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Add effective rank analysis, weight distribution evolution, Muon/SFT training figures

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- Add SVD effective rank (ER@99%) and condition number analysis across
  three checkpoints (kami-15bt, norm-15bt, muon-25bt) in appendix
- Add post-training weight distribution analysis as §5.4, verifying
  residual scaling persists throughout training
- Add per-component weight std table in appendix showing Muon constrains
  early-stage drift vs Normal init
- Add Muon optimizer training dynamics figure (muon_pt.png)
- Add SFT training metrics figure (sft_metrics.png) on mixed CN-EN data
- Restructure: move effective rank to appendix, weight distribution to
  §5.4, delete standalone §6, tighten abstract to prose-only
- Update conclusion to reference new analyses
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@@ -32,14 +32,23 @@ We present {\sc AstrAI}, an open-source framework for end-to-end training
of a 1.2B-parameter Transformer on $\sim$20B tokens. The pipeline covers
JSON-driven BBPE preprocessing with multi-strategy packing, HDF5/mmap
storage backends, and a companion SFT pipeline ({\sc Alembic}) with MinHash
deduplication and LLM-as-Judge scoring. The 24-layer decoder uses GQA, SwiGLU,
RoPE, and RMSNorm, trained with a hybrid Muon/AdamW optimizer and cosine scheduling under DDP/FSDP.
A focused BF16 stability analysis shows that GPT-2 residual scaling
($\sigma = 0.02/\sqrt{2L}$) reduces per-block residual variance by a factor
of 48, containing post-24-layer variance at 1.34 versus 17.5 under standard
initialization. Empirically, this scaling yields a sustained loss advantage
over Kaiming initialization, with the gap peaking at $\Delta = 0.79$ in the
mid-training regime.
deduplication and LLM-as-Judge scoring. The 24-layer decoder uses GQA,
SwiGLU, RoPE, and RMSNorm, trained with a hybrid Muon/AdamW optimizer and
cosine scheduling under DDP/FSDP. A BF16 stability analysis shows that
GPT-2 residual scaling substantially reduces per-block residual variance
accumulation, keeping post-training variance well below the overflow
threshold of standard initialization; empirically this yields a sustained
loss advantage over Kaiming initialization throughout training.
Post-training weight distribution analysis across three
checkpoints---varying optimizer, initialization, and training
budget---confirms that residual-scaled projections maintain narrow
distributions throughout training, preserving the numerical stability
established at initialization. An SVD-based effective rank analysis
further reveals that the model operates near its representational
capacity, with attention Q/O projections consistently showing lower
utilization than K/V projections, a pattern stable across all
configurations and consistent with the low-rank structure induced by
grouped query attention.
\end{abstract}
% ======================================================================
@@ -264,6 +273,46 @@ initialization over $\sim$20B tokens.}
Figure~\ref{fig:ckpt_comparison} compares four configurations. The left panel shows training loss for Muon (Embedding Adam + 1D Adam), Muon (Embedding Muon + 1D Adam), Kaiming init, and Normal init; the center panel zooms in on the two current Muon variants; and the right panel shows gradient norms over optimizer steps. The older Kaiming and Normal initializations converge more slowly and plateau at higher loss. Between the current variants, using Adam for the embedding layer yields lower loss and more stable gradients than using Muon embeddings.
\begin{figure}[H]
\centering
\includegraphics[width=0.85\linewidth]{data/muon_pt.png}
\caption{Muon optimizer training dynamics: training loss and learning
rate schedule for the hybrid Muon/AdamW configuration across $\sim$25B
tokens. The Muon optimizer is applied to all 2D weight matrices
(attention projections and FFN layers), while AdamW handles 1D
parameters (embeddings, biases, and normalization scales). The cosine
schedule with 2\% warmup is visible in the lower panel.}
\label{fig:muon_pt}
\end{figure}
Figure~\ref{fig:muon_pt} shows the extended training dynamics of the
Muon optimizer configuration over $\sim$25B tokens. The loss curve
exhibits the expected power-law decay in the early phase (0--5B tokens),
followed by a gradual plateau as the model approaches convergence on
the pretraining distribution. The cosine learning rate schedule
reaches its minimum at the end of training, ensuring stable weight
updates in the final phase.
\begin{figure}[H]
\centering
\includegraphics[width=0.85\linewidth]{data/sft_metrics.png}
\caption{SFT training metrics over 1{,}000 fine-tuning steps on a
mixed Chinese--English instruction dataset: training loss, learning
rate, and gradient norm. The loss drops rapidly in the first 200
steps and then enters a slower decay phase. The learning rate
follows a cosine schedule with 2\% warmup. Gradient norms stabilize
after approximately 300 steps, indicating that the fine-tuning
process has reached a stable optimization regime.}
\label{fig:sft_metrics}
\end{figure}
Figure~\ref{fig:sft_metrics} shows the supervised fine-tuning metrics
for the 1K-step SFT checkpoint used in the IFD analysis
(Appendix~\ref{app:ifd}). The training loss on the mixed
Chinese--English instruction dataset decreases from $\sim$2.5 to
$\sim$1.6 over 1{,}000 steps, with the gradient norm converging to a
stable range after the warmup phase.
% ======================================================================
\section{Numerical Stability via Residual Scaling}
\label{sec:num-stability}
@@ -362,6 +411,60 @@ at 0.125B to a peak of 0.79 at 0.625B, then narrowing to 0.38 at 1B.
The widening mid-range gap aligns with the variance accumulation region
identified in the theoretical analysis (Section~\ref{sec:num-stability}).
\subsection{Post-Training Weight Distribution}
To verify that the residual-scaling constraint persists throughout
training---not just at initialization---we compare weight value
distributions across three checkpoints:
\texttt{kami-15bt} (Muon, 15B tokens), \texttt{norm-15bt} (Normal
init, 15B tokens), and \texttt{muon-25bt} (Muon, 25B tokens).
Figure~\ref{fig:ckpt_weight_density} shows the kernel density
estimates. Training preserves the bimodal structure: residual-scaled
projections ($\mathbf{W}_o$, $\mathbf{W}_{\text{down}}$) remain
narrowly concentrated near zero, while non-scaled weights broaden
from their initial $\mathcal{N}(0,0.02)$ distribution. Two trends
are visible in the per-component weight std
(Table~\ref{tab:weight_std}, Appendix~\ref{app:weight_std}):
\begin{itemize}[nosep]
\item \textbf{Training duration drives variance growth}: the
\texttt{muon-25bt} checkpoint exhibits the largest weight std
($\sim$0.024 for attention projections), exceeding both 15B
checkpoints ($\sim$0.015--0.021), reflecting continued weight
drift from $\sigma_0 = 0.02$ as training progresses.
\item \textbf{Muon constrains early-stage drift}: at equal training
budget (15B tokens), the Muon-trained \texttt{kami-15bt} shows
\emph{smaller} std ($\sim$0.015) than the Normal-init
\texttt{norm-15bt} ($\sim$0.021). Muon's orthogonalization
step normalizes the update direction, constraining per-step
weight movement. At 25B tokens, cumulative steps overtake
this per-step constraint, producing the largest overall std.
\end{itemize}
Critically, the residual-scaled projections remain bounded at
$\sigma \approx 0.003$ across all checkpoints regardless of training
duration, confirming that the $1/\sqrt{2L}$ scaling continues to
enforce its design constraint throughout training.
\begin{figure}[H]
\centering
\includegraphics[width=0.95\linewidth]{data/ckpt_weight_density.png}
\caption{Weight value density estimates across three checkpoints,
grouped by component. Residual-scaled projections
($\mathbf{W}_o$, $\mathbf{W}_{\text{down}}$) maintain narrow
distributions near zero, while non-scaled weights show broader
spread.}
\label{fig:ckpt_weight_density}
\end{figure}
\begin{figure}[H]
\centering
\includegraphics[width=0.95\linewidth]{data/ckpt_weight_density_per_run.png}
\caption{Per-checkpoint weight density breakdowns.}
\label{fig:ckpt_weight_density_per_run}
\end{figure}
% ======================================================================
\section{Conclusion}
% ======================================================================
@@ -369,11 +472,17 @@ identified in the theoretical analysis (Section~\ref{sec:num-stability}).
We have described the end-to-end pipeline for training a 1.2B Transformer with
{\sc AstrAI}: data preprocessing with JSON-driven tokenization and packing,
a 24-layer GQA-SwiGLU architecture, callback-based training with a hybrid
Muon/AdamW optimizer under DDP/FSDP executors, and cosine scheduling. We further analyzed numerical stability
under BF16, showing that GPT-2 residual scaling ($\sigma_o = 0.02/\sqrt{2L}$)
reduces per-block residual variance by a factor of 48, keeping post-24-layer
variance at $1.34$ versus $17.5$ without scaling. The complete framework and model
weights are available at \url{https://github.com/ViperEkura/AstrAI}.
Muon/AdamW optimizer under DDP/FSDP executors, and cosine scheduling. We
further analyzed numerical stability under BF16, showing that GPT-2 residual
scaling ($\sigma_o = 0.02/\sqrt{2L}$) reduces per-block residual variance
by a factor of 48, keeping post-24-layer variance at $1.34$ versus $17.5$
without scaling. Post-training weight distribution analysis confirms that
this scaling constraint persists throughout training, with residual-scaled
projections maintaining narrow distributions ($\sigma \approx 0.003$)
regardless of optimizer or training duration. An SVD-based effective rank
analysis (Appendix~\ref{app:eff_rank}) further reveals that the model
operates near its representational capacity. The complete framework and
model weights are available at \url{https://github.com/ViperEkura/AstrAI}.
% ======================================================================
\appendix
@@ -553,32 +662,104 @@ This stability justifies using the base-model IFD as a data
selection signal without re-evaluating after fine-tuning.
% ======================================================================
\section{Weight Distribution by Component}
\label{app:weight_dist}
\section{Effective Rank Analysis}
\label{app:eff_rank}
% ======================================================================
Figure~\ref{fig:weight_dist} shows the distribution of weight
magnitudes at initialization, grouped by component type. Embeddings
and non-residual-scaled projections (QKV, attention output, FFN
gate/up) follow $\mathcal{N}(0, 0.02)$, producing near-identical
bell curves centered at zero. The residual-scaled projections
(output projection $\mathbf{W}_o$ and FFN down-projection
$\mathbf{W}_{\text{down}}$) use $\sigma = 0.02 / \sqrt{2L} \approx 0.0029$,
visible as the narrow, sharply peaked distribution concentrated
near zero. This factor-48 variance reduction is the mechanism by
which GPT-2 residual scaling prevents BF16 underflow in deep
Transformers (Section~\ref{sec:num-stability}).
To assess how well the trained parameters utilize their allocated
capacity, we perform an SVD-based effective rank analysis on three
checkpoints: \texttt{kami-15bt} (Muon, 15B tokens), \texttt{norm-15bt}
(Normal init, 15B tokens), and \texttt{muon-25bt} (Muon, 25B tokens).
For each 2D weight matrix $\mathbf{W} \in \mathbb{R}^{m\times n}$ with
SVD $\mathbf{W} = \mathbf{U}\boldsymbol{\Sigma}\mathbf{V}^{\mkern-1mu\mathsf{T}}$,
we compute the effective rank at 99\% energy:
\begin{equation}
\text{ER@99\%} = \frac{1}{\min(m,n)} \min_k \left\{ k \;\middle|\; \frac{\sum_{i=1}^{k} \sigma_i^2}{\sum_{i=1}^{\min(m,n)} \sigma_i^2} \ge 0.99 \right\}.
\end{equation}
\begin{figure}[H]
Table~\ref{tab:eff_rank} summarizes the results. All three checkpoints
exhibit a high overall ER@99\% ($\sim$90\%), indicating that the 1.2B
model operates close to its representational capacity. Key findings:
\begin{itemize}[nosep]
\item Q/O projections show lower ER@99\% ($\sim$0.73--0.77) and
high condition numbers ($\kappa > 10^4$), consistent with the
low-rank structure of GQA (24 query heads sharing 4 KV heads).
\item K/V projections, FFN layers, and embeddings maintain high
ER@99\% ($\sim$0.96--0.98) and low condition numbers
($\kappa < 10$).
\item The overall ER@99\% varies by less than 0.01 across
checkpoints, indicating these properties are determined
primarily by architecture rather than optimizer or training
duration.
\end{itemize}
\begin{table}[H]
\centering
\includegraphics[width=0.85\linewidth]{data/weight_dist_by_component.png}
\caption{Weight distribution by component at initialization.
Each panel shows the histogram of weight values for a specific
module group (embedding, attention projections, FFN projections,
output projections). The narrow peaks correspond to the
residual-scaled $\mathbf{W}_o$ and $\mathbf{W}_{\text{down}}$
projections.}
\label{fig:weight_dist}
\end{figure}
\caption{SVD effective rank (ER@99\%) and mean condition number
($\kappa$) by component across three checkpoints.}
\label{tab:eff_rank}
\small
\begin{tabular}{@{}lcccccc@{}}
\toprule
& \multicolumn{3}{c}{\textbf{ER@99\%}}
& \multicolumn{3}{c}{\textbf{Cond.\ Number $\kappa$}} \\
\cmidrule(lr){2-4} \cmidrule(lr){5-7}
\textbf{Component} & \textbf{kami} & \textbf{norm} & \textbf{muon}
& \textbf{kami} & \textbf{norm} & \textbf{muon} \\
\midrule
attn.k\_proj & 0.967 & 0.971 & 0.960 & 8.4 & 5.4 & 5.7 \\
attn.o\_proj & 0.766 & 0.708 & 0.730 & 31{,}589 & 57{,}284 & 22{,}838 \\
attn.q\_proj & 0.754 & 0.764 & 0.756 & 46{,}644 & 32{,}072 & 14{,}472 \\
attn.v\_proj & 0.976 & 0.976 & 0.971 & 2.5 & 2.4 & 2.9 \\
embed\_tokens & 0.985 & 0.987 & 0.984 & 4.9 & 1.9 & 3.5 \\
lm\_head & 0.969 & 0.981 & 0.980 & 21.0 & 13.5 & 16.1 \\
mlp.down & 0.961 & 0.961 & 0.963 & 6.4 & 7.2 & 7.2 \\
mlp.gate & 0.966 & 0.967 & 0.965 & 5.7 & 5.3 & 5.7 \\
mlp.up & 0.967 & 0.968 & 0.965 & 4.9 & 5.9 & 4.9 \\
\midrule
\textbf{Overall ER@99\%} & \textbf{0.909} & \textbf{0.903} & \textbf{0.903} & & & \\
\bottomrule
\end{tabular}
\end{table}
% ======================================================================
\section{Per-Component Weight Statistics}
\label{app:weight_std}
% ======================================================================
Table~\ref{tab:weight_std} reports the weight standard deviation by
component for each checkpoint, supplementing the post-training weight
distribution analysis in Section~\ref{sec:num-stability}.
\begin{table}[H]
\centering
\caption{Weight std by component across checkpoints. Non-scaled
weights broaden with training duration; Muon constrains drift at
equal token count (15B) but is overtaken by longer training (25B).
Residual-scaled projections ($\mathbf{W}_o$, $\mathbf{W}_{\text{down}}$)
remain bounded.}
\label{tab:weight_std}
\small
\begin{tabular}{@{}lccc@{}}
\toprule
\textbf{Component} & \textbf{kami-15bt} & \textbf{norm-15bt} & \textbf{muon-25bt} \\
& (Muon, 15B) & (Normal, 15B) & (Muon, 25B) \\
\midrule
attn.q\_proj & 0.0154 & 0.0207 & 0.0230 \\
attn.k\_proj & 0.0153 & 0.0206 & 0.0238 \\
attn.v\_proj & 0.0146 & 0.0202 & 0.0244 \\
attn.o\_proj$^*$ & 0.0148 & 0.0084 & 0.0177 \\
mlp.up & 0.0153 & 0.0204 & 0.0237 \\
mlp.gate & 0.0155 & 0.0204 & 0.0235 \\
mlp.down$^*$ & 0.0100 & 0.0089 & 0.0180 \\
embed\_tokens & 0.0205 & 0.0205 & 0.0239 \\
lm\_head & 0.0224 & 0.0257 & 0.0298 \\
\bottomrule
\end{tabular}
\\[2pt]
\footnotesize $^*$Residual-scaled projection ($\sigma_0 = 0.02/\sqrt{2L}$).
\end{table}
% ======================================================================
\begin{thebibliography}{99}