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# Adversarially-Robust Inference on Trees via Belief Propagation

*Proceedings of Thirty Seventh Conference on Learning Theory*, PMLR 247:2389-2417, 2024.

#### Abstract

We introduce and study the problem of posterior inference on tree-structured graphical models in the presence of a malicious adversary who can corrupt some observed nodes. In the well-studied \emph{broadcasting on trees} model, corresponding to the ferromagnetic Ising model on a $d$-regular tree with zero external field, when a natural signal-to-noise ratio exceeds one (the celebrated \emph{Kesten-Stigum threshold}), the posterior distribution of the root given the leaves is bounded away from $\mathrm{Ber}(1/2)$, and carries nontrivial information about the sign of the root. This posterior distribution can be computed exactly via dynamic programming, also known as belief propagation. We first confirm a folklore belief that a malicious adversary who can corrupt an inverse-polynomial fraction of the leaves of their choosing makes this inference impossible. Our main result is that accurate posterior inference about the root vertex given the leaves \emph{is} possible when the adversary is constrained to make corruptions at a $\rho$-fraction of randomly-chosen leaf vertices, so long as the signal-to-noise ratio exceeds $O(\log d)$ and $\rho \leq c \varepsilon$ for some universal $c > 0$. Since inference becomes information-theoretically impossible when $\rho \gg \varepsilon$, this amounts to an information-theoretically optimal fraction of corruptions, up to a constant multiplicative factor. Furthermore, we show that the canonical belief propagation algorithm performs this inference.