A note on optimal probability lower bounds for centered random variables

In this note we obtain lower bounds for $\P(\xi\geq 0)$ and $\P(\xi>0)$ under assumptions on the moments of a centered random variable $\xi$. The obtained estimates are shown to be optimal and improve results from the literature. The results are applied to obtain probability lower bounds for second order Rademacher chaos.Comment: Some typos corrected. To appear in Colloquium Mathematicu

Learning with Semi-Definite Programming: new statistical bounds based on fixed point analysis and excess risk curvature

Many statistical learning problems have recently been shown to be amenable to Semi-Definite Programming (SDP), with community detection and clustering in Gaussian mixture models as the most striking instances [javanmard et al., 2016]. Given the growing range of applications of SDP-based techniques to machine learning problems, and the rapid progress in the design of efficient algorithms for solving SDPs, an intriguing question is to understand how the recent advances from empirical process theory can be put to work in order to provide a precise statistical analysis of SDP estimators. In the present paper, we borrow cutting edge techniques and concepts from the learning theory literature, such as fixed point equations and excess risk curvature arguments, which yield general estimation and prediction results for a wide class of SDP estimators. From this perspective, we revisit some classical results in community detection from [gu\'edon et al.,2016] and [chen et al., 2016], and we obtain statistical guarantees for SDP estimators used in signed clustering, group synchronization and MAXCUT

Semidefinite Relaxation Bounds for Indefinite Homogeneous Quadratic Optimization

This paper studies the relationship between the optimal value of a homogeneous quadratic optimization problem and that of its Semidefinite Programming (SDP) relaxation. We consider two quadratic optimization models: (1) min{x ∗ Cx | x ∗ Akx ≥ 1, k = 0, 1,..., m, x ∈ F n} and (2) max{x ∗ Cx | x ∗ Akx ≤ 1, k = 0, 1,..., m, x ∈ F n}, where F is either the real field R or the complex field C, and Ak, C are symmetric matrices. For the minimization model (1), we prove that, if the matrix C and all but one of Ak’s are positive semidefinite, then the ratio between the optimal value of (1) and its SDP relaxation is upper bounded by O(m 2) when F = R, and by O(m) when F = C. Moreover, when two or more of Ak’s are indefinite, this ratio can be arbitrarily large. For the maximization model (2), we show that, if C and at most one of Ak’s are indefinite while other Ak’s are positive semidefinite, then the ratio between the optimal value of (2) and its SDP relaxation is bounded from below by O(1 / log m) for both the real and complex case. This result improves the bound based on the so-called approximate S-Lemma of Ben-Tal et al. [3]. When two or more of Ak in (2) are indefinite, we derive a general bound in terms of the problem data and the SDP solution. For both optimization models, we present examples to show that the derive