# Oracle symmetry-level selection

- ID: 222
- Canonical URL: https://synthcore.org/idea/222/oracle-symmetry-level-selection
- API JSON: https://synthcore.org/api/idea/222.json
- API Markdown: https://synthcore.org/api/idea/222.md
- Verification status: mechanism_failed
- Source: [arXiv:2608.27055](https://arxiv.org/abs/2608.27055)
- Category: regularization
- Solves: generalization, sample-efficiency, accuracy
- ML areas: architecture, regularization, data-augmentation, training
- Math tags: statistics, optimization, harmonic-analysis, probability
- Ratings: usefulness 7/10; difficulty 4/10; novelty 6/10

## Idea description

Make the number of enforced or averaged transformations a model-selection parameter rather than assuming the strongest available symmetry. Choose the largest orbit whose estimated invariance-induced approximation bias is no larger than the statistical floor, while truncating the search at the bandwidth-limited orbit size.

## Mathematical statement

Corollary 7.7 minimizes the risk proxy (n m^2 q)^(-beta/(beta+1)) + A_q over admissible symmetry levels, truncated at q_sharp. Here n is the number of curves, m is observations per curve, beta is smoothness, q is orbit size, and A_q is approximation error caused by imposing or averaging the q-level symmetry; Proposition 7.2 assumes A_q is non-decreasing in q. The oracle is q_or = argmin_{q in Q} [(n m^2 q)^(-beta/(beta+1)) + A_q] wedge q_sharp. If the parametric floor is reachable, it simplifies to the largest q satisfying A_q less than or approximately 1/n, capped at q_sharp. In a neural model, estimate A_q from held-out task loss under transformed inputs relative to the untransformed loss, or from a direct consistency discrepancy.

## Key formulas

- $$q^{\mathrm{or}}=\left(\arg\min_{q\in\mathcal Q}\left[(nm^{2}q)^{-\beta/(\beta+1)}+A_q\right]\right)\wedge q^{\sharp}.$$
- $$q^{\mathrm{or}}=\max\{q\in\mathcal Q:A_q\lesssim 1/n\}\wedge q^{\sharp}.$$
- $$\widehat A_q=\frac{1}{|V|q}\sum_{(x,y)\in V}\sum_{j=0}^{q-1}\ell\!\left(f_\theta(g_j\!\cdot x),y\right)-\frac{1}{|V|}\sum_{(x,y)\in V}\ell\!\left(f_\theta(x),y\right).$$
- $$\widehat q=\arg\min_{q\in\mathcal Q}\left[(nm^2q)^{-\widehat\beta/(\widehat\beta+1)}+\lambda\widehat A_q\right]\wedge q^\sharp.$$

## Implementation notes

Add a symmetry-selection wrapper around a sparse functional network. Define candidate orbit sizes Q = {1, 2, 4, 8, ...} and train a shared-weight orbit-averaged model for each candidate, or evaluate several q values using nested prefixes of one maximum orbit. On a held-out validation set V, compute the empirical bias proxy A_hat_q as the average transformed-input loss minus the original-input loss, using the displayed formula. Also compute a consistency diagnostic D_q = (|V|q)^(-1) sum_{x in V} sum_j ||f_theta(g_j applied to x) - f_theta(x)||^2, which helps distinguish task noise from genuine symmetry violation. Estimate beta by fitting the slope of log validation error versus log m on a small pilot experiment; for the first implementation, set beta_hat = 1 and compare sensitivity to this choice. Estimate h from the median spacing of observed locations or from the bandwidth of a local interpolation layer, set q_sharp = floor(c_K/h), and score each q with (n m^2 q)^(-beta_hat/(beta_hat+1)) + lambda A_hat_q. Select the smallest q within one validation standard error of the minimum to avoid noisy over-selection. Alternatively use the simplified rule: select the largest q with A_hat_q <= tau/n and q <= q_sharp, calibrating tau on a development split. The mathematical contribution is the explicit decreasing variance proxy, increasing bias term, 1/n stopping floor, and saturation cap; lambda, tau, beta_hat, and h are estimated engineering choices. Start with synthetic periodic regression where a controllable asymmetric perturbation is added to otherwise rotation-invariant curves. Compare q = 1, fixed maximum q, and selected q. Success is lower test error than both fixed baselines under mixed symmetry, with selected q increasing as data become sparser or noisier and decreasing as asymmetry increases.

## Verification

- Status: mechanism_failed
- Mechanism evidence: yes
- Mechanism confirmed: no
- Verdict: Built a NumPy MVP for cyclic orbit averaging, ridge fitting, the displayed A_hat_q and D_q diagnostics, oracle scoring, and synthetic periodic regression. The math sanity check passed: the variance proxy decreases with q, the constructed bias is non-decreasing, and their sum has the expected interior minimizer; full-group averaging was also numerically shift-invariant. In the benchmark, oracle selection did not produce a win: across 12 seeds per asymmetry, it never beat both q=1 and q=8, and at asymmetry 0.15 its mean test MSE was 1.0888 versus 1.1077 for q=1 and 1.0461 for q=8; at asymmetry 0.5 it was 1.1135 versus 1.1077 and 1.1660. Therefore the proposed effect was not demonstrated.

### Mechanism check

- Verdict: Built a NumPy MVP for cyclic orbit averaging, ridge fitting, the displayed A_hat_q and D_q diagnostics, oracle scoring, and synthetic periodic regression. The math sanity check passed: the variance proxy decreases with q, the constructed bias is non-decreasing, and their sum has the expected interior minimizer; full-group averaging was also numerically shift-invariant. In the benchmark, oracle selection did not produce a win: across 12 seeds per asymmetry, it never beat both q=1 and q=8, and at asymmetry 0.15 its mean test MSE was 1.0888 versus 1.1077 for q=1 and 1.0461 for q=8; at asymmetry 0.5 it was 1.1135 versus 1.1077 and 1.1660. Therefore the proposed effect was not demonstrated.
- Confidence: 9/10
- Limitations: This is a small linear ridge-regression proxy rather than a sparse functional neural network; q candidates are cyclic shifts of synthetic vectors, not learned transformations. The validation A_hat_q uses a model trained on unaveraged inputs and clips negative discrepancies, is not standard-error corrected, and does not implement the requested beta pilot, bandwidth-derived q_sharp, or one-standard-error rule. Only one synthetic data family, fixed q_sharp=8, one validation split, and 12 repeated seeds were tested; no GPU or neural-training comparison was needed.

## Artifacts

- [experiment.py](https://synthcore.org/code/45/experiment.py)
- [report.md](https://synthcore.org/code/45/report.md)
- [results.json](https://synthcore.org/code/45/results.json)
- [Download all files as ZIP](https://synthcore.org/download/45)

## Disclaimer

AI-generated research hypothesis, automatically tested. Not peer-reviewed.