Near-optimal Sample Complexity Bounds for Robust Learning of Gaussian Mixtures via Compression Schemes
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Bibliographic record
Abstract
We introduce a novel technique for distribution learning based on a notion of sample compression . Any class of distributions that allows such a compression scheme can be learned with few samples. Moreover, if a class of distributions has such a compression scheme, then so do the classes of products and mixtures of those distributions. As an application of this technique, we prove that ˜Θ( kd 2 /ε 2 ) samples are necessary and sufficient for learning a mixture of k Gaussians in R d , up to error ε in total variation distance. This improves both the known upper bounds and lower bounds for this problem. For mixtures of axis-aligned Gaussians, we show that Õ( kd /ε 2 ) samples suffice, matching a known lower bound. Moreover, these results hold in an agnostic learning (or robust estimation) setting, in which the target distribution is only approximately a mixture of Gaussians. Our main upper bound is proven by showing that the class of Gaussians in R d admits a small compression scheme.
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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 | 0.001 |
| 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 | 0.002 | 0.001 |
| 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