Tight analyses for subgradient descent I: Lower bounds
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Bibliographic record
Abstract
Consider the problem of minimizing functions that are Lipschitz and convex, but not necessarily differentiable. We construct a function from this class for which the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>T</mml:mi> <mml:mi>þ</mml:mi> </mml:mrow> </mml:math> iterate of subgradient descent has error <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>Ω</mml:mi> <mml:mo>(</mml:mo> <mml:mo form="prefix">log</mml:mo> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>T</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>/</mml:mo> <mml:msqrt> <mml:mi>T</mml:mi> </mml:msqrt> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> . This matches a known upper bound of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>O</mml:mi> <mml:mo>(</mml:mo> <mml:mo form="prefix">log</mml:mo> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>T</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>/</mml:mo> <mml:msqrt> <mml:mi>T</mml:mi> </mml:msqrt> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> . We prove analogous results for functions that are additionally strongly convex. There exists such a function for which the error of the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>T</mml:mi> <mml:mi>þ</mml:mi> </mml:mrow> </mml:math> iterate of subgradient descent has error <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>Ω</mml:mi> <mml:mo>(</mml:mo> <mml:mo form="prefix">log</mml:mo> <mml:mo>(</mml:mo> <mml:mi>T</mml:mi> <mml:mo>)</mml:mo> <mml:mo>/</mml:mo> <mml:mi>T</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> , matching a known upper bound of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>O</mml:mi> <mml:mo>(</mml:mo> <mml:mo form="prefix">log</mml:mo> <mml:mo>(</mml:mo> <mml:mi>T</mml:mi> <mml:mo>)</mml:mo> <mml:mo>/</mml:mo> <mml:mi>T</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> . These results resolve a question posed by Shamir (2012).
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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 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.000 |
| Bibliometrics | 0.000 | 0.001 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.003 | 0.001 |
| Open science | 0.001 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.001 | 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