Dynamic Regret Bounds without Lipschitz Continuity: Online Convex Optimization with Multiple Mirror Descent Steps
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Bibliographic record
Abstract
We study the dynamic regret in online convex optimization (OCO), where the cost functions are revealed sequentially over time. Prior studies on the dynamic regret of OCO algorithms often require the cost functions to be Lipschitz continuous. However, the costs functions that arise in many applications may not satisfy this condition. In this work, we analyze the performance of Online Multiple Mirror Descent (OMMD), which can handle non-Lipschitz cost functions. OMMD is based on mirror descent but uses multiple mirror descent steps per online round. We first derive two upper bounds on the dynamic regret based on the path length and squared path length, and we further derive a third upper bound based on the cumulative optimal cost, which can be much smaller than the path length or squared path length especially when the sequence of minimizers fluctuates over time. We show that the dynamic regret of OMMD scales linearly with the minimum among the path length, squared path length, and cumulative optimal cost.
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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 | 0.002 | 0.003 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.000 |
| Bibliometrics | 0.001 | 0.003 |
| Science and technology studies | 0.001 | 0.001 |
| Scholarly communication | 0.001 | 0.001 |
| Open science | 0.002 | 0.001 |
| Research integrity | 0.000 | 0.001 |
| Insufficient payload (model declined to judge) | 0.004 | 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