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# Parallels Between Phase Transitions and Circuit Complexity?

*Proceedings of Thirty Third Conference on Learning Theory*, PMLR 125:2910-2946, 2020.

#### Abstract

In many natural average-case problems, there are or there are believed to be critical values in the parameter space where the structure of the space of solutions changes in a fundamental way. These phase transitions are often believed to coincide with drastic changes in the computational complexity of the associated problem. In this work, we study the circuit complexity of inference in the broadcast tree model, which has important applications in phylogenetic reconstruction and close connections to community detection. We establish a number of qualitative connections between phase transitions and circuit complexity in this model. Specifically we show that there is a $\mathbf{TC}^0$ circuit that competes with the Bayes optimal predictor in some range of parameters above the Kesten-Stigum bound. We also show that there is a $16$ label broadcast tree model beneath the Kesten-Stigum bound in which it is possible to accurately guess the label of the root, but beating random guessing is $\mathbf{NC}^1$-hard on average. The key to locating phase transitions is often to study some intrinsic notions of complexity associated with belief propagation \— e.g. where do linear statistics fail, or when is the posterior sensitive to noise? Ours is the first work to study the complexity of belief propagation in a way that is grounded in circuit complexity.