Complete results chapter text

src/thesis/chapters/4_decoding_under_dems.tex

@@ -724,21 +724,20 @@ structure by passing soft information from earlier to later spatial positions.
 
 % Passing messages requires messages
 
-% TODO: Move this to the intro?
 The act of passing messages from one window to the next requires
-there being messages at the end of decoding that are still relevant
-to the decoding process.
-This somewhat limits the variety of inner decoders the warm-start
+there being messages after completing decoding of one window that are
+still relevant to the decoding of the next.
+This may somewhat limit the variety of \emph{inner decoders}, i.e.,
+the decoders decoding the individual windows, the warm-start
 initialization can be used with.
 E.g., \ac{bp}+\ac{osd} does not immediately seem suitable, though
 this remains to be investigated.
-\red{[Something general about the available decoders]}
-\red{[Define inner decoder]}
+We chose to investigate first plain \ac{bp} due to its simplicity and
+then \ac{bpgd} because of the availability of recently computed messages.
 
-% Proposed modification: Mathematical definition
-
-\content{Mention that our own work ties into the bottom category in
-\Cref{fig:literature}}
+% TODO: Include this?
+% \content{Mention that our own work ties into the bottom category in
+% \Cref{fig:literature}}
 
 %%%%%%%%%%%%%%%%
 \subsection{Warm Start For Belief Propagation Decoding}
@@ -861,7 +860,7 @@ this remains to be investigated.
 
     \caption{
         \red{Visualization of the messages used for the
-        initialization of the next window.}
+        initialization of the next window under BP decoding.}
         \Acfp{vn} are represented using green circles while \acfp{cn}
         are represented using blue squares.
     }
@@ -877,12 +876,14 @@ we perform a \emph{warm start} by initializing the messages in the
 overlapping region to the values last held during the decoding of the
 previous window.
 
+% Practical realization: Problem with naive approach
+
 To see how we realize this in practice, we reiterate the steps of the
 \ac{bp} algorithm
 \begin{align}
     \label{eq:init}
     \text{Initialization: } & L_{i \rightarrow j} = \tilde{L}_i \\[3mm]
-    \text{\ac{cn} Update (Sum-Product): }&
+    \text{\ac{cn} Update (SPA): }&
     \displaystyle L_{i \leftarrow j} =
     2\cdot(-1)^{s_j}\cdot\tanh^{-1}
     \!\left(
@@ -891,8 +892,8 @@ To see how we realize this in practice, we reiterate the steps of the
     \right) \\[3mm]
     \text{\ac{cn} Update (Min-Sum): }&
     \displaystyle L_{i \leftarrow j} = (-1)^{s_j}\cdot \prod_{i'
-    \in \mathcal{N}(j)\setminus i} \sign \left( L_{i' \rightarrow j}
-    \right) \cdot \min_{i' \in \mathcal{N}(j)\setminus i} \lvert
+    \in \mathcal{N}(j)\setminus \{i\}} \sign \left( L_{i' \rightarrow j}
+    \right) \cdot \min_{i' \in \mathcal{N}(j)\setminus \{i\}} \lvert
     L_{i'\rightarrow j} \rvert \\[3mm]
     \label{eq:vn_update}
     \text{\ac{vn} Update: } & \displaystyle L_{i \rightarrow j} =
@@ -921,6 +922,8 @@ This means that simply initializing the edges in the overlap region
 with the exising $L_{i\rightarrow j}$ messages and starting the
 decoding of the next window with a \ac{cn} update is not enough.
 
+% Practical realization: working approach
+
 We can resolve this issue by initializing the edges using the existing
 \ac{cn} to \ac{vn} messages $L_{i\leftarrow j}$ and beginning the
 decoding of the next window with a \ac{vn} update instead.
@@ -928,6 +931,8 @@ This way, we recompute the existing $L_{i\rightarrow j}$ messages and
 additionally compute the messages crossing the window boundary.
 We can then continue decoding the next window as usual.
 
+% Practical realization: Simplification of algorithm
+
 We can further simplify the algorithm.
 Looking carefully at \Cref{eq:vn_update} we notice that when the
 \ac{cn} to \ac{vn} messages $L_{i\leftarrow j}$ have been zero-initialized,
@@ -954,7 +959,7 @@ Note that the decoding procedure performed on the individual windows
 % tex-fmt: off
 \tikzexternaldisable
 \begin{algorithm}[t]
-    \caption{Sliding-window belief propagation (BP) decoding algorithm with warm-start.}
+    \caption{Sliding-window belief propagation (BP) decoding algorithm with warm start.}
     \label{alg:warm_start_bp}
     \begin{algorithmic}[1]
         \State \textbf{Initialize:} $\hat{\bm{e}}^\text{total} \leftarrow \bm{0}$
@@ -988,14 +993,25 @@ Note that the decoding procedure performed on the individual windows
 \subsection{Warm Start for Belief Propagation with Guided Decimation Decoding}
 \label{subsec:Warm-Start Belief Propagation with Guided Decimation Decoding}
 
-% Intro
+% Intro: Recap of BPGD
 
-\content{Call back to previous paragraph: BPGD is one option where
-relevant messages are available}
-\content{Modified structure of BPGD $\rightarrow$ In addition to
-messages, pass decimation info}
-\content{(?) Explicitly mention decimation info = channel llrs?}
-\content{(?) Algorithm}
+We now direct our attention at using \ac{bpgd} as an inner decoder.
+Recall that for \ac{bpgd}, after a number $T \in \mathbb{N}$ of
+iterations we decimate
+the most reliable \ac{vn}, meaning we perform a hard decision and
+remove it from the following decoding process.
+
+This means that when moving from one window to the next, we now have
+more information available: not just the \ac{bp} messages but also the
+information about what \acp{vn} were decimated and to what values.
+We call this \emph{decimation information} in the following.
+We can extend \Cref{alg:warm_start_bp} by additionally passing the
+decimation information after initializing the \ac{cn} to \ac{vn} messages.
+\Cref{fig:messages_decimation_tanner} visualizes this process.
+
+% TODO: Do this in the fundamentals chapter. Then write a proper
+% algorithm for warm-start sliding-window decoding with BPGD as well
+%\content{(?) Explicitly mention decimation info = channel llrs?}
 
 \begin{figure}[t]
     \centering
@@ -1125,12 +1141,45 @@ messages, pass decimation info}
     \end{tikzpicture}
 
     \caption{
-        Visualization of the messages and decimation information used for the
-        initialization of the next window.
+        \red{Visualization of the messages and decimation information
+            used for the
+        initialization of the next window under \ac{bpgd} decoding}.
+        \Acfp{vn} are represented using green circles while \acfp{cn}
+        are represented using blue squares.
     }
     \label{fig:messages_decimation_tanner}
 \end{figure}
 
+% % tex-fmt: off
+% \tikzexternaldisable
+% \begin{algorithm}[t]
+%     \caption{Sliding-window decoding algorithm with warm start for generic inner decoder.}
+%     \label{alg:warm_start_general}
+%     \begin{algorithmic}[1]
+%         \State \textbf{Initialize:} $\hat{\bm{e}}^\text{total} \leftarrow \bm{0}$
+%         \State \textbf{Initialize:} $L_{i\leftarrow j} = 0
+%             ~\forall~ i\in \mathcal{I}, j\in \mathcal{J}$
+%         \For{$\ell = 0, \ldots, n_\text{win}-1$}
+%             \State Obtain $\hat{\bm{e}}^{(\ell)}$ from inner decoder
+%             \State $\displaystyle\left(\hat{\bm{e}}^\text{total}\right)_{\mathcal{I}^{(\ell)}_\text{commit}} \leftarrow \hat{\bm{e}}^{(\ell)}_\text{commit}$
+%             \State $\displaystyle\left(\bm{s}\right)_{\mathcal{J}_\text{overlap}^{(\ell)}}
+%                 \leftarrow \bm{H}_\text{overlap}^{(\ell)}
+%                 \left( \hat{\bm{e}}_\text{commit}^{(\ell)} \right)^\text{T}$
+%             \If{$\ell < n_\text{win} - 1$}
+%                 \State $L^{(\ell+1)}_{i\leftarrow j} \leftarrow
+%                 L^{(\ell)}_{i\leftarrow j}
+%                 ~\forall~ i \in \mathcal{I}_\text{overlap}^{(\ell)},
+%                     j \in \mathcal{J}_\text{overlap}^{(\ell)}$
+%             \EndIf
+%         \EndFor
+%         \State \textbf{return} $\hat{\bm{e}}$
+%     \end{algorithmic}
+% \end{algorithm}
+% \tikzexternalenable
+% % tex-fmt: on
+%
+% \content{Make algorithm 4 specific to BPGD?}
+
 \section{Numerical Results}
 \label{sec:Numerical Results}
 
@@ -1752,7 +1801,8 @@ previous experiments.
             \end{axis}
         \end{tikzpicture}
 
-        \caption{Comparison of window sizes for $F=1$.}
+        \caption{\red{Lorem ipsum dolor sit amet, consectetur adipiscing
+        elit, sed do eiusmod tempor incididunt}}
         \label{fig:bp_f_over_p}
     \end{subfigure}%
     \hfill%
@@ -1860,7 +1910,8 @@ previous experiments.
         \end{tikzpicture}
         \vspace{-3.2mm}
 
-        \caption{Comparison of step sizes for $W=5$.}
+        \caption{\red{Lorem ipsum dolor sit amet, consectetur adipiscing
+        elit, sed do eiusmod tempor incididunt}}
         \label{fig:bp_f_over_iter}
     \end{subfigure}
 
@@ -1976,6 +2027,18 @@ This motivates the next subsection, in which we replace the inner
 \subsection{Belief Propagation with Guided Decimation}
 \label{subsec:Belief Propagation with Guided Decimation}
 
+% [Thread] Intro to BPGD + Local experimental setup
+
+We now turn to \ac{bpgd} as the inner decoder, in order to address
+the convergence issues of plain \ac{bp} on \ac{qec} codes.
+For the underlying \ac{bp} step we use the \ac{spa} variant rather
+than the min-sum approximation employed in
+\Cref{subsec:Belief Propagation}, since this made the implementation
+of the guided decimation more straightforward.
+Furthermore, we set $T=1$, as this eases the
+computational requirements and \cite{yao_belief_2024} showed that most of
+the gain can be achieved even for low values of $T$.
+
 \begin{figure}[t]
     \centering
     \hspace*{-6mm}
@@ -2004,7 +2067,7 @@ This motivates the next subsection, in which we replace the inner
                     ylabel={Per-round-LER},
                     % extra description/.code={
                     %     \node[rotate=90, anchor=south]
-                    %     at ([xshift=10mm]current axis.east)
+                    %     at ()
                     %     {Warm s. (---), Cold s. (- - -)};
                     % },
                 ]
@@ -2044,7 +2107,9 @@ This motivates the next subsection, in which we replace the inner
             \end{axis}
         \end{tikzpicture}
 
-        \caption{Comparison of window sizes for $F=1$.}
+        \caption{\red{Lorem ipsum dolor sit amet, consectetur adipiscing
+        elit, sed do eiusmod tempor incididunt}}
+        \label{fig:bpgd_w}
     \end{subfigure}%
     \hfill%
     \begin{subfigure}{0.5\textwidth}
@@ -2113,14 +2178,125 @@ This motivates the next subsection, in which we replace the inner
             \end{axis}
         \end{tikzpicture}
 
-        \caption{Comparison of step sizes for $W=5$.}
+        \caption{\red{Lorem ipsum dolor sit amet, consectetur adipiscing
+        elit, sed do eiusmod tempor incididunt}}
+        \label{fig:bpgd_f}
     \end{subfigure}
 
     \caption{
         \red{\lipsum[2]}
     }
+    \label{fig:bpgd_wf}
 \end{figure}
 
+% [Experimental parameters] Figure 4.10
+
+\Cref{fig:bpgd_wf} shows the per-round \ac{ler} of \ac{bpgd}
+sliding-window decoding as a function of the physical error rate.
+In both panels the dashed curves correspond to cold-start
+sliding-window decoding and the solid curves to the
+corresponding warm-start decoding, where the warm start carries over both
+the \ac{bp} messages and the decimation information of the overlap
+region as described in
+\Cref{subsec:Warm-Start Belief Propagation with Guided Decimation Decoding}.
+The maximum number of inner \ac{bp} iterations was set to
+$n_\text{iter} = 5000$.
+This value was chosen to be at least as large as the number of
+\acp{vn} in any single window, since with one \ac{bp} iteration
+between consecutive decimations ($T = 1$ in the notation of
+\Cref{alg:bpgd}) this is the maximum number of inner iterations
+that can occur before every \ac{vn} in the window has been decimated.
+A preliminary investigation showed that \ac{bpgd} only delivers its
+intended performance gain once most \acp{vn} have actually been decimated,
+which motivated this choice.
+The physical error rate was swept from $p = 0.001$ to $p = 0.004$
+in steps of $0.0005$.
+\Cref{fig:bpgd_w} sweeps over the window size with
+$W \in \{3, 4, 5\}$ at fixed step size $F = 1$, and
+\Cref{fig:bpgd_f} sweeps over the step size with
+$F \in \{1, 2, 3\}$ at fixed window size $W = 5$.
+
+% [Description] Figure 4.10
+
+In both panels, every curve again exhibits the expected monotonic
+increase of the per-round \ac{ler} with the physical error rate.
+Across both panels and across all parameter choices, the warm-start
+curves lie above the corresponding cold-start curves, i.e.,
+the warm-start variant performsworse than its cold-start counterpart.
+This is the opposite of what we observed for plain \ac{bp}, where
+warm-start improved upon cold-start at every parameter setting.
+The gap between the warm- and cold-start curves additionally widens
+as the physical error rate decreases:
+at the lowest sampled rate $p = 0.001$, the per-round \ac{ler} of the
+warm-start runs is more than two orders of magnitude above that of
+the corresponding cold-start runs.
+In \Cref{fig:bpgd_w}, larger window sizes yield lower per-round
+\acp{ler} for both warm- and cold-start, and the spacing between the
+cold-start curves shrinks as $W$ grows.
+In \Cref{fig:bpgd_f}, the cold-start curves follow the previously seen
+ordering with $F = 1$ at the bottom and $F = 3$ at the top.
+The warm-start curves, however, exhibit the opposite ordering:
+$F = 1$ now yields the highest per-round \ac{ler}, $F = 2$ lies below
+it, and $F = 3$ is the lowest of the three warm-start curves.
+
+% [Interpretation] Figure 4.10
+
+The fact that warm-start sliding-window decoding now performs worse
+than its cold-start counterpart is surprising in light of the results
+for plain \ac{bp}, where the warm-start modification was uniformly beneficial.
+The dependence on the window size in \Cref{fig:bpgd_w} is, on its own,
+consistent with the same explanation that we gave for
+\Cref{fig:whole_vs_cold}: larger windows expose the inner decoder to
+a larger fraction of the constraints encoded in the detector error
+matrix at the time of decoding, and this benefits both warm- and
+cold-start decoding.
+The dependence on the step size in \Cref{fig:bpgd_f}, however, is the
+opposite of the corresponding dependence under plain \ac{bp}
+(\Cref{fig:bp_f_over_p}): for warm-start, smaller $F$ now hurts
+rather than helps, even though smaller $F$ implies a larger overlap
+in both cases.
+
+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}
+messages on the edges of the overlap region but also the decimation
+information.
+Because we run with an iteration budget large enough to decimate
+every \ac{vn} in a window, by the time window $\ell$ ends, all
+of its \acp{vn} have already been hard-decided.
+For the \acp{vn} that lie in the overlap region with window $\ell + 1$
+this hard decision is then carried into the next window through the
+warm-start initialization, and the next window thus begins decoding
+with a substantial fraction of its \acp{vn} already frozen, before
+its own parity checks have had any chance to influence the
+corresponding bit estimates.
+This identifies one of two competing effects on the warm-start performance.
+The larger the overlap, the more such prematurely frozen \acp{vn} the
+next window inherits, which hurts performance.
+On the other hand, a larger window still exposes the inner decoder to
+a larger set of constraints, which helps performance.
+The two effects together are consistent with what we observe in
+\Cref{fig:bpgd_wf}.
+Increasing $W$ at fixed $F$ enlarges both the overlap and the window
+itself, and the benefit due to the larger $W$ dominates.
+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}
@@ -2191,7 +2367,9 @@ This motivates the next subsection, in which we replace the inner
             \end{axis}
         \end{tikzpicture}
 
-        \caption{Comparison of window sizes for $F=1$.}
+        \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}
@@ -2262,16 +2440,120 @@ This motivates the next subsection, in which we replace the inner
             \end{axis}
         \end{tikzpicture}
 
-        \caption{
-            Comparison of step sizes for $W=5$.
-        }
+        \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 our earlier
+hypothesis  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.
+
+% [Thread] Turn to previous way of warm-start
+
+The natural consequence of the previous diagnosis is to drop the
+problematic part of the warm-start initialization for \ac{bpgd} and
+to carry over only the \ac{bp} messages on the edges of the overlap
+region, as in \Cref{fig:messages_tanner}, while leaving the channel
+\acp{llr} of the next window in their original cold-start state.
+Note that some information about the previous window's decimation
+state is still implicitly carried over through the \ac{bp} messages,
+since the decimation decisions were made based on the messages themselves.
+
 \begin{figure}[t]
     \centering
     \hspace*{-6mm}
@@ -2340,7 +2622,9 @@ This motivates the next subsection, in which we replace the inner
             \end{axis}
         \end{tikzpicture}
 
-        \caption{Comparison of window sizes for $F=1$.}
+        \caption{\red{Lorem ipsum dolor sit amet, consectetur adipiscing
+        elit, sed do eiusmod tempor incididunt}}
+        \label{fig:bpgd_msg_W}
     \end{subfigure}%
     \hfill%
     \begin{subfigure}{0.5\textwidth}
@@ -2409,14 +2693,73 @@ This motivates the next subsection, in which we replace the inner
             \end{axis}
         \end{tikzpicture}
 
-        \caption{Comparison of step sizes for $W=5$.}
+        \caption{\red{Lorem ipsum dolor sit amet, consectetur adipiscing
+        elit, sed do eiusmod tempor incididunt}}
+        \label{fig:bpgd_msg_F}
     \end{subfigure}
 
     \caption{
         \red{\lipsum[2]}
     }
+    \label{fig:bpgd_msg}
 \end{figure}
 
+% [Experimental parameters] Figure 4.12
+
+\Cref{fig:bpgd_msg} repeats the experiment of \Cref{fig:bpgd_wf}
+with the modified warm-start procedure that carries over only the
+\ac{bp} messages.
+All other experimental parameters are unchanged: the maximum number
+of inner \ac{bp} iterations is $n_\text{iter} = 5000$, and the
+physical error rate is swept from $p = 0.001$ to $p = 0.004$ in steps
+of $0.0005$.
+The cold-start curves (dashed) are identical to those in
+\Cref{fig:bpgd_wf}.
+The warm-start curves are shown with solid lines.
+\Cref{fig:bpgd_msg_W} sweeps over the window size with
+$W \in \{3, 4, 5\}$ at fixed step size $F = 1$, and
+\Cref{fig:bpgd_msg_F} sweeps over the step size with
+$F \in \{1, 2, 3\}$ at fixed window size $W = 5$.
+
+% [Description] Figure 4.12
+
+The warm-start curves now lie below their cold-start counterparts
+across both panels and across the entire physical error rate range,
+in contrast to \Cref{fig:bpgd_wf}.
+In \Cref{fig:bpgd_msg_W}, larger window sizes again yield lower
+per-round \acp{ler} for both warm- and cold-start, and the warm-start
+advantage over cold-start is more pronounced for $W \in \{4, 5\}$
+than for $W = 3$, where the warm- and cold-start curves nearly coincide.
+In \Cref{fig:bpgd_msg_F}, smaller step sizes again yield lower
+per-round \acp{ler} for both warm- and cold-start, and the warm-start
+advantage over cold-start is most pronounced for $F = 1$ and shrinks
+as $F$ grows.
+
+% [Description] Interpretation 4.12
+
+Removing the channel \acp{llr} from the warm-start initialization lifts
+the warm-start regression observed in \Cref{fig:bpgd_wf},
+and warm-start now consistently outperforms cold-start.
+The dependence on the window size and the step size also recovers
+the qualitative behavior we observed for plain \ac{bp} in
+\Cref{fig:whole_vs_cold_vs_warm,fig:bp_f_over_p}: a larger overlap
+between consecutive windows, achieved either by enlarging $W$ or by
+decreasing $F$, both improves the absolute decoding performance and
+increases the warm-start advantage over cold-start.
+This is consistent with the original effective-iterations picture.
+Without the premature hard decisions from carried-over decimation
+information, the warm-start initialization once again amounts to
+additional \ac{bp} iterations on the \acp{vn} of the overlap region,
+and the larger the overlap, the more such effective iterations are gained.
+
+% [Thread] As before, view max iter behavior
+
+Finally, we repeat the iteration-budget sweep of \Cref{fig:bpgd_iter}
+with the message-only warm-start procedure.
+This serves both to verify that the premature hard decision effect
+does not reappear at any iteration count and to compare the warm- and
+cold-start curves across the entire range of $n_\text{iter}$ available to us.
+
 \begin{figure}[t]
     \centering
     \hspace*{-6mm}
@@ -2487,7 +2830,9 @@ This motivates the next subsection, in which we replace the inner
             \end{axis}
         \end{tikzpicture}
 
-        \caption{Comparison of window sizes for $F=1$.}
+        \caption{\red{Lorem ipsum dolor sit amet, consectetur adipiscing
+        elit, sed do eiusmod tempor incididunt}}
+        \label{fig:bpgd_msg_iter_W}
     \end{subfigure}%
     \hfill%
     \begin{subfigure}{0.48\textwidth}
@@ -2558,13 +2903,76 @@ This motivates the next subsection, in which we replace the inner
             \end{axis}
         \end{tikzpicture}
 
-        \caption{
-            Comparison of step sizes for $W=5$.
-        }
+        \caption{\red{Lorem ipsum dolor sit amet, consectetur adipiscing
+        elit, sed do eiusmod tempor incididunt}}
+        \label{fig:bpgd_msg_iter_F}
     \end{subfigure}
 
     \caption{
         \red{\lipsum[2]}
     }
+    \label{fig:bpgd_msg_iter}
 \end{figure}
 
+% [Experimental parameters] Figure 4.13
+
+\Cref{fig:bpgd_msg_iter} repeats the experiment of
+\Cref{fig:bpgd_iter} with the modified warm-start procedure that
+carries over only the \ac{bp} messages.
+All other experimental parameters are unchanged: the physical error
+rate is fixed at $p = 0.0025$ and the iteration budget is swept over
+$n_\text{iter} \in \{32, 128, 256, 512, 1024, 1536, 2048, 2560,
+3072, 3584, 4096\}$.
+The cold-start curves (dashed) are identical to those in
+\Cref{fig:bpgd_iter}.
+\Cref{fig:bpgd_msg_iter_W} sweeps over the window size with
+$W \in \{3, 4, 5\}$ at fixed step size $F = 1$, and
+\Cref{fig:bpgd_msg_iter_F} sweeps over the step size with
+$F \in \{1, 2, 3\}$ at fixed window size $W = 5$.
+
+% [Description] Figure 4.13
+
+The warm-start curves now again lie consistently below their cold-start
+counterparts across both panels and across the entire range of
+$n_\text{iter}$, contrary to \Cref{fig:bpgd_iter}.
+The warm-start curves furthermore track the overall shape of the
+corresponding cold-start curves closely, including the iteration
+count at which they drop sharply and the level at which they plateau.
+The warm-start improvement over cold-start grows with the window size
+in \Cref{fig:bpgd_msg_iter_W} and shrinks with the step size in
+\Cref{fig:bpgd_msg_iter_F}, with the largest gap visible at $W = 5$
+and at $F = 1$, respectively.
+
+% [Interpretation] Figure 4.13
+
+These observations match our expectations.
+With only the \ac{bp} messages carried over, the warm-start
+initialization no longer freezes any \acp{vn} in the next window
+The dependence of this benefit on $W$ and $F$ also recovers the
+pattern observed for plain \ac{bp} in
+\Cref{fig:whole_vs_cold_vs_warm,fig:bp_f_over_p}:
+larger overlap, achieved by larger $W$ or smaller $F$, yields more
+effective extra iterations and therefore a larger warm-start gain.
+
+% BPGD conclusion
+
+We conclude our investigation into the performance of warm-start
+sliding-window decoding under \ac{bpgd} by summarizing our findings.
+Warm-starting the inner decoder still provides a consistent
+performance gain when the inner decoder is upgraded from plain
+\ac{bp} to its guided-decimation variant, but only if some care is
+taken in choosing what to carry over.
+Passing the channel \acp{llr} along with the \ac{bp} messages,
+as suggested by naively carrying over the warm-start idea to \ac{bpgd},
+leads to premature hard decisions on \acp{vn} in the overlap region.
+This leads to warm-start initialization actually worsening the
+performance compared to cold-start initialization.
+Restricting the warm start to the \ac{bp} messages alone removes
+this effect and recovers a consistent warm-start improvement over
+cold-start that follows the same behavior as for plain \ac{bp} with
+regard to overlap.
+A second observation specific to \ac{bpgd} is that its iteration
+requirements are substantially larger than those of plain \ac{bp}:
+the per-round \ac{ler} drops sharply only once the iteration budget
+is on the order of the number of \acp{vn} in each window.
+