The algorithmic hardness threshold for continuous random energy models
Bibliographic record
Abstract
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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How this classification was reachedexpand
Full frame machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.003 | 0.021 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.002 | 0.003 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.002 | 0.005 |
| Scholarly communication | 0.004 | 0.010 |
| Open science | 0.004 | 0.005 |
| Research integrity | 0.003 | 0.007 |
| Insufficient payload (model declined to judge) | 0.008 | 0.001 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from itClassification
machine, unvalidatedMachine predicted; a candidate call from one source (direct Gemma or distilled Codex), not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".