JG1310/repro-gauss-universality-bundle

Reproduction bundle — Characterization of Gaussian Universality Breakdown in High-Dimensional Empirical Risk Minimization

ICML 2026 · OpenReview UHQDfvZBFi · arXiv 2604.03146 Authors: Mohamed Chiheb Yaakoubi, Cosme Louart, Malik Tiomoko, Zhenyu Liao.

Independent, CPU-only reproduction. Every formula is implemented from the equations printed in the paper; no code from the authors' repository is reused. Their released solver (github.com/cosmital/Empirical-risk-minimization-asymptotics) is invoked once, purely as an external cross-check, in outputs/authors_solver_crosscheck.txt.

Logbook: https://huggingface.co/spaces/JG1310/repro-characterization-of-gaussian-universality-breakdown-in-high-dimensional

Outcome

Claim	Verdict
Claim 1 — Theorem 4.3/4.4 min–max characterization of the mean and covariance	Verified, with the validity boundary of the covariance half mapped to the paper's own Corollary 2.2 (‖C_x‖ = O(1))
Claim 2 — Theorem 6.1, score = non-Gaussian xᵀμ* convolved with independent centered Gaussian α*z	Verified; 20×–222× closer to the empirical law than a fitted Gaussian

Incidental findings: two statement-level errors in Corollary 5.1 (a missing factor ½; C_x where Σ_x is required when E[x] ≠ 0), and paper Figure 2 (MNIST) is not reproducible from the text — Appendix G specifies only the digit merge and its representation sentence is truncated mid-clause.

Layout

Path	What it is
ermlib.py	Core library: losses + proximal operators, quadratic and pseudo-Huber regularizers, data models (bimodal mixtures, teacher–student, MNIST), the ERM solver, and the Theorem 4.4 fixed-point solver.
exp1_thm44_lemmas.py	16 executable lemma-level checks of the Theorem 4.3/4.4 derivation (prox identities, Eq. 15, closed-form α*², the μ* formula, Corollary 5.1, Lemma 3.2, fixed-point stability). Exits non-zero if any check fails.
exp2_convergence.py	Theorem 4.3 n-sweep: ‖μ̂ − μ*‖ and tr(Cx Ĉ) − α*², two designs × two losses, n = 50…1600.
exp3_thm33.py	Theorem 3.3: ERM under a pseudo-Huber ρ vs its quadratic surrogate ρ_q.
exp4_elastic_sweep.py	Paper Figures 1 (top) and 3 at the paper's own scale (p=30, n=50, λ=5).
exp5_score_thm61.py	Theorem 6.1: score law vs empirical, vs the Gaussian score-universality hypothesis.
exp6_mnist.py	Paper Figure 2: MNIST {3,6} vs {4,7}, raw 784-d pixels and 13×13 downsampled.
exp7_confinement.py	Paper Figure 4 / Theorem 7.1 / Corollary 7.2: subspace confinement of μ*.
exp8_alpha_diagnostic.py	The decisive experiment. 4 designs × 4 sizes, 20 000 exact-ridge solves per point, isolating the α* regime boundary.
exp9_noise_floor.py	Measures the fixed-point solver's own Monte-Carlo resolution floor, so the measured ‖μ̂ − μ*‖ gap can be interpreted.
make_figures2.py	Builds every Plotly figure + its raw CSV into figures/.
run_rest.sh	Runs experiments 3–8 in sequence.
outputs/	All JSON results, logs, and raw score samples (.npz).
figures/	Interactive HTML figures with matching CSV data tables.

Rerun

pip install numpy scipy scikit-learn plotly
python3 exp1_thm44_lemmas.py          # ~1 min, exits non-zero if any check fails
python3 exp2_convergence.py 3000      # ~50 min on 16 CPU cores
./run_rest.sh                         # exp3–exp8
python3 exp9_noise_floor.py           # ~30 min
python3 make_figures2.py

MNIST is read from mnist.npz (standard 70k MNIST, uint8 28×28). Set the MNIST constant at the top of exp6_mnist.py to point at your copy.

Total wall-clock for the full suite is roughly 5 hours on one 16-core CPU. Everything is CPU-only; no GPU is used or required.

Note on outputs/exp2.log

This log file was appended by two successive runs of exp2_convergence.py with different print formats, so it interleaves two schemas; exp2_convergence.json holds the authoritative final numbers. One row in the older text (l2, n=1600) shows a diverged solver value (alpha_*^2 = 9.4e7) from a run that did not converge at that size; it is superseded and is not used anywhere in the logbook. The α* evidence in the logbook comes from exp8_alpha_diagnostic.py, which uses closed-form exact ridge and is free of that failure mode.