A Two-Parameter Weibull Framework for Diagnosing Transformer Weight Distributions
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Computer Science > Machine Learning
arXiv:2605.18898 (cs)
[Submitted on 17 May 2026]
Title:A Two-Parameter Weibull Framework for Diagnosing Transformer Weight Distributions
Authors:Tiexin Ding
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Abstract:We apply the Weibull distribution -- a two-parameter family from extreme-value theory -- as a diagnostic framework for element-wise weight magnitude distributions in transformers. At initialization, i.i.d. Gaussian weights give |w| ~ HalfNormal, yielding k ~ 1.20 via middle-80% probability-plot fit (the protocol used throughout this work). This anchor makes k a principled, architecture-independent measuring stick for training dynamics; fitting each weight matrix independently at every layer at every checkpoint enables per-component, per-layer, and per-step diagnostics that aggregate statistics cannot resolve.
Applying this framework to 12 model entries spanning 7 architectural families (Pythia, OLMo-1/2, LLaMA-3, Mistral, Qwen2.5/3) reveals three findings. First, FFN modules and the attention output projection W_o -- the Transmission Class -- fall in a narrow k band: median terminal k in [1.186, 1.204] across 12 entries (cross-family CV = 0.51%), shared across SwiGLU/GeLU activations, Pre-LN/QK-Norm placements, and 70M-14B sizes. Second, the attention input projections W_q, W_k -- the Selection Class -- depart from the Weibull family, with severity shaped by storage: separately-stored Q/K (OLMo-1, OLMo-2) yields k in [0.76, 0.99] (deep); GQA models yield k in [1.10, 1.16] (mild); Pythia's merged W_qkv occupies a transitional zone tracking training budget T/tau monotonically. Third, lambda grows substantially during training and scales with sqrt(eta/lambda_wd) within the Pythia family (Pearson r = 0.94, three Transmission kinds), directionally consistent with Fan et al. (2025). The two parameters carry independent information: k labels the functional class, lambda labels training progress.
We release npm-weibull-py v0.4 (Python library) and DATABASE_v9_1 at this https URL .
| Comments: | 27 pages, 14 figures. Companion library npm-weibull-py and benchmark database available at this https URL |
| Subjects: | Machine Learning (cs.LG) |
| Cite as: | arXiv:2605.18898 [cs.LG] |
| (or arXiv:2605.18898v1 [cs.LG] for this version) | |
| https://doi.org/10.48550/arXiv.2605.18898
arXiv-issued DOI via DataCite
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