Second Order Methods for Optimizing Convex Matrix Functions and Sparse Covariance Clustering
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Bibliographic record
Abstract
A variety of first-order methods have recently been proposed for solving matrix optimization problems arising in machine learning. The premise for utilizing such algorithms is that second order information is too expensive to employ, and so simple first-order iterations are likely to be optimal. In this paper, we argue that second-order information is in fact efficiently accessible in many matrix optimization problems, and can be effectively incorporated into optimization algorithms. We begin by reviewing how certain Hessian operations can be conveniently represented in a wide class of matrix optimization problems, and provide the first proofs for these results. Next we consider a concrete problem, namely the minimization of the ℓ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> regularized Jeffreys divergence, and derive formulae for computing Hessians and Hessian vector products. This allows us to propose various second order methods for solving the Jeffreys divergence problem. We present extensive numerical results illustrating the behavior of the algorithms and apply the methods to a speech recognition problem. We compress full covariance Gaussian mixture models utilized for acoustic models in automatic speech recognition. By discovering clusters of (sparse inverse) covariance matrices, we can compress the number of covariance parameters by a factor exceeding 200, while still outperforming the word error rate (WER) performance of a diagonal covariance model that has 20 times less covariance parameters than the original acoustic model.
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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 | 0.000 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.000 | 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