Bibliographic record
Abstract
We study the problem of solving matrix games of the form $\min_{\mathbf{p}\inΔ}\max_{\mathbf{w}\in\mathcal{W}}\mathbf{p}^{\top}A\mathbf{w}$, where $A$ is a matrix and $Δ$ is the probability simplex. This problem encapsulates canonical tasks such as finding a linear separator and computing Nash equilibria in zero-sum games. However, perhaps surprisingly, its inherent complexity (as formalized in the standard framework of oracle complexity) is not well understood. In this work, we first identify different oracle models that are implicitly used by prior algorithms, corresponding to multiplying the matrix $A$ by a vector from either one or both sides. We then prove complexity lower bounds for algorithms under both access models, which in particular imply a separation between them. As our main result, we prove that in the general $\ell_p$/simplex setting where $\mathcal{W}$ is an $\ell_p$ ball for $p\in[1,\infty]$, any algorithm that utilizes two-sided matrix-vector multiplications requires $\tildeΩ(ε^{-2/3})$ iterations to return an $ε$-suboptimal solution. For any $p\in[1,\infty]$, this is either the first lower bound for such problems, or an exponential improvement over the previously best-known results. Moreover, for the canonical tasks of finding a linear separator and computing a Nash equilibrium, our lower bounds match (up to log factors) recent algorithms of Karmarkar, O'Carroll and Sidford (2026), thereby resolving their oracle complexities in a natural setting.
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How this classification was reachedexpand
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 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.001 |
| Science and technology studies | 0.000 | 0.001 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.002 | 0.002 |
| Research integrity | 0.000 | 0.001 |
| Insufficient payload (model declined to judge) | 0.000 | 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 teacher head, 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".