A polynomial expansion line search for large-scale unconstrained minimization of smooth L2-regularized loss functions, with implementation in Apache Spark
Bibliographic record
Abstract
In large-scale unconstrained optimization algorithms such as limited memory BFGS (LBFGS), a common subproblem is a line search minimizing the loss function along a descent direction. Commonly used line searches iteratively find an approximate solution for which the Wolfe conditions are satisfied, typically requiring multiple function and gradient evaluations per line search, which is expensive in parallel due to communication requirements. In this paper we propose a new line search approach for cases where the loss function is analytic, as in least squares regression, logistic regression, or low rank matrix factorization. We approximate the loss function by a truncated Taylor polynomial, whose coefficients may be computed efficiently in parallel with less communication than evaluating the gradient, after which this polynomial may be minimized with high accuracy in a neighbourhood of the expansion point. Our Polynomial Expansion Line Search (PELS) was implemented in the Apache Spark framework and used to accelerate the training of a logistic regression model on binary classification datasets from the LIBSVM repository with LBFGS and the Nonlinear Conjugate Gradient (NCG) method. In large-scale numerical experiments in parallel on a 16-node cluster with 256 cores using the URL, KDDA, and KDDB datasets, the PELS approach produced significant convergence improvements compared to the use of classical Wolfe line searches. For example, to reach the final training label prediction accuracies, LBFGS using PELS had speedup factors of 1.8--2 over LBFGS using a Wolfe line search, measured by both the number of iterations and the time required, due to the better accuracy of step sizes computed in the line search. PELS has the potential to significantly accelerate large-scale regression and factorization computations, and is applicable to continuous optimization problems with smooth loss functions.
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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 | 0.001 | 0.004 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.001 | 0.001 |
| Scholarly communication | 0.001 | 0.001 |
| Open science | 0.002 | 0.001 |
| Research integrity | 0.001 | 0.002 |
| Insufficient payload (model declined to judge) | 0.009 | 0.004 |
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".