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+Alpaca-GPT4 dataset~\cite{alpaca} using both the pretrained base model (after 15B tokens of pretraining) and a supervised fine-tuned checkpoint (after 1K SFT steps). Figure~\ref{fig:ifd} shows the distribution. @@ -184,13 +185,59 @@ $5.3\times$ more than unconditional loss, confirming that SFT teaches instruction following rather than merely improving generic language modeling. Detailed analysis is provided in Appendix~\ref{app:ifd}. +\subsubsection{IFD vs.\ Loss Ratio} + +We further define the \emph{loss ratio}---the fraction of +conditional loss retained after SFT---as: +\begin{equation} +\text{Loss Ratio} = \frac{L_{\text{cond}}^{\text{ckpt}}}{L_{\text{cond}}^{\text{base}}}. +\end{equation} +Table~\ref{tab:ifd_lossratio_corr} reports the pairwise correlations. + +\begin{table}[H] +\centering +\caption{Pairwise correlations among IFD and Loss Ratio.} +\label{tab:ifd_lossratio_corr} +\small +\begin{tabular}{@{}lcc@{}} +\toprule +\textbf{Pair} & \textbf{Pearson $r$} & \textbf{Spearman $\rho$} \\ +\midrule +IFD\textsubscript{base} vs.\ Loss Ratio & $+0.10$ & $+0.05$ \\ +IFD\textsubscript{ckpt} vs.\ Loss Ratio & $+0.90$ & $+0.91$ \\ +IFD\textsubscript{base} vs.\ IFD\textsubscript{ckpt} & $+0.38$ & $+0.49$ \\ +\bottomrule +\end{tabular} +\end{table} + +The near-perfect correlation between IFD\textsubscript{ckpt} and +Loss Ratio ($r = 0.90$) reflects a mathematical near-identity: +both are dominated by $L_{\text{cond}}^{\text{ckpt}}$ in the +numerator. Consequently, IFD\textsubscript{ckpt} is +redundant---it essentially measures how much the conditional loss +has dropped after SFT, i.e., the learning speed of each sample. In contrast, IFD\textsubscript{base} and Loss +Ratio are nearly orthogonal ($r = 0.10$), forming a complementary +two-dimensional screening space: IFD\textsubscript{base} measures +``how hard does the base model find this,'' while Loss Ratio +measures ``how much did SFT improve it.'' Samples with high +IFD\textsubscript{base} \emph{and} low Loss Ratio are the most +informative for training. + +\begin{figure}[H] +\centering +\includegraphics[width=0.80\linewidth]{data/ifd_both_vs_lossratio.png} +\caption{IFD\textsubscript{base} vs.\ Loss Ratio (left), +IFD\textsubscript{ckpt} vs.\ Loss Ratio (right).} +\label{fig:ifd_lossratio} +\end{figure} + % ====================================================================== \section{Model Architecture} % ====================================================================== The model is a 24-layer decoder-only Transformer with Grouped Query Attention -(GQA)~\cite{ainslie2023gqa} and SwiGLU feed-forward blocks~\cite{shazeer2020glu}, -with Rotary Position Embedding (RoPE)~\cite{su2024roformer}. +(GQA)~\cite{ainslie2023gqa}, SwiGLU feed-forward blocks~\cite{shazeer2020glu}, +and Rotary Position Embedding (RoPE)~\cite{su2024roformer}. Table~\ref{tab:model_config} summarizes the configuration. \begin{table}[H] @@ -210,12 +257,49 @@ Norm & RMSNorm ($\epsilon=10^{-5}$) & RoPE $\theta$ & 10,000 \\ \end{tabular} \end{table} -Each decoder block $\ell$ computes: +With Grouped Query Attention~\cite{ainslie2023gqa} ($n_q = 24$ query +heads, $n_{kv} = 4$ key/value heads, group size $g = n_q / n_{kv} = 6$): \begin{equation} -\mathbf{h}_\ell = \mathbf{x}_\ell + \operatorname{GQA}\bigl(\operatorname{RMSNorm}(\mathbf{x}_\ell)\bigr), \qquad -\mathbf{x}_{\ell+1} = \mathbf{h}_\ell + \operatorname{MLP}\bigl(\operatorname{RMSNorm}(\mathbf{h}_\ell)\bigr), +\begin{aligned} +\operatorname{GQA}(\mathbf{X}) &= \operatorname{Concat}\bigl(\operatorname{head}_1,\dots,\operatorname{head}_{n_q}\bigr)\mathbf{W}_O,\\[2mm] +\operatorname{head}_i &= \operatorname{Attn}\Bigl( +\mathbf{X}\mathbf{W}_Q^{(i)},\, +\mathbf{X}\mathbf{W}_K^{(\lfloor i / g \rfloor)},\, +\mathbf{X}\mathbf{W}_V^{(\lfloor i / g \rfloor)} +\Bigr), +\end{aligned} +\end{equation} +where $\operatorname{Attn}(\mathbf{Q},\mathbf{K},\mathbf{V}) = +\operatorname{Softmax}(\mathbf{Q}\mathbf{K}^{\mkern-1mu\mathsf{T}} / \sqrt{d_h})\mathbf{V}$. +Rotary Position Embedding (RoPE)~\cite{su2024roformer} encodes position +$m$ by rotating pairs of hidden dimensions: +\begin{equation} +\operatorname{RoPE}(\mathbf{x}_m)_i = +\begin{cases} +x_{m,i}\cos(m\theta_{j}) - x_{m,i+1}\sin(m\theta_{j}), & i = 2j,\\[2mm] +x_{m,i-1}\sin(m\theta_{j}) + x_{m,i}\cos(m\theta_{j}), & i = 2j+1, +\end{cases} +\end{equation} +with frequency $\theta_j = 10000^{-2j/d}$ for $j = 0,\dots,d/2-1$. +The SwiGLU~\cite{shazeer2020glu} feed-forward applies a gated Swish +non-linearity: +\begin{equation} +\operatorname{MLP}(\mathbf{x}) = \mathbf{W}_{\text{down}}\Bigl( +\mathbf{W}_{\text{up}}\mathbf{x} \odot +\operatorname{SiLU}\bigl(\mathbf{W}_{\text{gate}}\mathbf{x}\bigr) +\Bigr), +\label{eq:swiglu} +\end{equation} +where $\operatorname{SiLU}(z) = z / (1 + e^{-z})$. + +Each decoder block $\ell$ then applies pre-norm residual connections: +\begin{equation} +\begin{aligned} +\mathbf{h}_\ell &= \mathbf{x}_\ell + \operatorname{GQA}\bigl(\operatorname{RMSNorm}(\mathbf{x}_\ell)\bigr),\\[2mm] +\mathbf{x}_{\ell+1} &= \mathbf{h}_\ell + \operatorname{MLP}\bigl(\operatorname{RMSNorm}(\mathbf{h}_\ell)\bigr). +\end{aligned} +\label{eq:decoder_block} \end{equation} -where $\operatorname{MLP}(\mathbf{x}) = \mathbf{W}_{\text{down}}(\mathbf{W}_{\text{up}}\mathbf{x} \odot \operatorname{SiLU}(\mathbf{W}_{\text{gate}}\mathbf{x}))$. \subsection{Initialization} @@ -589,6 +673,12 @@ in the high-IFD tail. % ====================================================================== \begin{thebibliography}{99} +\bibitem{alpaca} +R.~Taori, I.~Gulrajani, T.~Zhang, Y.~Dubois, X.~Li, C.~Guestrin, +P.~Liang, T.~B.~Hashimoto. +Alpaca: A strong, replicable instruction-following model. +\textit{Stanford Center for Research on Foundation Models (CRFM)}, 2023. + \bibitem{ainslie2023gqa} J.~Ainslie, J.~Lee-Thorp, M.~de Jong, Y.~Zemlyanskiy, F.~Lebr\'on, S.~Sanghai. GQA: Training generalized multi-query transformer models from multi-head