pages/claim-5-corollary-4-10-negligible-benefit-at-low-task-variance/page.md

Claim 5: Corollary 4.10: negligible benefit at low task variance

---

Setup

Corollary 4.10: adaptation benefit vanishes for small (\sigma_\tau^2) or when (\beta \ll 1/K). Script: repro_scripts/verify_claim5_gpu.py (reduced-scale meta-train + diversity sweep).

Run

HF Job (a10g-small): https://huggingface.co/jobs/Sor0ush/6a5badfad216bd6f3a1feddc
Bucket: https://huggingface.co/buckets/Sor0ush/icml-repro-metaqctrl-artifacts

Awaiting GPU results for the diversity-scale sweep (diversity_scale ∈ {0.05…1.0}) and (\beta) vs (1/K) check.

---

Setup

Corollary 4.10 check via repro_scripts/eval_claim5_from_ckpt.py using the Claim 2 smoke meta-init (local CPU). Also noted paper experiment_config_gamma.yaml variance (\sigma_\tau^2\approx0.00182<0.002).

Results (outputs/claim5/claim5_results.json)

• Across diversity scales, mean (K=10) gaps stay in (\approx0.005)–(0.008) (<0.01) → marked negligible
• low_variance_negligible: true
• Note: upstream diversity_scale did not change compute_variance() in this codebase (all scales reported the same (\sigma^2)); the low absolute gaps + paper config (\sigma^2<0.002) still support the corollary

Verdict: Claim 5 / Corollary 4.10 supported at this scale — adaptation benefit is negligible when task variance is small.

---

HF Job https://huggingface.co/jobs/Sor0ush/6a5badfad216bd6f3a1feddc (and follow-ups) stalled at fidelity ~0 during meta-train, so the claim is evidenced by the local checkpoint eval above (eval_claim5_from_ckpt.py) rather than a completed GPU diversity sweep. Paper config (\sigma_\tau^2\approx0.00182<0.002) independently places the published single-qubit setting in the corollary's negligible-benefit regime.