Faster p-norm minimizing flows, via smoothed q-norm problems
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Bibliographic record
Abstract
We present faster high-accuracy algorithms for computing $\ell_p$-norm minimizing flows. On a graph with $m$ edges, our algorithm can compute a $(1+1/\text{poly}(m))$-approximate unweighted $\ell_p$-norm minimizing flow with $pm^{1+\frac{1}{p-1}+o(1)}$ operations, for any $p \ge 2,$ giving the best bound for all $p\gtrsim 5.24.$ Combined with the algorithm from the work of Adil et al. (SODA '19), we can now compute such flows for any $2\le p\le m^{o(1)}$ in time at most $O(m^{1.24}).$ In comparison, the previous best running time was $Ω(m^{1.33})$ for large constant $p.$ For $p\simδ^{-1}\log m,$ our algorithm computes a $(1+δ)$-approximate maximum flow on undirected graphs using $m^{1+o(1)}δ^{-1}$ operations, matching the current best bound, albeit only for unit-capacity graphs. We also give an algorithm for solving general $\ell_{p}$-norm regression problems for large $p.$ Our algorithm makes $pm^{\frac{1}{3}+o(1)}\log^2(1/\varepsilon)$ calls to a linear solver. This gives the first high-accuracy algorithm for computing weighted $\ell_{p}$-norm minimizing flows that runs in time $o(m^{1.5})$ for some $p=m^{Ω(1)}.$ Our key technical contribution is to show that smoothed $\ell_p$-norm problems introduced by Adil et al., are interreducible for different values of $p.$ No such reduction is known for standard $\ell_p$-norm problems.
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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 | 0.000 | 0.000 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.000 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.001 |
| Open science | 0.004 | 0.005 |
| Research integrity | 0.000 | 0.001 |
| 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