Near-optimal algorithm with complexity separation for strongly convex-strongly concave composite saddle point problems
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Bibliographic record
Abstract
In this work, we revisit the saddle point problem minxmaxyp(x)+R(x,y)−q(y), where the function R(x,y) is LR-smooth, μx-strongly convex, and μy-strongly concave, and the functions p(x),q(y) are convex and Lp,Lq-smooth, respectively. We develop a new algorithm that achieves separation of complexities with respect to the computation of the gradients ∇R(x,y) and ∇p(x), ∇q(y). In particular, our algorithm requires O((LRμxμy+Lpμx+Lqμy4LRμx+LRμy+Lpμx+Lqμy)logLRmin{μx,μy}log1ε) computations of the gradient ∇R(x,y) and O((Lpμx+Lqμy)log1ε) computations of the gradients ∇p(x), ∇q(y) to find an ϵ-accurate solution to the problem. Moreover, under the condition LR≥(μx+μy)μxμyμxLq+μyLp, the algorithm becomes optimal (up to logarithmic factors), i.e. it cannot be improved due to the existing lower complexity bounds. To the best of our knowledge, our algorithm is the first to achieve near-optimal complexity separation in the case when μx≠μy.
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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 | 0.001 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.001 |
| Science and technology studies | 0.001 | 0.000 |
| Scholarly communication | 0.001 | 0.001 |
| Open science | 0.001 | 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