Nesterov Acceleration of Alternating Least Squares for Canonical Tensor Decomposition: Momentum Step Size Selection and Restart Mechanisms
Bibliographic record
Abstract
We present Nesterov-type acceleration techniques for Alternating Least Squares (ALS) methods applied to canonical tensor decomposition. While Nesterov acceleration turns gradient descent into an optimal first-order method for convex problems by adding a momentum term with a specific weight sequence, a direct application of this method and weight sequence to ALS results in erratic convergence behaviour. This is so because the tensor decomposition problem is non-convex and ALS is accelerated instead of gradient descent. Instead, we consider various restart mechanisms and suitable choices of momentum weights that enable effective acceleration. Our extensive empirical results show that the Nesterov-accelerated ALS methods with restart can be dramatically more efficient than the stand-alone ALS or Nesterov accelerated gradient methods, when problems are ill-conditioned or accurate solutions are desired. The resulting methods perform competitively with or superior to existing acceleration methods for ALS, including ALS acceleration by NCG, NGMRES, or LBFGS, and additionally enjoy the benefit of being much easier to implement. We also compare with Nesterov-type updates where the momentum weight is determined by a line search, which are equivalent or closely related to existing line search methods for ALS. On a large and ill-conditioned 71$\times$1000$\times$900 tensor consisting of readings from chemical sensors to track hazardous gases, the restarted Nesterov-ALS method shows desirable robustness properties and outperforms any of the existing methods by a large factor. There is clear potential for extending our Nesterov-type acceleration approach to accelerating other optimization algorithms than ALS applied to other non-convex problems, such as Tucker tensor decomposition. Our Matlab code is available at https://github.com/hansdesterck/nonlinear-preconditioning-for-optimization.
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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 | 0.005 |
| Meta-epidemiology (narrow) | 0.002 | 0.000 |
| 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 | 0.001 | 0.001 |
| Research integrity | 0.001 | 0.002 |
| Insufficient payload (model declined to judge) | 0.004 | 0.002 |
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".