A Continuous-Time Gradient-Tracking Algorithm for Directed Networks
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Bibliographic record
Abstract
In this letter, we consider the problem of unconstrained convex optimization over directed networks and design a continuous-time (CT) gradient-tracking dynamics to address it. First, we establish that the optimum of the distributed optimization problem (DOP) is contained in the equilibrium of the designed dynamics under appropriate initialization. Subsequently, we construct a novel Lyapunov function to establish exponential convergence to the equilibrium. Specifically, for the Lyapunov function, we rely on the Lyapunov-like equations associated with the asymmetric graph Laplacians of the directed networks. As a result of the convergence analysis, we obtain sufficiency conditions on the gains involved in the dynamics. Additionally, we also present an adaptive variant of the designed gradient-tracking dynamics which converges to the aforementioned equilibrium asymptotically.
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Full frame distilled prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. Learned from the 10,348 direct Codex labels and 10,348 direct Gemma labels. Candidate is the union of thresholded teacher heads; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels or direct frontier model labels.
Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.001 | 0.000 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.000 | 0.001 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.002 | 0.001 |
| Open science | 0.001 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.000 | 0.000 |
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 it