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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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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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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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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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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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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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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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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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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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]

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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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 ,

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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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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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.

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

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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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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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

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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