# Diophantine spherical probe schedule

- ID: 2301
- Canonical URL: https://synthcore.org/idea/2301/diophantine-spherical-probe-schedule
- API JSON: https://synthcore.org/api/idea/2301.json
- API Markdown: https://synthcore.org/api/idea/2301.md
- Verification status: unverified
- Source: [arXiv:2608.14607](https://arxiv.org/abs/2608.14607)
- Category: training
- Solves: sample-efficiency, stability, accuracy
- ML areas: optimizer, data-augmentation, training-dynamics
- Math tags: dynamical-systems, harmonic-analysis, statistics, linear-algebra
- Ratings: usefulness 6/10; difficulty 5/10; novelty 7/10

## Idea description

Replace independently sampled unit-sphere perturbations or augmentation directions by a deterministic measure-preserving image of a Kronecker flow. Use the resulting directions cyclically for gradient perturbations, adversarial training, random-feature estimation, or spherical data augmentation. The schedule should reduce directional bias at a predictable polynomial rate while eliminating batch-to-batch randomness.

## Mathematical statement

Let $\mathbb{T}^{d}=\mathbb{R}^{d}/\mathbb{Z}^{d}$, let $\theta(t)=\theta_0+t\omega\pmod{\mathbb{Z}^{d}}$ be a continuous torus flow, and let $\Phi_d:\mathbb{T}^{d}\to\mathbb{S}^{d}$ be a measure-preserving map from torus measure to normalized spherical measure. Assume the frequency is Diophantine: there are $\gamma>0$ and $\tau\geq0$ such that $|k\cdot\omega|\geq\gamma\|k\|^{-\tau}$ for every nonzero $k\in\mathbb{Z}^{d}$. For every $C^s$ observable $\varphi$ on the sphere, $0<s\leq1$, the time-average error is bounded by $C\|\varphi\|_{C^s}T^{-\vartheta}$, where $\vartheta=s/((1+s)(\tau+d))$. If the observable vanishes near the polar degeneracy of the map, the exponent improves to $s/(\tau+d)$. The bound follows by Fourier expansion and the mode estimate $1/(T|k\cdot\omega|)$, combined with the Diophantine lower bound.

## Key formulas

- $$\theta(t)=\theta_0+t\omega\pmod{\mathbb{Z}^{d}},\qquad u(t)=\Phi_d(\theta(t))\in\mathbb{S}^{d}.$$
- $$|k\cdot\omega|\geq\gamma\|k\|^{-\tau}\quad\forall k\in\mathbb{Z}^{d}\setminus\{0\},$$
- $$\left|\frac{1}{T}\int_0^T\varphi(u(t))\,dt-\int_{\mathbb{S}^{d}}\varphi\,d\sigma_d\right|\leq C\|\varphi\|_{C^s}T^{-s/((1+s)(\tau+d))},$$
- $$\left|\frac{1}{T}\int_0^T e^{2\pi i k\cdot(\theta_0+t\omega)}dt\right|\leq\frac{1}{\pi T|k\cdot\omega|}\leq\frac{\|k\|^{\tau}}{\pi\gamma T},\qquad \vartheta_{\mathrm{away\ from\ cut}}=\frac{s}{\tau+d}.$$

## Implementation notes

1. Integration point: use this schedule wherever a model currently draws unit vectors uniformly from $\mathbb{S}^{d}$. The cheapest target is zeroth-order optimization: evaluate the loss at $w+\delta u_n$ and $w-\delta u_n$, form a directional gradient estimate, and average a short deterministic block before updating. The same generator can produce deterministic embedding, image, or adversarial augmentation directions. 2. Pseudocode: choose dimension $d$, phase $\theta_0$, step $\Delta t$, perturbation radius $\delta$, and a candidate frequency $\omega$; at iteration $n$, compute $\theta_n=(\theta_0+n\Delta t\omega)\bmod1$, map $u_n=\Phi_d(\theta_n)$, evaluate $L_+=L(w+\delta u_n)$ and $L_-=L(w-\delta u_n)$, set $g_n=(L_+-L_-)u_n/(2\delta)$, average $m$ successive estimates, and update $w\leftarrow w-\eta\bar g$. Before training, reject candidates with small $|k\cdot\omega|$ for low-order integer vectors $k$. 3. Computed from the paper: the flow, spherical map, Diophantine screening rule, and predicted exponent. Estimated empirically: the constant $C$, effective smoothness $s$, gradient variance, and actual discrepancy. 4. First cheap experiment: train a small MLP on MNIST with IID, resonant, and Diophantine spherical directions for zeroth-order perturbation estimates. For fixed smooth test observables, plot absolute averaging error against probe count $N$ on log-log axes. For $s=1$, the predicted slope is approximately $-1/[2(\tau+d)]$ with a polar cutoff and $-1/(\tau+d)$ away from the cutoff; measured slopes should agree within 20%. Deliberately introducing a low-order resonance should produce plateaus or a visibly worse slope when $\min_{\|k\|\leq K}|k\cdot\omega|$ is small.

## Disclaimer

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