Sample-Efficient Learning of Mixtures
Bibliographic record
Abstract
We consider PAC learning of probability distributions (a.k.a. density estimation), where we are given an i.i.d. sample generated from an unknown target distribution, and want to output a distribution that is close to the target in total variation distance. Let F be an arbitrary class of probability distributions, and let Fk denote the class of k-mixtures of elements of F. Assuming the existence of a method for learning F with sample complexity m(ε), we provide a method for learning Fk with sample complexity O((k.log k .m(ε))/(ε2)). Our mixture learning algorithm has the property that, if the F-learner is proper and agnostic, then the Fk-learner would be proper and agnostic as well. This general result enables us to improve the best known sample complexity upper bounds for a variety of important mixture classes. First, we show that the class of mixtures of k axis-aligned Gaussians in Rd is PAC-learnable in the agnostic setting with O((kd)/(ε4)) samples, which is tight in k and d up to logarithmic factors. Second, we show that the class of mixtures of k Gaussians in Rd is PAC-learnable in the agnostic setting with sample complexity Õ((kd2)/(ε4)), which improves the previous known bounds of Õ((k3.d2)/(ε4)) and Õ(k4.d4/ε2) in its dependence on k and d. Finally, we show that the class of mixtures of k log-concave distributions over Rd is PAC-learnable using Õ(k.d((d+5)/2)ε(-(d+9)/2)) samples.
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How this classification was reachedexpand
Full frame machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.006 | 0.031 |
| Meta-epidemiology (narrow) | 0.002 | 0.002 |
| Meta-epidemiology (broad) | 0.004 | 0.003 |
| Bibliometrics | 0.002 | 0.002 |
| Science and technology studies | 0.001 | 0.003 |
| Scholarly communication | 0.003 | 0.009 |
| Open science | 0.005 | 0.005 |
| Research integrity | 0.003 | 0.007 |
| Insufficient payload (model declined to judge) | 0.003 | 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 source (direct Gemma or distilled Codex), 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".