Add text for second BPGD plot

src/thesis/chapters/4_decoding_under_dems.tex

@@ -2253,7 +2253,7 @@ opposite of the corresponding dependence under plain \ac{bp}
 rather than helps, even though smaller $F$ implies a larger overlap
 in both cases.
 
-This inversion provides the clue to what is going wrong.
+This inversion provides a clue to what is going wrong.
 Recall from
 \Cref{subsec:Warm-Start Belief Propagation with Guided Decimation Decoding}
 that the warm start for \ac{bpgd} carries over not only the \ac{bp}
@@ -2281,6 +2281,19 @@ Decreasing $F$ at fixed $W$, by contrast, enlarges only the overlap
 without enlarging the window, so the freezing effect is no longer
 offset and warm-start performance worsens with smaller $F$.
 
+% [Thread] Test hypothesis by carying number of iterations
+
+The hypothesis from the previous paragraph is straightforward to test.
+If the warm-start regression in \Cref{fig:bpgd_wf} is indeed caused by
+the decimation state being carried across the window boundary, then
+reducing the maximum number of inner \ac{bp} iterations
+$n_\text{iter}$ should reduce the maximum number of \acp{vn} that can
+be decimated before window $\ell$ commits, and the warm-start
+performance should approach that of warm-start under plain \ac{bp} as
+$n_\text{iter}$ is lowered.
+We therefore now vary $n_\text{iter}$ at fixed window parameters and
+fixed physical error rate.
+
 \begin{figure}[t]
     \centering
     \hspace*{-6mm}
@@ -2353,6 +2366,7 @@ offset and warm-start performance worsens with smaller $F$.
 
         \caption{\red{Lorem ipsum dolor sit amet, consectetur adipiscing
         elit, sed do eiusmod tempor incididunt}}
+        \label{fig:bpgd_iter_W}
     \end{subfigure}%
     \hfill%
     \begin{subfigure}{0.48\textwidth}
@@ -2425,13 +2439,107 @@ offset and warm-start performance worsens with smaller $F$.
 
         \caption{\red{Lorem ipsum dolor sit amet, consectetur adipiscing
         elit, sed do eiusmod tempor incididunt}}
+        \label{fig:bpgd_iter_F}
     \end{subfigure}
 
     \caption{
         \red{\lipsum[2]}
     }
+    \label{fig:bpgd_iter}
 \end{figure}
 
+% [Experimental parameters] Figure 4.11
+
+\Cref{fig:bpgd_iter} shows the per-round \ac{ler} of \ac{bpgd}
+sliding-window decoding as a function of the maximum number of inner
+\ac{bp} iterations $n_\text{iter}$.
+The dashed colored curves correspond to cold-start sliding-window
+decoding and the solid colored curves to warm-start, again carrying
+over both the \ac{bp} messages and the channel \acp{llr} on the
+overlap region.
+The physical error rate is fixed at $p = 0.0025$ and the iteration
+budget is swept over $n_\text{iter} \in \{32, 128, 256, 512, 1024,
+1536, 2048, 2560, 3072, 3584, 4096\}$.
+\Cref{fig:bpgd_iter_W} sweeps over the window size with
+$W \in \{3, 4, 5\}$ at fixed step size $F = 1$, and
+\Cref{fig:bpgd_iter_F} sweeps over the step size with
+$F \in \{1, 2, 3\}$ at fixed window size $W = 5$.
+
+% [Description] Figure 4.11
+
+For low iteration budgets, all curves in both panels behave similarly
+to the plain-\ac{bp} curves in
+\Cref{fig:bp_w_over_iter,fig:bp_f_over_iter}.
+The per-round \ac{ler} decreases gradually with $n_\text{iter}$, and
+the warm-start curves lie below their cold-start counterparts at
+matching window parameters.
+As $n_\text{iter}$ continues to grow, however, the cold-start curves
+undergo a sharp drop, after which they lie roughly an order of
+magnitude below the warm-start curves, and eventually settle into a
+flat plateau.
+The warm-start curves first reach a minimum at an intermediate
+iteration count, then turn upwards, and finally also approach a
+plateau, albeit at a substantially higher per-round \ac{ler}.
+The warm-start curves are also less smooth than the cold-start ones
+at certain points.
+
+In \Cref{fig:bpgd_iter_W}, the iteration count at which the
+cold-start curves drop sharply increases with the window size, from
+roughly $n_\text{iter} \approx 2000$ for $W = 3$, to
+$\approx 2500$ for $W = 4$, to $\approx 3000$ for $W = 5$.
+The corresponding warm-start curves reach their minima at
+approximately the same iteration counts, and from there onwards begin
+to worsen.
+At the largest sampled iteration budget, the cold-start curves have
+plateaued at per-round \acp{ler} of order $10^{-3}$ while the
+warm-start curves have grown to per-round \acp{ler} above
+$4 \times 10^{-2}$.
+In \Cref{fig:bpgd_iter_F}, the cold-start curves drop sharply at
+roughly the same iteration count for all three step sizes, while
+the warm-start curves now show a clear reordering as $n_\text{iter}$
+grows.
+At low iteration budgets the warm-start ordering matches the
+cold-start ordering, with $F = 1$ best and $F = 3$ worst, but at the
+largest iteration budget this ordering is fully inverted: warm-start
+$F = 1$ is now the worst and $F = 3$ the best.
+
+% [Interpretation] Figure 4.11
+
+The cold-start behavior matches the preliminary investigation that
+motivated our choice of $n_\text{iter} = 5000$ in \Cref{fig:bpgd_wf}.
+At low iteration budgets the inner decoder has not yet had time to
+decimate a substantial fraction of the \acp{vn}, so its behavior
+remains close to that of plain \ac{bp}.
+Once the iteration budget is large enough for the decimation effects
+to become pronounced, the per-round \ac{ler} drops sharply and
+\ac{bpgd} delivers its intended performance gain.
+Once every \ac{vn} in a window has been decimated, no further
+iterations can change the outcome, which is why each cold-start curve
+reaches a flat plateau.
+
+The warm-start curves exhibit the same two regimes, but with the
+opposite outcome in the second one, which is exactly what the
+hypothesis from the previous paragraph predicts.
+At low $n_\text{iter}$, decimation has not yet taken hold and the
+warm-start initialization carries forward only the \ac{bp} messages
+in any meaningful sense, so the warm-start variant outperforms its
+cold-start counterpart for the same reason as in the plain-\ac{bp}
+investigation.
+As $n_\text{iter}$ grows past the point where decimation begins to
+matter, the decimation information carried over starts to impede the
+decoding performance.
+
+The same mechanism explains the inversion of the step-size ordering
+in \Cref{fig:bpgd_iter_F}.
+At low iteration budgets, the ordering is set by the same overlap
+argument as for plain \ac{bp}: smaller $F$ implies a larger overlap
+between consecutive windows, more shared messages, and therefore
+better warm-start performance.
+At large iteration budgets, the ordering is set by the premature hard
+decisions of the \acp{vn}.
+We do not have a definitive explanation for the roughness visible in some
+of the warm-start curves and limit ourselves to noting it.
+
 \begin{figure}[t]
     \centering
     \hspace*{-6mm}