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

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