Spurious Stationarity and Hardness Results for Bregman Proximal-Type Algorithms
Bibliographic record
Abstract
Bregman proximal-type algorithms (BPs), such as mirror descent, have become popular tools in machine learning and data science for exploiting problem structures through non-Euclidean geometries. In this paper, we show that BPs can get trapped near a class of non-stationary points, which we term \emph{spurious stationary points}. Such stagnation can persist for any finite number of iterations if the gradient of the Bregman kernel is not Lipschitz continuous, even in convex problems. The root cause lies in a fundamental contrast in descent behavior between Euclidean and Bregman geometries: While Euclidean gradient descent ensures sufficient decrease near any non-stationary point, BPs may exhibit arbitrarily slow decrease around spurious stationary points. As a result, commonly used Bregman-based stationarity measure, such as relative change in terms of Bregman divergence, can vanish near spurious stationary points. This may misleadingly suggest convergence, even when the iterates remain far from any true stationary point. Our analysis further reveals that spurious stationary points are not pathological, but rather occur generically in a broad class of nonconvex problems with polyhedral constraints. Taken together, our findings reveal a serious blind spot in Bregman-based optimization methods and calls for new theoretical tools and algorithmic safeguards to ensure reliable convergence.
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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 | 0.007 | 0.044 |
| Meta-epidemiology (narrow) | 0.002 | 0.001 |
| Meta-epidemiology (broad) | 0.002 | 0.002 |
| Bibliometrics | 0.002 | 0.001 |
| Science and technology studies | 0.002 | 0.007 |
| Scholarly communication | 0.003 | 0.007 |
| Open science | 0.003 | 0.005 |
| Research integrity | 0.003 | 0.007 |
| Insufficient payload (model declined to judge) | 0.004 | 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".