From 0016df0004d39120f90d421ba74532a19b7c703b Mon Sep 17 00:00:00 2001 From: Andreas Tsouchlos Date: Sun, 3 May 2026 03:10:21 +0200 Subject: [PATCH] Add text for second BPGD plot --- src/thesis/chapters/4_decoding_under_dems.tex | 110 +++++++++++++++++- 1 file changed, 109 insertions(+), 1 deletion(-) diff --git a/src/thesis/chapters/4_decoding_under_dems.tex b/src/thesis/chapters/4_decoding_under_dems.tex index dcc098e..a7fdcf9 100644 --- a/src/thesis/chapters/4_decoding_under_dems.tex +++ b/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}