Эксперимент: Spectral-Certified Block-Diagonal Preconditioning (#1100)

{ "worked": true, "confidence": 8, "verdict": "Built a spectral certificate module and adaptive block-merging quadratic optimizer prototype. The mechanism manifested: in the 2x2 equality case, weak coupling produced distortion/epsilon² of 2.0000, 1.0000, 0.5000, and 0.2500 for gaps 0.5, 1, 2, and 4, matching 1/eta; strong coupling approached distortion/epsilon = 0.995 at epsilon=100, and zero-gap distortion equaled epsilon exactly. The rank-one Frobenius check had 0/100 violations, while adaptive merging retained blocks at coupling 0.03, merged one pair at 0.3, and merged all blocks at 0.8; this is promising evidence for certificate-controlled merging, not a demonstrated large-scale optimizer win.", "metrics": { "baseline": "Fixed 2x2 block-diagonal preconditioner: 16 stored matrix entries; residual quadratic loss after 8 steps was 0.0, 1.03e-10, and 2.03e-5 at couplings 0.03, 0.3, and 0.8.", "idea": "Adaptive certificate: 16, 24, and 64 stored entries at couplings 0.03, 0.3, and 0.8; residual loss was 0.0, 2.22e-14, and 0.0 respectively. Certificate predictions matched the exact 2x2 shifts to numerical precision and had no rank-one violations." }, "how_to_run": "/home/maxwelhelp/main/bin/python3 experiment.py && /home/maxwelhelp/main/bin/python3 mini_optimizer.py", "files": [ "experiment.py", "mini_optimizer.py", "results.json", "optimizer_results.json", "run_output.txt", "optimizer_run.txt" ], "limitations": "Only tiny synthetic Hermitian quadratics were tested; no MLP, CIFAR-10, empirical-Fisher sketch estimation, randomized power iteration error analysis, CUDA benchmark, wall-clock/FLOP measurement, noisy online certificates, or meaningful large-scale memory comparison was performed. The adaptive method does not preserve memory savings when the certificate merges all blocks." }