✗ Failed on benchmark 2026

Pre-Training Depth Feasibility Certificates

Implementation & benchmark of arXiv:2607.23390 — When Can Depth Replace Precision? A Resource Theory of Quantized Neural Computation

Usefulness7/10

Difficulty5/10

Novelty7/10

Training Speedup Scalability Sample efficiency Quantization Distillation Training Inference speedup Approximation theory Optimization Numerical analysis Dynamical systems

Source paper: When Can Depth Replace Precision? A Resource Theory of Quantized Neural Computation arXiv:2607.23390 ⓘ · analyzed Aug 30, 2026

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 Not peer-reviewed.

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.

Formulas

$$\left|E^{\mathrm{impl}}_{D,s}(F^{\star})-E_{\Omega,\infty}(F^{\star})\right|\leq\frac{C_{\mathrm{syn}}}{D}+C_{\mathrm{meta}}2^{-s/m}+\mathcal{A}_{D},\qquad E_{\Omega,\infty}(F^{\star})=\operatorname{dist}(F^{\star},\mathscr{R}_{\Omega,\mathrm{rel}}).$$

$$R_{D,s}=\frac{C_{\mathrm{syn}}}{D}+C_{\mathrm{meta}}2^{-s/m}+\mathcal{A}_{D},\qquad R_{D,s}^{(\vartheta)}=C_{\vartheta}D^{-\vartheta}+C_1D^{-1}+C_{\mathrm{meta}}2^{-s/m}+\mathcal{A}_{D}.$$

$$U+R_{D,s}\leq\varepsilon_H\Rightarrow\text{feasible},\qquad L-R_{D,s}>\varepsilon_H\Rightarrow\text{impossible};\quad\text{otherwise unresolved}.$$

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.

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

Failed on benchmark

Stage 1 · Toy mechanism gate: Passed ✓

Stage 2 · Mechanism transferred to benchmark: Not reproduced ✗

Stage 2 · Practical benchmark result: No measurable effect

Methodology: 8 paired seeds, permutation test (p<0.05) against a learning-rate-tuned baseline. How verification works

Stage 1 — Mechanism check agent confidence 8/10

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.

Agent confidence: 8/10

Baseline: Exhaustive evaluation: 90/90 candidates evaluated; 24 successful and 66 doomed.
Idea: Certificate screening: 68 unresolved candidates evaluated, 22 feasible candidates certified, 22 evaluations avoided, 100% feasible-certificate precision, 100% successful-candidate retention, 0/66 doomed candidates rejected.

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.

How to run: python3 certificate_experiment.py

Stage 2 — Benchmark vs. tuned baseline (arXiv:2607.23390) Sep 2, 2026

No significant difference vs. the tuned baseline (−-0.0%, p=1, wins 0 / 8 paired seeds).

Benchmark: Tabular regression (Friedman)
Model: variable_mlp
Paired seeds: 8
Baseline mean: 11.3431
Idea mean: 11.3431
Effect (Δ): 0 (−-0.0%; negative = idea better)
Wins: 0 / 8 paired seeds
p-value: 1 (permutation test, 20 000 shuffles)
Smallest detectable effect: ±0.0%
Mechanism: Not confirmed ✗
Practical effect: No effect
Baseline tuning: swept over 12 configs

Limitations:

Only the registered tabular track was tested. Actual integer quantization, activation and weight bit widths, codebook metadata, GPU-hour savings, bootstrap confidence intervals, and the vision, sequence, and dynamics tracks were not tested. Calibration used a small 400/400 split and produced a very loose floor bracket.

How to run: python3 bench_depth_certificate.py

Verdict computed by deterministic test code from paired-seed statistics — not by the language model.

Stage 2 — Benchmark Aug 31, 2026 latest Failed ✗

Agent confidence: 10/10

Implemented and ran the depth-feasibility certificate on the registered tabular Friedman regression track. The certificate selected depth 4 at lr=0.006, identical to the tuned baseline, yielding delta_mean=0 and permutation p_value=1.0. The mechanism signature was not confirmed: observed trained-model depth slope was -0.38086177111636127 versus the predicted -1.0.

Baseline: Best depth=4, lr=0.006; full 8-seed MSE mean=11.343093633651733, std=1.098638538871677.
Certificate: Floor bracket [-19.786727674992903,5.600807448724559], C_syn=70.65927382249097, chosen depth=4.
Idea: Depth=4, lr=0.006; full 8-seed MSE mean=11.343093633651733, std=1.098638538871677; paired delta=0.0, p=1.0.
Mechanism signature: Predicted exponent=-1.0; observed slope=-0.38086177111636127; confirmed=false.

Benchmark result