2016/10/17 by Fan Zhou, Zhou Fan, Andrea Montanari +2 · 1 citation
Computer Science · Engineering · Mathematics · #Complexity and Algorithms in Graphs #Discrete Mathematics (cs.DM) #FOS: Computer and information sciences #FOS: Mathematics #Machine Learning (stat.ML) #Optimization and Control (math.OC) #Sparse and Compressive Sensing Techniques #Stochastic Gradient Optimization Techniques #cs.DM #math.OC #stat.ML
paper · pdf · doi:10.48550/arxiv.1610.05350
48 pages, 1 pdf figure
arxiv created 2016/10/17 · openalex publication_date 2016/10/17 · arxiv updated 2016/10/19 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Several probabilistic models from high-dimensional statistics and machine learning reveal an intriguing --and yet poorly understood-- dichotomy. Either simple local algorithms succeed in estimating the object of interest, or even sophisticated semi-definite programming (SDP) relaxations fail. In order to explore this phenomenon, we study a classical SDP relaxation of the minimum graph bisection problem, when applied to Erdős-Renyi random graphs with bounded average degree d>1, and obtain several types of results. First, we use a dual witness construction (using the so-called non-backtracking matrix of the graph) to upper bound the SDP value. Second, we prove that a simple local algorithm approximately solves the SDP to within a factor 2d2/(2d2+d-1) of the upper bound. In particular, the local algorithm is at most 8/9 suboptimal, and 1+O(1/d) suboptimal for large degree. We then analyze a more sophisticated local algorithm, which aggregates information according to the harmonic measure on the limiting Galton-Watson (GW) tree. The resulting lower bound is expressed in terms of the conductance of the GW tree and matches surprisingly well the empirically determined SDP values on large-scale Erdős-Renyi graphs. We finally consider the planted partition model. In this case, purely local algorithms are known to fail, but they do succeed if a small amount of side information is available. Our results imply quantitative bounds on the threshold for partial recovery using SDP in this model.