk Measures and Nonlinear Expectations

We show that 1 in the family of convex risk measures, only coherent risk measures satisfy Jensen’s inequality; 2 coherent risk measures are always bounded by the corresponding Choquet ex[r]

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Regularized Bundle Methods for Convex and Non-Convex Risks

Regularized Bundle Methods for Convex and Non-Convex Risks

where t is the number of cutting planes (it is equal to the iteration number in CRBM) and D is the dimensionality of w. In addition, the computational cost for solving the dual program is usually quadratic or cubic in t. These costs may be prohibitive especially in situations where the objective is hard to optimize and the algorithm requires a large number of iterations to converge (e.g. weak regularization), where t may become very large. For instance, in experiments of training a linear SVM for adult data set (Teo et al., 2007), CRBM requires thousands of iterations for small values of λ. To overcome such an issue and to make our NRBM practical for large scale and difficult optimization problems we propose a limited memory mechanism. It is based on the use of a cutting plane aggregation method which allows drastically limiting the number of CPs in the working set at the price of a less accurate underestimator approximation. Note that such a limited memory variant may be used with convex and non-convex risks. Also, this limited memory variant applied to convex risks may be shown to inherit the convergence rate (w.r.t. the number of iterations) of CRBM, while the cost of every iteration does not depend on the iteration number anymore.
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fis convex if its epigraph is a convex set, andf is closed

fis convex if its epigraph is a convex set, andf is closed

The concept of the subgradient is a simple generalization of the gradient for nondifferentiable convex functions. Lemma 1. Let f : S → R ∪ { } + ∞ be a convex function defined on a convex set S ⊆ R n , and x ′ ∈ int S . Let { } x k be a sequence such that x k → x ′ , where x k = x ′ + ε k s k ,

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Convex functions

Convex functions

First Set of criteria for convexity.. LISTGFHGURES Figure 1... To show that f is convex, we have to verify that the following inequality holds.. i displays a geometric interpTetatio[r]

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On convex permutations

On convex permutations

The class of permutations order isomorphic to finite generic subsets of a circle has been studied by Vatter and Waton [12]. Here we consider a much larger class, the class of all convex permutations, i.e., those permutations order isomorphic to a finite convex generic set. For example, the permutation 1243 is convex, as can be seen from Figure 1. Note that the “standard” drawing of 1243 — the drawing with integer coordinates on the right of Figure 1 — is not convex. There are far fewer permutations whose standard drawing is convex than there are convex permutations 1 . It is clear that the set of all convex permuta-
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Some properties of harmonic convex and harmonic quasi-convex functions

Some properties of harmonic convex and harmonic quasi-convex functions

In this paper, we introduce a new class of convex function which is known as harmonically convex function. It is shown that harmonically log-convex function implies that harmonically convex functions which implies that harmonically quasi-convex functions. Results proved in this paper may stimulate further research in this field.

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Unsupervised Data Classification for Convex and Non Convex Classes

Unsupervised Data Classification for Convex and Non Convex Classes

In this experiment, two types of simulation are carried out. For both simulations the classes have convex form, we have three classes in each case, and the data have been generated by a Gaussian distribution routine through M atlab. The difference between the two simulations can be seen in the number of data, the level of overlapping between the classes and the noisy data introduced in the second simulation. For both simulations we set S Split =0.8, S Clean =10 and S Merge =0.2

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Minimum Risk Training of Approximate CRF Based NLP Systems

Minimum Risk Training of Approximate CRF Based NLP Systems

Motivated by the recently proposed method of Stoy- anov et al. (2011) for minimum-risk training of CRF-based systems, we revisited three NLP do- mains that can naturally be modeled with approx- imate CRF-based systems. These include appli- cations that have not been modeled with CRFs before (the ConVote corpus), as well as applica- tions that have been modeled with loopy CRFs trained to minimize the approximate log-likelihood (semi-structured information extraction and collec- tive multi-label classification). We show that (i) the NLP models are improved by moving to richer CRFs that require approximate inference, and (ii) empirical performance is always significantly im- proved by training to reduce the loss that would be achieved by approximate inference, even compared to another state-of-the-art training method (softmax- margin) that also considers loss and uses approxi- mate inference. The general software package that implements the algorithms in this paper is avail- able at http://www.clsp.jhu.edu/˜ves/ software.html.
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On Hadamard and Fej\'{e}r-Hadamard inequalities for Caputo $\small{k}$-fractional derivatives

On Hadamard and Fej\'{e}r-Hadamard inequalities for Caputo $\small{k}$-fractional derivatives

In this paper, in Section 2 we define Caputo k-fractional derivatives and utilize them to give the Hadamard inequality for functions whose nth derivatives are convex. We also find the bound of a difference of this inequality. In Section 3 we derive the Fej´ er–Hadamard inequality via Caputo k– fractional derivatives and find bounds of a difference of this inequality. We also deduce some related results.

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On Convexity and Approximating the Perimeter of an Ellipse

On Convexity and Approximating the Perimeter of an Ellipse

x ∈Ν ⊂ S be a dense subset and assume x 0 to be a point of intS. Denote by C x ( ) 0 the family of all convex, closed subsets of S, containing x 0 in their relative interior with respect to S. Endow the nonempty family C x ( ) 0 with the order relation defined by set inclusion relation. Every totally ordered family ( ) ( ) 0

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Convex Quadratic Optimization Based on Generator Matrix in Credit Risk Transfer Process

Convex Quadratic Optimization Based on Generator Matrix in Credit Risk Transfer Process

Abstract. In this paper, the generator matrix is calculated based on the grade transfer matrix over a period of time in the quantitative analysis of credit risk, which has become one of the hot issues that many scholars pay attention to and study.From the mathematical model point of view combined with the actual situation, the calculation of the generated matrix can be regarded as optimization problems of the matrix logarithm. This article is based on Prof Kreinin and Prof Sidelnikova’s classical optimization model, and we use the combination model of Markov chain and the effective set method of convex quadratic optimization to solve the problem. We mainly make the feasible domain satisfy the applicable conditions by reducing dimensions and correcting variables and theoretically verify the convergence and feasibility of the optimization system. Meanwhile, we verify the data published by Standard and Poor’s Credit Review by Matlab.
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