Learnability can be independent of set theory (invited paper)
Why this work is in the frame
A frame that forgets how it found something cannot be audited. These are the routes that admitted this work.
Bibliographic record
Abstract
A fundamental result in statistical learning theory is the equivalence of PAC learnability of a class with the finiteness of its Vapnik-Chervonenkis dimension. However, this clean result applies only to binary classification problems. In search for a similar combinatorial characterization of learnability in a more general setting, we discovered a surprising independence of set theory for some basic general notion of learnability. Consider the following statistical estimation problem: given a family F of real valued random variables over some domain X and an i.i.d. sample drawn from an unknown distribution P over X, find f in F such that its expectation w.r.t. P is close to the supremum expectation over all members of F. This Expectation Maximization (EMX) problem captures many well studied learning problems. Surprisingly, we show that the EMX learnability of some simple classes depends on the cardinality of the continuum and is therefore independent of the set theory ZFC axioms. Our results imply that that there exist no "finitary" combinatorial parameter that characterizes EMX learnability in a way similar to the VC-dimension characterization of binary classification learnability.
Fetched live from OpenAlex and de-inverted. Abstracts are not stored in this database: the inverted indexes are 8.6 GB of the frame’s 9.3 GB of text, and the host has 13 GB free.
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 | 0.000 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.000 | 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