Robust recovery for stochastic block models

TL;DR

This paper introduces an efficient algorithm for robust weak recovery in stochastic block models that matches the best known guarantees, demonstrating no additional cost for robustness and employing novel convex optimization techniques.

Contribution

The work presents the first robust recovery algorithm for stochastic block models that handles complex optimization landscapes and introduces a general success probability boosting technique.

Findings

Algorithm achieves statistical guarantees comparable to non-robust algorithms.

First to handle push-out effects in non-asymptotic settings.

Provides a framework potentially applicable to other robust matrix estimation problems.

Abstract

We develop an efficient algorithm for weak recovery in a robust version of the stochastic block model. The algorithm matches the statistical guarantees of the best known algorithms for the vanilla version of the stochastic block model. In this sense, our results show that there is no price of robustness in the stochastic block model. Our work is heavily inspired by recent work of Banks, Mohanty, and Raghavendra (SODA 2021) that provided an efficient algorithm for the corresponding distinguishing problem. Our algorithm and its analysis significantly depart from previous ones for robust recovery. A key challenge is the peculiar optimization landscape underlying our algorithm: The planted partition may be far from optimal in the sense that completely unrelated solutions could achieve the same objective value. This phenomenon is related to the push-out effect at the BBP phase transition for PCA. To the best of our knowledge, our algorithm is the first to achieve robust recovery in the presence of such a push-out effect in a non-asymptotic setting. Our algorithm is an instantiation of a framework based on convex optimization (related to but distinct from sum-of-squares), which may be useful for other robust matrix estimation problems. A by-product of our analysis is a general technique that boosts the probability of success (over the randomness of the input) of an arbitrary robust weak-recovery algorithm from constant (or slowly vanishing) probability to exponentially high probability.