# Pre-Training Depth Feasibility Certificates

- ID: 1319
- Canonical URL: https://synthcore.org/idea/1319/pre-training-depth-feasibility-certificates
- API JSON: https://synthcore.org/api/idea/1319.json
- API Markdown: https://synthcore.org/api/idea/1319.md
- Verification status: failed_benchmark
- Source: [arXiv:2607.23390](https://arxiv.org/abs/2607.23390)
- Category: training
- Solves: speedup, scalability, sample-efficiency
- ML areas: quantization, distillation, training, inference-speedup
- Math tags: approximation-theory, optimization, numerical-analysis, dynamical-systems
- Ratings: usefulness 7/10; difficulty 5/10; novelty 7/10

## Idea description

Use computable upper and lower error bounds to reject quantized-depth configurations that cannot reach the desired accuracy before training. The planner separates irreducible library mismatch from finite-depth synthesis, codebook metadata, and execution errors, then selects the smallest depth and metadata budget whose estimated bound passes the target.

## Mathematical statement

For target map F^star, operation-library family Omega, depth D, and metadata precision s, the paper defines the structural floor E_{Omega,infty}(F^star)=dist(F^star,R_{Omega,rel}), where R_{Omega,rel} is the closed relaxed reachable set. The integrated bound is |E_impl_{D,s}(F^star)-E_{Omega,infty}(F^star)| <= C_syn/D + C_meta 2^{-s/m} + A_D. For a theta-Hölder time-dependent field, replace C_syn/D by C_theta D^{-theta}+C_1/D. Given an empirically or formally obtained floor bracket [L,U] and finite-resource radius R_{D,s}, the decision rules are U+R_{D,s} <= epsilon_H implies feasible, while L-R_{D,s} > epsilon_H implies impossible; otherwise the configuration is unresolved. Here epsilon_H is the high-precision teacher error, m is the number of metadata dimensions, and A_D is an execution-arithmetic penalty estimated from quantization and saturation tests.

## Key formulas

- $$\left|E^{\mathrm{impl}}_{D,s}(F^{\star})-E_{\Omega,\infty}(F^{\star})\right|\leq\frac{C_{\mathrm{syn}}}{D}+C_{\mathrm{meta}}2^{-s/m}+\mathcal{A}_{D},\qquad E_{\Omega,\infty}(F^{\star})=\operatorname{dist}(F^{\star},\mathscr{R}_{\Omega,\mathrm{rel}}).$$
- $$R_{D,s}=\frac{C_{\mathrm{syn}}}{D}+C_{\mathrm{meta}}2^{-s/m}+\mathcal{A}_{D},\qquad R_{D,s}^{(\vartheta)}=C_{\vartheta}D^{-\vartheta}+C_1D^{-1}+C_{\mathrm{meta}}2^{-s/m}+\mathcal{A}_{D}.$$
- $$U+R_{D,s}\leq\varepsilon_H\Rightarrow\text{feasible},\qquad L-R_{D,s}>\varepsilon_H\Rightarrow\text{impossible};\quad\text{otherwise unresolved}.$$

## Implementation notes

Add a calibration stage before quantization-aware training or knowledge distillation. Choose a teacher network F_H and a family of candidate low-bit architectures indexed by depth D, activation or weight bit width b, codebook metadata bits s, and execution mode. On a held-out calibration set, estimate each candidate's teacher discrepancy E_impl(D,s). Fit the finite-resource radius R_{D,s} by regressing observed discrepancies across several small depths against 1/D and 2^{-s/m}; estimate A_D by running the same network with and without simulated integer arithmetic, including saturation diagnostics. Use bootstrap confidence intervals to form a conservative bracket [L,U] for the structural floor. For every candidate compute upper=U+R and lower=L-R. Keep candidates satisfying upper<=epsilon_H, reject those satisfying lower>epsilon_H, and allocate more calibration data to unresolved cases. Pseudocode is: `for D,s,b in candidates: R=Csyn/D+Cmeta*2**(-s/m)+A[D,b]; upper=U+R; lower=L-R; if upper<=eps_teacher: keep; elif lower>eps_teacher: reject; else: calibrate_more`. The constants are estimated empirically rather than assumed. First test on CIFAR-10 or WikiText-2 with a full-precision ResNet-18 or small Transformer teacher, comparing bound-guided selection against training every 2-, 4-, and 8-bit candidate. Measure GPU-hours avoided, correlation between predicted feasibility and final accuracy, and the smallest depth reaching teacher error. Success means rejecting at least 30% of doomed configurations while retaining at least 95% of successful ones.

## Verification

- Status: failed_benchmark
- Mechanism evidence: yes
- Mechanism confirmed: no
- Practical verdict: no_effect
- Verdict: Built a readable certificate-planning MVP with calibration, structural-floor bracketing, resource-radius computation, and feasible/impossible/unresolved decisions. Empirical sweeps confirmed the predicted depth exponent (-0.9989 vs -1), metadata halving law (maximum absolute ratio error 0.0217), and threshold boundary (predicted D=11.51, first passing grid D=16). The certificate avoided 22 of 90 evaluations and retained all successful candidates, but rejected 0/66 doomed candidates because the conservative lower bound could not exceed the target in this toy setting; therefore the mathematical mechanism manifested, but strong practical screening benefit was not demonstrated.

### Mechanism check

- Verdict: Built a readable certificate-planning MVP with calibration, structural-floor bracketing, resource-radius computation, and feasible/impossible/unresolved decisions. Empirical sweeps confirmed the predicted depth exponent (-0.9989 vs -1), metadata halving law (maximum absolute ratio error 0.0217), and threshold boundary (predicted D=11.51, first passing grid D=16). The certificate avoided 22 of 90 evaluations and retained all successful candidates, but rejected 0/66 doomed candidates because the conservative lower bound could not exceed the target in this toy setting; therefore the mathematical mechanism manifested, but strong practical screening benefit was not demonstrated.
- Confidence: 8/10
- Limitations: This is a synthetic scalar-error experiment, not CIFAR-10/WikiText-2 or neural-network quantization-aware training. The structural floor, constants, arithmetic penalty, and target are simulated rather than learned from real teacher/student models; no GPU-hours, wall-clock speedup, accuracy calibration, bootstrap confidence intervals, or real integer saturation tests were measured.

### Practical benchmark

- Paired seeds: 8
- Baseline mean: 0
- Idea mean: 0
- p-value: 1
- Paired wins: 0/8
- Benchmark verdict: no measurable effect

## Artifacts

- [bench_depth_certificate.py](https://synthcore.org/code/520/bench_depth_certificate.py)
- [bench_report.json](https://synthcore.org/code/520/bench_report.json)
- [certificate_experiment.py](https://synthcore.org/code/520/certificate_experiment.py)
- [report.md](https://synthcore.org/code/520/report.md)
- [report_bench_2026-08-31T164709.md](https://synthcore.org/code/520/report_bench_2026-08-31T164709.md)
- [results.json](https://synthcore.org/code/520/results.json)
- [Download all files as ZIP](https://synthcore.org/download/520)

## Disclaimer

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