Low Degree Hardness for Broadcasting on Trees

TL;DR

This paper proves that below the Kesten-Stigum bound, low-degree polynomial algorithms cannot effectively infer the root in broadcasting on trees, highlighting inherent computational hardness in this regime.

Contribution

It establishes the first low-degree polynomial lower bounds for broadcasting on trees outside product measure settings, confirming computational hardness below the Kesten-Stigum bound.

Findings

Any $O(\log n)$ degree polynomial has vanishing correlation with the root below the Kesten-Stigum bound.

The result extends low-degree hardness to non-product measure settings in broadcasting problems.

Supports the conjecture that efficient algorithms cannot succeed below the Kesten-Stigum threshold.

Abstract

We study the low-degree hardness of broadcasting on trees. Broadcasting on trees has been extensively studied in statistical physics, in computational biology in relation to phylogenetic reconstruction and in statistics and computer science in the context of block model inference, and as a simple data model for algorithms that may require depth for inference. The inference of the root can be carried by celebrated Belief Propagation (BP) algorithm which achieves Bayes-optimal performance. Despite the fact that this algorithm runs in linear time (using real operations), recent works indicated that this algorithm in fact requires high level of complexity. Moitra, Mossel and Sandon constructed a chain for which estimating the root better than random (for a typical input) is $NC1$ complete. Kohler and Mossel constructed chains such that for trees with $N$ leaves, recovering the root better than random requires a polynomial of degree $N^{\Omega(1)}$. Both works above asked if such complexity bounds hold in general below the celebrated {\em Kesten-Stigum} bound. In this work, we prove that this is indeed the case for low degree polynomials. We show that for the broadcast problem using any Markov chain on trees with $n$ leaves, below the Kesten Stigum bound, any $O(\log n)$ degree polynomial has vanishing correlation with the root. Our result is one of the first low-degree lower bound that is proved in a setting that is not based or easily reduced to a product measure.