Article Dans Une Revue Mathematical Statistics and Learning Année : 2020

The algorithmic hardness threshold for continuous random energy models

Résumé

We prove an algorithmic hardness result for finding low-energy states in the so-called continuous random energy model (CREM) , introduced by Bovier and Kurkova in 2004 as an extension of Derrida’s generalized random energy model . The CREM is a model of a randomenergy landscape (X_v)_{v \in \{0,1\}^N} on the discrete hypercube with built-in hierarchical structure, and can be regarded as a toy model for strongly correlated random energy landscapes such as the family of p -spin models including the Sherrington–Kirkpatrick model. The CREM is parameterized by an increasing function A \colon [0,1]\to[0,1] , which encodes the correlations between states. We exhibit an algorithmic hardness threshold x_* , which is explicit in terms of A . More precisely, we obtain two results: First, we show that a renormalization procedure combined with a greedy search yields for any \varepsilon > 0 a linear-time algorithm which finds states v \in \{0,1\}^N with X_v \ge (x_*-\varepsilon) N . Second, we show that the value x_* is essentially best-possible: for any \varepsilon > 0 , any algorithm which finds states v with X_v \ge (x_*+\varepsilon)N requires exponentially many queries in expectation and with high probability. We further discuss what insights this study yields for understanding algorithmic hardness thresholds for random instances of combinatorial optimization problems.

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hal-04853633 , version 1 (07-07-2025)

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Louigi Addario-Berry, Pascal Maillard. The algorithmic hardness threshold for continuous random energy models. Mathematical Statistics and Learning, 2020, 2 (1), pp.77-101. ⟨10.4171/MSL/12⟩. ⟨hal-04853633⟩
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