Size and Depth Separation in Approximating Natural Functions with Neural Networks
Bibliographic record
Abstract
When studying the expressive power of neural networks, a main challenge is to understand how the size and depth of the network affect its ability to approximate real functions. However, not all functions are interesting from a practical viewpoint: functions of interest usually have a polynomially-bounded Lipschitz constant, and can be computed efficiently. We call functions that satisfy these conditions ``benign, and explore the benefits of size and depth for approximation of benign functions with ReLU networks. As we show, this problem is more challenging than the corresponding problem for non-benign functions. We give complexity-theoretic barriers to showing depth-lower-bounds: Proving existence of a benign function that cannot be approximated by polynomial-sized networks of depth $4$ would settle longstanding open problems in computational complexity. It implies that beyond depth $4$ there is a barrier to showing depth-separation for benign functions, even between networks of constant depth and networks of nonconstant depth. We also study size-separation, namely, whether there are benign functions that can be approximated with networks of size $O(s(d))$, but not with networks of size $O(s'(d))$. We show a complexity-theoretic barrier to proving such results beyond size $O(d\log^2(d))$, but also show an explicit benign function, that can be approximated with networks of size $O(d)$ and not with networks of size $o(d/\log d)$. For approximation in the $L_\infty$ sense we achieve such separation already between size $O(d)$ and size $o(d)$. Moreover, we show superpolynomial size lower bounds and barriers to such lower bounds, depending on the assumptions on the function. Our size-separation results rely on an analysis of size lower bounds for Boolean functions, which is of independent interest: We show linear size lower bounds for computing explicit Boolean functions (such as set disjointness) with neural networks and threshold circuits.
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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 | 0.002 | 0.020 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.001 | 0.003 |
| Scholarly communication | 0.002 | 0.009 |
| Open science | 0.002 | 0.004 |
| Research integrity | 0.002 | 0.004 |
| Insufficient payload (model declined to judge) | 0.004 | 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".