full_coverage_results/appendix_fitted_params.tex

\begin{table}[ht]
\centering
\caption{Chinchilla fitted parameters: $L(N,D)= E + A\,N^{-\alpha} + B\,D^{-\beta}$. Final epoch.}
\label{tab:chinchilla-params}
\resizebox{\textwidth}{!}{%
\begin{tabular}{llccccccccc}
\toprule
\textbf{Dataset} & \textbf{Design} & $E$ & $A$ & $\alpha$ & $B$ & $\beta$ & $R^2_{\text{train}}$ & $R^2_{\mathcal{H}}$ & $R^2_{\mathcal{H}_{\mathrm{col}}}$ & $R^2_{\mathcal{H}_{\mathrm{nc}}}$ \\
\midrule
Wikipedia      & CO & $1.8570$ & $4.26\times10^{6}$ & $0.9077$ & $2.99\times10^{4}$ & $0.5181$ & 0.9919 & 0.8644 & 0.9664 & 0.6329 \\
               & NC & $0.00\times10^{0}$ & $239.1559$ & $0.2600$ & $5.02\times10^{4}$ & $0.5522$ & 0.9374 & 0.8981 & 0.9286 & 0.8288 \\
\midrule
Cosmopedia     & CO & $1.3668$ & $4249.5355$ & $0.4708$ & $8.53\times10^{6}$ & $0.8738$ & 0.9860 & 0.9501 & 0.9614 & 0.9368 \\
               & NC & $1.1740$ & $871.9566$ & $0.3698$ & $1.37\times10^{7}$ & $0.9018$ & 0.9503 & 0.9516 & 0.9558 & 0.9466 \\
\midrule
Red Pajama     & CO & $2.4002$ & $1.87\times10^{5}$ & $0.7062$ & $2.68\times10^{4}$ & $0.5175$ & 0.9931 & 0.8803 & 0.9670 & 0.6918 \\
               & NC & $0.2424$ & $118.9487$ & $0.2089$ & $5.02\times10^{4}$ & $0.5570$ & 0.9436 & 0.9160 & 0.9394 & 0.8647 \\
\midrule
Pes2o          & CO & $2.1168$ & $1.66\times10^{4}$ & $0.5657$ & $2.45\times10^{5}$ & $0.6673$ & 0.9827 & 0.8957 & 0.9244 & 0.8570 \\
               & NC & $0.00\times10^{0}$ & $68.7992$ & $0.1744$ & $2.73\times10^{7}$ & $0.9506$ & 0.9350 & 0.9453 & 0.9569 & 0.9295 \\
\midrule
C4             & CO & $0.4104$ & $47.5945$ & $0.1611$ & $3.59\times10^{7}$ & $0.9707$ & 0.9794 & 0.9451 & 0.9600 & 0.9275 \\
               & NC & $1.1306$ & $82.6447$ & $0.2157$ & $3.82\times10^{4}$ & $0.5796$ & 0.9342 & 0.9539 & 0.9625 & 0.9437 \\
\bottomrule
\end{tabular}%
}
\end{table}


\begin{table}[ht]
\centering
\caption{Kaplan fitted parameters: $L(N,D)=\bigl[(N_c/N)^{\alpha_N/\alpha_D} + D_c/D\bigr]^{\alpha_D}$. Final epoch.}
\label{tab:kaplan-params}
\resizebox{\textwidth}{!}{%
\begin{tabular}{llccccccccc}
\toprule
\textbf{Dataset} & \textbf{Design} & $N_c$ & $D_c$ & $\alpha_N$ & $\alpha_D$ & $\alpha_N/\alpha_D$ & $R^2_{\text{train}}$ & $R^2_{\mathcal{H}}$ & $R^2_{\mathcal{H}_{\mathrm{col}}}$ & $R^2_{\mathcal{H}_{\mathrm{nc}}}$ \\
\midrule
Wikipedia      & CO & $6.78\times10^{8}$ & $6.90\times10^{8}$ & $0.3821$ & $0.5630$ & $0.6787$ & 0.9849 & 0.6121 & 0.8619 & 0.0451 \\
               & NC & $1.11\times10^{10}$ & $2.11\times10^{7}$ & $0.2078$ & $2.0000$ & $0.1039$ & 0.9272 & 0.8845 & 0.9090 & 0.8285 \\
\midrule
Cosmopedia     & CO & $1.17\times10^{9}$ & $1.15\times10^{8}$ & $0.2736$ & $0.8412$ & $0.3253$ & 0.9847 & 0.9421 & 0.9589 & 0.9227 \\
               & NC & $2.53\times10^{9}$ & $6.15\times10^{7}$ & $0.2268$ & $1.0755$ & $0.2109$ & 0.9496 & 0.9586 & 0.9608 & 0.9560 \\
\midrule
Red Pajama     & CO & $3.41\times10^{9}$ & $10^{9}$ & $0.2958$ & $0.5125$ & $0.5771$ & 0.9866 & 0.6656 & 0.8882 & 0.1813 \\
               & NC & $1.89\times10^{11}$ & $3.74\times10^{7}$ & $0.1621$ & $1.4110$ & $0.1149$ & 0.9197 & 0.8747 & 0.9036 & 0.8113 \\
\midrule
Pes2o          & CO & $5.32\times10^{9}$ & $2.10\times10^{8}$ & $0.2360$ & $0.6847$ & $0.3447$ & 0.9810 & 0.8242 & 0.8815 & 0.7469 \\
               & NC & $4.67\times10^{10}$ & $2.76\times10^{7}$ & $0.1688$ & $1.4352$ & $0.1176$ & 0.9364 & 0.9427 & 0.9537 & 0.9277 \\
\midrule
C4             & CO & $1.89\times10^{11}$ & $3.74\times10^{7}$ & $0.1407$ & $1.1925$ & $0.1180$ & 0.9801 & 0.9425 & 0.9559 & 0.9267 \\
               & NC & $1.89\times10^{11}$ & $3.74\times10^{7}$ & $0.1403$ & $1.1003$ & $0.1276$ & 0.9278 & 0.9402 & 0.9498 & 0.9289 \\
\bottomrule
\end{tabular}%
}
\end{table}


\begin{table}[ht]
\centering
\caption{Droppo--Elibol fitted parameters. Final epoch.}
\label{tab:droppo-params}
\resizebox{\textwidth}{!}{%
\begin{tabular}{llcccccccccc}
\toprule
\textbf{Dataset} & \textbf{Design} & $L_\infty$ & $N_C$ & $D_C$ & $\alpha_N$ & $\alpha_D$ & $\alpha$ & $R^2_{\text{train}}$ & $R^2_{\mathcal{H}}$ & $R^2_{\mathcal{H}_{\mathrm{col}}}$ & $R^2_{\mathcal{H}_{\mathrm{nc}}}$ \\
\midrule
Wikipedia      & CO & $2.0943$ & $4.64\times10^{7}$ & $7.72\times10^{8}$ & $0.7349$ & $0.5050$ & $0.7201$ & 0.9917 & 0.8104 & 0.9669 & 0.4555 \\
               & NC & $0.1138$ & $1.36\times10^{8}$ & $2.57\times10^{7}$ & $0.2982$ & $0.8525$ & $2.0000$ & 0.9566 & 0.9157 & 0.9396 & 0.8611 \\
\midrule
Cosmopedia     & CO & $0.5095$ & $8.02\times10^{7}$ & $3.34\times10^{7}$ & $0.3585$ & $1.0949$ & $1.4756$ & 0.9860 & 0.9597 & 0.9659 & 0.9524 \\
               & NC & $0.5559$ & $8.02\times10^{7}$ & $3.34\times10^{7}$ & $0.3291$ & $1.1579$ & $1.4807$ & 0.9540 & 0.9628 & 0.9663 & 0.9587 \\
\midrule
Red Pajama     & CO & $1.9954$ & $3.69\times10^{6}$ & $2.61\times10^{7}$ & $1.0163$ & $0.6746$ & $1.9179$ & 0.9933 & 0.9150 & 0.9626 & 0.8115 \\
               & NC & $0.0544$ & $1.61\times10^{9}$ & $1.97\times10^{7}$ & $0.2151$ & $0.7760$ & $2.0000$ & 0.9594 & 0.9185 & 0.9385 & 0.8748 \\
\midrule
Pes2o          & CO & $1.0470$ & $8.02\times10^{7}$ & $3.34\times10^{7}$ & $0.3630$ & $0.9827$ & $1.3926$ & 0.9830 & 0.8936 & 0.9164 & 0.8629 \\
               & NC & $0.5344$ & $1.10\times10^{8}$ & $1.52\times10^{7}$ & $0.2636$ & $1.3121$ & $1.8328$ & 0.9428 & 0.9536 & 0.9636 & 0.9402 \\
\midrule
C4             & CO & $0.4676$ & $3.48\times10^{8}$ & $1.49\times10^{7}$ & $0.2054$ & $1.6378$ & $1.7943$ & 0.9807 & 0.9450 & 0.9549 & 0.9333 \\
               & NC & $0.0503$ & $1.23\times10^{10}$ & $7.80\times10^{6}$ & $0.1527$ & $1.2661$ & $1.9840$ & 0.9404 & 0.9542 & 0.9631 & 0.9436 \\
\bottomrule
\end{tabular}%
}
\end{table}


\begin{table}[ht]
\centering
\caption{Parametric-Epoch fitted parameters. Epoch-Expanded mode.}
\label{tab:epoch-params}
\resizebox{\textwidth}{!}{%
\begin{tabular}{llccccccccccc}
\toprule
\textbf{Dataset} & \textbf{Design} & $A$ & $\alpha$ & $B$ & $\beta$ & $E$ & $R^*_D$ & $R^*_N$ & $R^2_{\text{train}}$ & $R^2_{\mathcal{H}}$ & $R^2_{\mathcal{H}_{\mathrm{col}}}$ & $R^2_{\mathcal{H}_{\mathrm{nc}}}$ \\
\midrule
Wikipedia      & CO & $4.22\times10^{5}$ & $0.7462$ & $2.19\times10^{4}$ & $0.5150$ & $1.5380$ & $1.0744$ & $0.1000$ & 0.9853 & 0.8666 & 0.9521 & 0.6905 \\
               & NC & $194.4072$ & $0.2440$ & $9.09\times10^{4}$ & $0.6014$ & $1.86\times10^{-11}$ & $1.4687$ & $0.1000$ & 0.9585 & 0.9155 & 0.9317 & 0.8817 \\
\midrule
Cosmopedia     & CO & $7.22\times10^{4}$ & $0.6565$ & $1.45\times10^{6}$ & $0.7866$ & $1.7525$ & $1.4995$ & $0.1529$ & 0.9841 & 0.9496 & 0.9646 & 0.9335 \\
               & NC & $303.6931$ & $0.2902$ & $1.46\times10^{6}$ & $0.7898$ & $0.6179$ & $1.2855$ & $0.1000$ & 0.9656 & 0.9684 & 0.9719 & 0.9646 \\
\midrule
Red Pajama     & CO & $1.15\times10^{5}$ & $0.6669$ & $1.26\times10^{4}$ & $0.4853$ & $2.1637$ & $1.1624$ & $0.1000$ & 0.9879 & 0.8905 & 0.9604 & 0.7560 \\
               & NC & $97.2407$ & $0.1916$ & $4.86\times10^{4}$ & $0.5664$ & $5.23\times10^{-12}$ & $1.6241$ & $0.1000$ & 0.9644 & 0.9290 & 0.9427 & 0.9023 \\
\midrule
Pes2o          & CO & $3.07\times10^{4}$ & $0.5995$ & $4.75\times10^{5}$ & $0.7225$ & $2.2649$ & $1.8972$ & $0.1194$ & 0.9862 & 0.9232 & 0.9495 & 0.8911 \\
               & NC & $70.3865$ & $0.1766$ & $8.92\times10^{5}$ & $0.7619$ & $2.25\times10^{-11}$ & $2.1882$ & $0.1000$ & 0.9616 & 0.9464 & 0.9513 & 0.9405 \\
\midrule
C4             & CO & $1.68\times10^{6}$ & $0.8795$ & $1.84\times10^{7}$ & $0.9630$ & $3.0884$ & $0.8006$ & $0.5642$ & 0.9781 & 0.8183 & 0.8571 & 0.7753 \\
               & NC & $34.4084$ & $0.1358$ & $1.82\times10^{5}$ & $0.6845$ & $2.94\times10^{-11}$ & $2.3521$ & $0.1000$ & 0.9361 & 0.9570 & 0.9590 & 0.9547 \\
\bottomrule
\end{tabular}%
}
\end{table}