Generalized Gauss inequalities via semidefinite programming

Abstract

A sharp upper bound on the probability of a random vector falling outside a polytope, based solely on the first and second moments of its distribution, can be computed efficiently using semidefinite programming. However, this Chebyshev-type bound tends to be overly conservative since it is determined by a discrete worst-case distribution. In this paper we obtain a less pessimistic Gauss-type bound by imposing the additional requirement that the random vector's distribution must be unimodal. We prove that this generalized Gauss bound still admits an exact and tractable semidefinite representation. Moreover, we demonstrate that both the Chebyshev and Gauss bounds can be obtained within a unified framework using a generalized notion of unimodality. We also offer new perspectives on the computational solution of generalized moment problems, since we use concepts from Choquet theory instead of traditional duality arguments to derive semidefinite representations for worst-case probability bounds. Recent generalizations of the Chebyshev inequality (1) provide upper bounds on the probability of a multivariate random vector ξ ∈ R n falling outside a prescribed confidence region Ξ ⊆ R n if only a few low-order moments of ξ

A distributionally robust perspective on uncertainty quantification and chance constrained programming

“…If we intersect the nested moment ambiguity set from Example 1 with the structural ambiguity set of all unimodal distributions, then we recover an ambiguity set that has been studied in [38] and is closely related to a tightened Chebyshev-type inequality due to Gauss [21].…”

“…Applying a coordinate shift and employing Theorem 5 allows us to reformulate the uncertainty quantification problem (1) over the Gauss ambiguity set as a semi-infinite program. Thereby, we recover a generalized multivariate Gauss inequality that was discovered in [38]. Since g α (z) = α α+2 zz can be computed explicitly, we may use standard robust optimization techniques to further simplify this semi-infinite program to the finite convex program…”

“…If α is a rational number exceeding 1, that is, if α = p/q for some p, q ∈ N with p ≥ q, then the epigraph of this term can be expressed through O(p) second-order conic constraints, see e.g. [38,Lemma 4.2]. In analogy to the formulation (23) of Theorem 5, the tractability of (28) depends on the functional form of g α (·).…”

“…[38]. To our best knowledge, the complexity of the joint chance constrained program (2) has not been settled for S(x) = S and for P generated by the intersection of a conic moment ambiguity set and the structural ambiguity set P α of all unimodal distributions.…”

“…Ambiguity sets of special interest include the Markov ambiguity set containing all distributions with known mean and support [48], the Chebyshev ambiguity set containing all distributions with known bounds on the first and second-order moments [12,14,22,31,39,46,49,51,52], the Gauss ambiguity set containing all unimodal distributions from within the Chebyshev ambiguity set [38,41], various generalized Chebyshev ambiguity sets that specify asymmetric moments [12,13,35], higher-order moments [7,30,45] or marginal moments [17,18], the median-absolute deviation ambiguity set containing all symmetric distributions with known median and mean absolute deviation [24], the Huber ambiguity set containing all distributions with known upper bound on the expected Huber loss function [15,48], the Wasserstein ambiguity set containing all distributions that are close to the empirical distribution with respect to the Wasserstein metric [19,34,40], the KullbackLeibler divergence ambiguity set and likelihood ratio ambiguity set [10,26,27,31,47] containing all distributions that are sufficiently likely to have generated a given data set, the Hoeffding ambiguity set containing all component-wise independent distributions with a box support [3,8,10], the Bernstein ambiguity set containing all distributions from within the Hoeffding ambiguity set subject to marginal moment bounds [36], several φ-divergence-based ambiguity sets [2,…”