Add correlation analysis interpretation to IFD bias section

main.tex

@@ -141,9 +141,14 @@ quality scores that can be used to filter low-quality samples.
 
 Instruction Fulfillment Difficulty (IFD)~\cite{li2023ifd} quantifies
 how challenging an instruction is for a model by comparing conditional
-and unconditional losses:
+and unconditional per-token losses over a response
+$\mathbf{y} = (y_1,\dots,y_T)$:
 \begin{equation}
-\mathrm{IFD} = \frac{L_{\text{cond}}}{L_{\text{uncond}}}.
+\begin{aligned}
+\mathrm{IFD} &= \frac{L_{\text{cond}}}{L_{\text{uncond}}},\\[2mm]
+L_{\text{cond}} &= -\frac{1}{T}\sum_{t=1}^T \log P(y_t \mid \mathbf{x}, y_{<t}),\\[2mm]
+L_{\text{uncond}} &= -\frac{1}{T}\sum_{t=1}^T \log P(y_t \mid y_{<t}).
+\end{aligned}
 \end{equation}
 An IFD $>1$ indicates the instruction increases the loss relative to
 unconditional generation (the model struggles to follow it), while
@@ -488,26 +493,98 @@ necessity, not a signal of instruction difficulty. Consequently, IFD,
 being a ratio of two such averages, inherits a systematic length bias:
 short responses inflate IFD variance.
 
-Figure~\ref{fig:length_bias} illustrates this artifact. Responses
+Figure~\ref{fig:length_bias} confirms this artifact across a 9-panel
-with $<20$ tokens (e.g., ``Paris,'' ``42'') show wildly scattered
+grid. The top row shows conditional loss, middle row unconditional
-$L_{\text{uncond}}$ values, producing spurious high or low IFD scores.
+loss, and bottom row IFD---each plotted against response length and
+loss magnitude. Short responses ($<20$ tokens, e.g., ``Paris,'' ``42'')
+produce wildly scattered $L_{\text{uncond}}$ values, which in turn
+generate spurious high or low IFD scores in the bottom panels.
 Longer responses ($>50$ tokens) converge toward the model's intrinsic
-mean loss, yielding stable IFD estimates.
+mean loss, yielding stable IFD estimates across both base and
+checkpoint models.
 
 \begin{figure}[H]
 \centering
-\includegraphics[width=0.75\linewidth]{data/length_vs_loss_ifd.png}
+\includegraphics[width=0.80\linewidth]{data/ifd_length_grid.png}
-\caption{Response length vs.\ unconditional loss and IFD. Short
+\caption{Response length vs.\ conditional loss, unconditional loss,
-responses produce high-variance $L_{\text{uncond}}$ estimates,
+and IFD. Short responses produce high-variance $L_{\text{uncond}}$
-inflating IFD noise.}
+estimates, inflating IFD noise.}
 \label{fig:length_bias}
 \end{figure}
 
-This bias is well-known in the IFD literature but often omitted.
+\paragraph{Distribution summary.}
-For practitioners, a simple mitigation is to filter samples with
+Over the full 3000-sample set, the base model's conditional loss
-$<20$ response tokens before computing IFD. In our Alpaca-style
+has median $2.56$, unconditional loss median $2.80$, and IFD median
-dataset, this removes approximately $5$--$8\%$ of samples and
+$0.95$, concentrated in the $0.6$--$1.1$ range with a slight left
-substantially reduces false positives in the high-IFD tail.
+skew (cond $<$ uncond for most samples).
+
+\paragraph{Correlation analysis.}
+Table~\ref{tab:corr_bias} reports Pearson $r$ and Spearman $\rho$
+between key dimensions and the three IFD components.
+
+Three patterns stand out:
+
+\begin{enumerate}[nosep]
+  \item \textbf{Instruction length is nearly independent}
+        ($r \approx 0$ for all three targets). The length of the
+        instruction text itself has no meaningful correlation with
+        either loss or IFD. The slight negative IFD correlation
+        ($r = -0.24$, $\rho = -0.35$) is an indirect artifact driven
+        by response length (longer instructions tend to elicit shorter
+        answers in our Alpaca distribution).
+
+  \item \textbf{Response length is the dominant confound.}
+        $L_{\text{uncond}}$ shows a strong negative monotonic trend
+        ($\rho = -0.70$), a direct consequence of the per-token
+        average variance scaling as $1/T$ (Section~\ref{sec:ifd_bias}).
+        $L_{\text{cond}}$ has a weaker negative correlation
+        ($r = -0.38$), because conditional generation already
+        constrains the output distribution regardless of length.
+        The net effect on IFD is a moderate positive bias
+        ($r = +0.31$, $\rho = +0.47$): long responses produce
+        higher IFD not because they are harder, but because
+        $L_{\text{uncond}}$ drops faster with length than
+        $L_{\text{cond}}$.
+
+  \item \textbf{The ratio (resp/inst) is collinear with response
+        length} and provides no independent information.
+        All three columns mirror those of response length with
+        slightly attenuated magnitudes. Filtering by response
+        length alone suffices.
+\end{enumerate}
+
+The consistently larger $\rho$ than $r$ across all rows confirms
+that the relationships are monotonic but nonlinear---steep at
+the short end and flat for long sequences, consistent with the
+$1/T$ variance decay predicted in Section~\ref{sec:ifd_bias}.
+
+\begin{table}[H]
+\centering
+\caption{Pearson $r$ and Spearman $\rho$ between sample dimensions and IFD components.}
+\label{tab:corr_bias}
+\small
+\begin{tabular}{@{}lcccccc@{}}
+\toprule
+ & \multicolumn{2}{c}{vs.\ $L_{\text{cond}}$}
+ & \multicolumn{2}{c}{vs.\ $L_{\text{uncond}}$}
+ & \multicolumn{2}{c}{vs.\ IFD} \\
+\cmidrule(lr){2-3} \cmidrule(lr){4-5} \cmidrule(lr){6-7}
+\textbf{Dimension} & $r$ & $\rho$ & $r$ & $\rho$ & $r$ & $\rho$ \\
+\midrule
+Instruction length & $-0.01$ & $+0.04$ & $+0.11$ & $+0.22$ & $-0.24$ & $-0.35$ \\
+Response length    & $-0.38$ & $-0.46$ & $-0.52$ & $-0.70$ & $+0.31$ & $+0.47$ \\
+Ratio (resp/inst)  & $-0.32$ & $-0.41$ & $-0.46$ & $-0.67$ & $+0.30$ & $+0.52$ \\
+\bottomrule
+\end{tabular}
+\end{table}
+
+\paragraph{Practical recommendation.}
+Filter samples with response length $<20$ or $>300$ tokens before
+computing IFD. This retains the middle interval where per-token
+loss averages are stable and IFD rankings are most reliable.
+In our Alpaca-style dataset, this removes approximately
+$5$--$8\%$ of samples and substantially reduces false positives
+in the high-IFD tail.
 
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