Decision Trees and Influences of Variables Over Product Probability Spaces
Why this work is in the frame
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Bibliographic record
Abstract
A celebrated theorem of Friedgut says that every function f : {0, 1} n → {0, 1} can be approximated by a function g : {0, 1} n → {0, 1} with $\|f-g\|_2^2 \leq \epsilon$ , which depends only on e O ( I f / ε) variables, where I f is the sum of the influences of the variables of f . Dinur and Friedgut later showed that this statement also holds if we replace the discrete domain {0, 1} n with the continuous domain [0, 1] n , under the extra assumption that f is increasing. They conjectured that the condition of monotonicity is unnecessary and can be removed. We show that certain constant-depth decision trees provide counter-examples to the Dinur–Friedgut conjecture. This suggests a reformulation of the conjecture in which the function g : [0, 1] n → {0, 1}, instead of depending on a small number of variables, has a decision tree of small depth. In fact we prove this reformulation by showing that the depth of the decision tree of g can be bounded by e O ( I f / ε 2 ) . Furthermore, we consider a second notion of the influence of a variable, and study the functions that have bounded total influence in this sense. We use a theorem of Bourgain to show that these functions have certain properties. We also study the relation between the two different notions of influence.
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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 | 0.001 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.001 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.001 |
| Open science | 0.001 | 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