Write conclusion to BP investigation. BP investigation now done

src/thesis/chapters/4_decoding_under_dems.tex

@@ -1914,37 +1914,50 @@ low physical error rates similarly mirrors the patterns already
 observed in
 \Cref{fig:whole_vs_cold_vs_warm,fig:bp_w_over_iter}.
 
-% TODO: Rephrase
-The coincidence of all three cold-start curves at
-$n_\text{iter} = 32$ is a direct consequence of the cold-start initialization.
-With each new window starting from a uniform prior regardless of $F$,
-the per-window decoding problem is essentially the same for every
-step size, and the corresponding \acp{ler} agree as long as the
-inner decoder has too few iterations to propagate information
-beyond the local syndrome structure within a single window.
-This is also the regime in which the warm-start advantage is most
-valuable, and indeed it is where the warm-start curves spread out
-most strongly with $F$.
-
-% TODO: Rephrase
-A noteworthy methodological point is that, in contrast to the window
-size $W$, the step size $F$ has no effect on decoding latency:
-the time at which the inner decoder for a given window can begin
-running is determined solely by when the syndromes for the rounds
+In contrast to the window size $W$, the step size $F$ has no effect
+on decoding latency.
+The time at which the inner decoder for a given window can begin
+decoding is determined solely by when the syndromes for the rounds
 covered by that window have been collected, which is independent of
 how much the window overlaps with its predecessor.
+Similarly, assuming the decoder is fast enough to keep up with the
+incoming syndrome measurements corresponding to the \acp{cn} of
+subsequent windows, the time at which decoding is complete depends only
+on the amount of time spent on decoding the very last window.
 A smaller $F$ thus only costs additional total compute and not
 additional latency, which is favorable for a warm-start
-sliding-window implementation:
-the regime in which the warm-start modification helps most --- large
-overlap and therefore small $F$ --- is precisely the regime in which
-the cost of that overlap shows up only in the compute budget and not
-in the latency budget.
+sliding-window implementation.
+This is especially favorable for our warm-start modification, as it
+works best where the overlap is largest, i.e., for low values of $F$.
 
-% At some later point
-\content{When looking at max iterations: Callback to diminishing
-    returns with growing window size: More iterations more beneficial
-than larger window (+1 for warm-start)}
+% Conclusion of BP investigation
+
+We conclude our investigation into the performance of warm-start
+sliding-window decoding under plain \ac{bp} by summarizing our findings.
+The warm-start modification raises the number of \ac{bp} iterations
+effectively spent on the \acp{vn} in an overlap region by reusing the
+messages from the previous window invocation instead of restarting
+from scratch.
+This explains why decoding performance improved monotonically with
+the size of the overlap, and consequently why both larger window
+sizes $W$ and smaller step sizes $F$ yielded lower per-round \acp{ler}.
+The warm-start gain over cold-start was most pronounced at low
+per-window iteration budgets,
+% and at low physical error rates, the
+% regimes
+the regime in which each additional iteration carries proportionally
+more information.
+Additionally, we would like to note that the warm-start modification
+incurs no computational cost relative to cold-start decoding.
+It changes neither the decoding latency nor the total compute, since
+both schemes process the same windows for the same number of
+iterations and differ only in the initialization of the \ac{bp}
+messages of each new window.
+We also observed that plain \ac{bp} did not saturate even at $4096$
+iterations, which we attribute to the short cycles in the underlying
+Tanner graph.
+This motivates the next subsection, in which we replace the inner
+\ac{bp} decoder by its guided-decimation variant.
 
 %%%%%%%%%%%%%%%%
 \subsection{Belief Propagation with Guided Decimation}