/pdf/convex-analysisnoer-san-nojin-zhan-nituite-5dl2rbmoyr.pdf

Convex Analysisの二,三の進展について

“Multivalued Analysis” is the theory of set-valued maps (called multifonctions) and has important applications in many different areas. Multivalued analysis is a remarkable mixture of many different parts of mathematics such as point-set topology, measure theory and nonlinear functional analysis. It is also closely related to “Nonsmooth Analysis” (Chapter 5) and in fact one of the main motivations behind the development of the theory, was in order to provide necessary analytical tools for the study of problems in nonsmooth analysis. It is not a coincidence that the development of the two fields coincide chronologically and follow parallel paths. Today multivalued analysis is a mature mathematical field with its own methods, techniques and applications that range from social and economic sciences to biological sciences and engineering. There is no doubt that a modern treatise on “Nonlinear Functional Analysis” can not afford the luxury of ignoring multivalued analysis. The omission of the theory of multifunctions will drastically limit the possible applications.

SET-Valued Analysis

Theory of multiobjective optimization

The goal of this paper is to analyze the intriguing instability of classifiers to adversarial perturbations (Szegedy et al., in: International conference on learning representations (ICLR), 2014). We provide a theoretical framework for analyzing the robustness of classifiers to adversarial perturbations, and show fundamental upper bounds on the robustness of classifiers. Specifically, we establish a general upper bound on the robustness of classifiers to adversarial perturbations, and then illustrate the obtained upper bound on two practical classes of classifiers, namely the linear and quadratic classifiers. In both cases, our upper bound depends on a distinguishability measure that captures the notion of difficulty of the classification task. Our results for both classes imply that in tasks involving small distinguishability, no classifier in the considered set will be robust to adversarial perturbations, even if a good accuracy is achieved. Our theoretical framework moreover suggests that the phenomenon of adversarial instability is due to the low flexibility of classifiers, compared to the difficulty of the classification task (captured mathematically by the distinguishability measure). We further show the existence of a clear distinction between the robustness of a classifier to random noise and its robustness to adversarial perturbations. Specifically, the former is shown to be larger than the latter by a factor that is proportional to $$\sqrt{d}$$
 (with d being the signal dimension) for linear classifiers. This result gives a theoretical explanation for the discrepancy between the two robustness properties in high dimensional problems, which was empirically observed by Szegedy et al. in the context of neural networks. We finally show experimental results on controlled and real-world data that confirm the theoretical analysis and extend its spirit to more complex classification schemes.

/pdf/analysis-of-classifiers-robustness-to-adversarial-jrvl63m6gu.pdf

Analysis of classifiers’ robustness to adversarial perturbations

Well, someone can decide by themselves what they want to do and need to do but sometimes, that kind of person will need some theory of vector optimization references. People with open minded will always try to seek for the new things and information from many sources. On the contrary, people with closed mind will always think that they can do it by their principals. So, what kind of person are you?

Theory Of Vector Optimization

In this paper we derive new fractional error bounds for polynomial systems with exponents explicitly determined by the dimension of the underlying space and the number/degree of the involved polynomials. Our major result extends the existing error bounds from the system involving only a single polynomial to a general polynomial system and do not require any regularity assumptions. In this way we resolve, in particular, some open questions posed in the literature. The developed techniques are largely based on variational analysis and generalized differentiation, which allow us to establish, e.g., a nonsmooth extension of the seminal ?ojasiewicz's gradient inequality to maxima of polynomials with explicitly determined exponents. Our major applications concern quantitative Holderian stability of solution maps for parameterized polynomial optimization problems and nonlinear complementarity systems with polynomial data as well as high-order semismooth properties of the eigenvalues of symmetric tensors.

http://www.optimization-online.org/DB_FILE//2013/01/3726.pdf

New fractional error bounds for polynomial systems with applications to Hölderian stability in optimization and spectral theory of tensors

The paper addresses parametric inequality systems described by polynomial functions in finite dimensions, where state-dependent infinite parameter sets are given by finitely many polynomial inequalities and equalities. Such systems can be viewed, in particular, as solution sets to problems of generalized semi-infinite programming with polynomial data. Exploiting the imposed polynomial structure together with powerful tools of variational analysis and semialgebraic geometry, we establish a far-going extension of the \L ojasiewicz gradient inequality to the general nonsmooth class of supremum marginal functions as well as higher-order (H\"older type) local error bounds results with explicitly calculated exponents. The obtained results are applied to higher-order quantitative stability analysis for various classes of optimization problems including generalized semi-infinite programming with polynomial data, optimization of real polynomials under polynomial matrix inequality constraints, and polynomial second-order cone programming. Other applications provide explicit convergence rate estimates for the cyclic projection algorithm to find common points of convex sets described by matrix polynomial inequalities and for the asymptotic convergence of trajectories of subgradient dynamical systems in semialgebraic settings.

Error Bounds for Parametric Polynomial Systems with Applications to Higher-Order Stability Analysis and Convergence Rates

An explicit bound for the Łojasiewicz exponent of real polynomials

In this paper, we study the existence of optimal solutions to a constrained polynomial optimization problem. More precisely, let $$f_0$$ f 0 and $$f_1, \ldots , f_p :{\mathbb {R}}^n \rightarrow {\mathbb {R}}$$ f 1 , ? , f p : R n ? R be convenient polynomial functions, and let $$S := \{x \in {\mathbb {R}}^n \ : \ f_i(x) \le 0, i = 1, \ldots , p\} \ne \emptyset .$$ S : = { x ? R n : f i ( x ) ≤ 0 , i = 1 , ? , p } ? ? . Under the assumption that the map $$(f_0, f_{1}, \ldots , f_{p}) :{\mathbb {R}}^n \rightarrow {\mathbb {R}}^{p + 1}$$ ( f 0 , f 1 , ? , f p ) : R n ? R p + 1 is non-degenerate at infinity, we show that if $$f_0$$ f 0 is bounded from below on $$S,$$ S , then $$f_0$$ f 0 attains its infimum on $$S.$$ S .

A Frank---Wolfe type theorem for nondegenerate polynomial programs

In this paper, we define the geometric and algebraic tangent cones at infinity of algebraic varieties and establish the following version at infinity of Whitney’s theorem [Local properties of analy...

Tien-Son Pham

Papers

New fractional error bounds for polynomial systems with applications to Hölderian stability in optimization and spectral theory of tensors

Error Bounds for Parametric Polynomial Systems with Applications to Higher-Order Stability Analysis and Convergence Rates

An explicit bound for the Łojasiewicz exponent of real polynomials

A Frank---Wolfe type theorem for nondegenerate polynomial programs

On tangent cones at infinity of algebraic varieties