Rate optimal Chernoff bound and application to community detection in the stochastic block models

Abstract

Chernoff coefficient is an upper bound of Bayes error probability in classification problem. In this paper, we will develop sharp Chernoff type bound on Bayes error probability. The new bound is not only an upper bound but also a lower bound of Bayes error probability up to a constant in a non-asymptotic setting. Moreover, we will apply this result to community detection in stochastic block model. As a clustering problem, the optimal error rate of community detection can be characterized by our Chernoff type bound. This can be formalized by deriving a minimax error rate over certain class of parameter space, then achieving such error rate by a feasible algorithm employ multiple steps of EM type updates.

Achieving the Bayes Error Rate in Synchronization and Block Models by SDP, Robustly

“…They also provide an exponential-time algorithm that achieves a matching upper bound (up to an o(1) factor in the exponent). Much research effort focuses on developing computationally feasible algorithms, and identifying the minimax rates in more general settings [26,27,59,60,61,63,64]. The monograph [25] provides a review on recent work on this front.…”

“…The assumption c 0 p ≤ q for Model 3 is common in the literature on minimax rates [26,27,63,64]. It stipulates that p and q are on the same order (but their difference can be vanishingly small).…”

Optimal Non-Convex Exact Recovery in Stochastic Block Model via Projected Power Method

“…Then, it is of interest to design computationally tractable methods that can achieve exact recovery under a condition on α and β that meets the information-theoretic limit. In the past years, many algorithms have been proposed to achieve this task, such as spectral clustering (McSherry, 2001;Su et al, 2019;Yun & Proutiere, 2014;2016), SDP-based approach (Amini et al, 2018;Fei & Chen, 2018;2020;Li et al, 2018), and likelihood-based approach (Amini et al, 2013;Gao et al, 2017;Zhang & Zhou, 2016;Zhou & Li, 2020). However, most of these algorithms have a time complexity that is at least quadratic in n, which usually does not scale well to large-scale problems.…”