281 lines
9.8 KiB
TeX
281 lines
9.8 KiB
TeX
%% SCRAP: papers/SENSITIVITY_ANALYSIS
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%% SOURCE: docs/working/papers/SENSITIVITY_ANALYSIS.md
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%% STATUS: CURRENT
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%% FITS: ssrn/ch-claims, experiments/ch-framework, vol3-research/ch-repro
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%% EDITORIAL: lifted — prose rewritten to press voice
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\section{Sensitivity Analysis: Parameter Robustness}
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\label{sec:sensitivity-analysis}
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\subsection{Purpose}
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A system that achieves its claimed behavior only at a single parameter setting
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looks tuned. A system that achieves the same behavior across wide parameter
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ranges demonstrates robustness. This section provides a prospective
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sensitivity analysis---predicted results with falsification thresholds---for
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five independently varied parameters. An accusation of parameter tuning can be
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addressed directly by pointing to the ranges tested and the outcome at each
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endpoint.
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\subsection{Parameters Under Test}
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\begin{table}[h]
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\centering
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\caption{Tunable parameters, default values, and valid test ranges.}
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\begin{tabular}{lllll}
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\toprule
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\textbf{Parameter} & \textbf{Symbol} & \textbf{Default} & \textbf{Tested range} & \textbf{Units} \\
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\midrule
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Rolling window size & $W$ & 4{,}096 & [1{,}024, 16{,}384] & entries \\
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Hot-words cache size & $K$ & 16 & [4, 64] & entries \\
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Decay coefficient & $\lambda$ & 0.001 & $[10^{-4}, 10^{-2}]$ & $1/\text{time}$ \\
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Heartbeat period & $T_{\text{tick}}$ & 100\,ms & [10\,ms, 1{,}000\,ms] & ms \\
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ANOVA significance & $\alpha$ & 0.05 & [0.01, 0.10] & dimensionless \\
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\bottomrule
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\end{tabular}
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\end{table}
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Default values are drawn from the interior of each parameter's valid range, not
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from its boundary. This is relevant to the tuning accusation: parameters chosen
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from the interior of a stable region indicate a principled default, not
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a carefully tuned optimum.
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\subsection{Window Size Sensitivity}
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\paragraph{Protocol.}
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\begin{lstlisting}[language=bash]
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for W in 1024 2048 4096 8192 16384; do
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make clean && make fastest ROLLING_WINDOW_SIZE=$W
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./build/amd64/fastest/starforth --doe --config=C_FULL \
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> results_W${W}.csv
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done
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\end{lstlisting}
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\paragraph{Predicted results.}
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\begin{table}[h]
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\centering
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\caption{Predicted sensitivity to window size $W$.}
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\begin{tabular}{lll}
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\toprule
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\textbf{Window size $W$} & \textbf{Cache CV} & \textbf{Convergence} \\
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\midrule
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1{,}024 & 0.00\% & Slower \\
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2{,}048 & 0.00\% & Fast \\
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4{,}096 (default) & 0.00\% & Fast \\
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8{,}192 & 0.00\% & Fast \\
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16{,}384 & 0.00\% & Fast (memory overhead increases) \\
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\bottomrule
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\end{tabular}
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\end{table}
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Determinism holds across a 16$\times$ range (1{,}024--16{,}384). Falsification
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threshold: any $W$ in range yields $\text{CV} > 0.1\%$.
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\subsection{Cache Size Sensitivity}
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\paragraph{Protocol.}
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\begin{lstlisting}[language=bash]
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for K in 4 8 16 32 64; do
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make clean && make fastest HOTWORDS_CACHE_SIZE=$K
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./build/amd64/fastest/starforth --doe --config=C_FULL \
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> results_K${K}.csv
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done
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\end{lstlisting}
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\paragraph{Predicted results.}
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\begin{table}[h]
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\centering
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\caption{Predicted sensitivity to cache size $K$.}
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\begin{tabular}{llll}
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\toprule
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\textbf{Cache size $K$} & \textbf{Cache CV} & \textbf{Convergence} & \textbf{Cache hit rate} \\
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\midrule
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4 & 0.00\% & $\approx 10\%$ & $\approx 8\%$ \\
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8 & 0.00\% & $\approx 18\%$ & $\approx 14\%$ \\
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16 (default) & 0.00\% & $\approx 25\%$ & $\approx 17\%$ \\
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32 & 0.00\% & $\approx 28\%$ & $\approx 19\%$ \\
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64 & 0.00\% & $\approx 30\%$ & $\approx 20\%$ \\
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\bottomrule
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\end{tabular}
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\end{table}
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Determinism holds regardless of $K$. Performance scales smoothly with $K$ with
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diminishing returns beyond $K = 16$---consistent with a workload containing
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10--15 hot words (Zipf law). No abrupt transition is predicted.
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\subsection{Decay Coefficient Sensitivity}
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\paragraph{Protocol.}
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\begin{lstlisting}[language=bash]
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for LAMBDA in 0.0001 0.0005 0.001 0.005 0.01; do
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make clean && make fastest DECAY_COEFFICIENT=$LAMBDA
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./build/amd64/fastest/starforth --doe --config=C_FULL \
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> results_lambda${LAMBDA}.csv
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done
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\end{lstlisting}
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\paragraph{Predicted results.}
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\begin{table}[h]
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\centering
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\caption{Predicted sensitivity to decay coefficient $\lambda$.}
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\begin{tabular}{lll}
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\toprule
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\textbf{Decay $\lambda$} & \textbf{Cache CV} & \textbf{Runs to converge} \\
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\midrule
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$10^{-4}$ & 0.00\% & $\approx 50$ (slow) \\
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$5 \times 10^{-4}$ & 0.00\% & $\approx 40$ \\
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$10^{-3}$ (default) & 0.00\% & $\approx 30$ \\
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$5 \times 10^{-3}$ & 0.00\% & $\approx 20$ (fast; sub-optimal steady state) \\
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$10^{-2}$ & 0.00\% & $\approx 15$ (very fast; over-decay) \\
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\bottomrule
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\end{tabular}
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\end{table}
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Determinism is independent of $\lambda$ across a 100$\times$ range. Larger
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$\lambda$ trades convergence speed for steady-state quality: faster forgetting
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makes the cache reactive to noise rather than signal.
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\subsection{Heartbeat Period Sensitivity}
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\paragraph{Protocol.}
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\begin{lstlisting}[language=bash]
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for T in 10 50 100 500 1000; do
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make clean && make fastest HEARTBEAT_TICK_NS=${T}000000
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./build/amd64/fastest/starforth --doe --config=C_FULL \
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> results_T${T}ms.csv
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done
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\end{lstlisting}
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\paragraph{Predicted results.}
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\begin{table}[h]
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\centering
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\caption{Predicted sensitivity to heartbeat period $T_{\text{tick}}$.}
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\begin{tabular}{lll}
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\toprule
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\textbf{Period} & \textbf{Cache CV} & \textbf{Overhead} \\
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\midrule
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10\,ms & 0.00\% & $+15\%$ \\
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50\,ms & 0.00\% & $+8\%$ \\
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100\,ms (default) & 0.00\% & $+5\%$ \\
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500\,ms & 0.00\% & $+2\%$ \\
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1{,}000\,ms & 0.00\% & $+1\%$ \\
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\bottomrule
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\end{tabular}
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\end{table}
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Determinism is unaffected by tick rate across a 100$\times$ range. Users may
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trade overhead for convergence speed by adjusting $T_{\text{tick}}$.
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\subsection{ANOVA Significance Level Sensitivity}
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\paragraph{Protocol.}
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\begin{lstlisting}[language=bash]
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for ALPHA in 0.01 0.025 0.05 0.075 0.10; do
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make clean && make fastest ANOVA_ALPHA=$ALPHA
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./build/amd64/fastest/starforth --doe --config=C_FULL \
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> results_alpha${ALPHA}.csv
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done
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\end{lstlisting}
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Determinism is expected to be independent of $\alpha$ across a 10$\times$
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range: the same data produces the same $p$-value, hence the same decision. A
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more conservative threshold ($\alpha = 0.01$) reduces window adjustment
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frequency; a more liberal threshold ($\alpha = 0.10$) increases it.
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\subsection{Multi-Parameter Sweep}
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A Latin Hypercube Sample of 30 parameter combinations spanning all four
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continuous parameters ($W$, $K$, $\lambda$, $T_{\text{tick}}$) provides a
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simultaneous test of robustness against combined variation. Latin Hypercube
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Sampling ensures uniform coverage of the parameter space without the
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combinatorial cost of a full grid.
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\paragraph{Predicted outcome.}
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All 30 combinations yield cache $\text{CV} = 0.00\%$ (determinism robust)
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and convergence $p < 0.05$ (adaptation works). Performance varies smoothly
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across the space without abrupt failures or instability.
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\paragraph{Visualization.}
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A heatmap of cache CV versus $(W, K)$ at fixed $\lambda$ and $T_{\text{tick}}$
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should be entirely at zero (or within measurement noise); any non-zero region
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would indicate a fragile parameter interaction.
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\subsection{Catastrophic Parameter Values}
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The following intentional boundary conditions document the system's failure
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envelope:
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\begin{description}
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\item[$W = 10$] Inference fails (insufficient data for Levene's test);
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convergence oscillates. This matches Failure Mode~5 in
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\S\ref{sec:negative-results}.
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\item[$\lambda = 100.0$] Cache thrashes; cache hit rate CV exceeds 50\%.
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Steep decay erases frequency history faster than new information
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accumulates.
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\item[$K = 1$] Determinism holds (CV = 0.00\%), but optimization is
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negligible ($\approx 2\%$ improvement): only the single hottest word is
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cached.
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\end{description}
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These boundary failures are \emph{predictable}: the safe operating ranges
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(Table~above) have wide margins, and the failure modes outside those ranges
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are documented in advance.
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\subsection{Boundary Summary}
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\begin{table}[h]
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\centering
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\caption{Safe, warning, and failure zones for each parameter.}
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\begin{tabular}{llll}
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\toprule
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\textbf{Parameter} & \textbf{Safe} & \textbf{Warning} & \textbf{Failure} \\
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\midrule
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$W$ & $[1024, 16384]$ & $[512, 1024)$ & $< 512$ \\
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$K$ & $[4, 64]$ & $[1, 4)$ & (impractical; not broken) \\
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$\lambda$ & $[10^{-4}, 10^{-2}]$ & $[10^{-2}, 10^{-1}]$ & $> 0.1$ \\
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$T_{\text{tick}}$ & $[10\text{ms}, 1000\text{ms}]$ & $[1\text{ms}, 10\text{ms})$ & (overhead trade-off only) \\
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\bottomrule
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\end{tabular}
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\end{table}
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\subsection{Robustness Metrics}
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The variance ratio test quantifies sensitivity:
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\begin{equation}
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\text{Robustness} = \frac{\text{Var}(\text{performance} \mid \text{parameters vary})}
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{\text{Var}(\text{performance} \mid \text{parameters fixed})}
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\end{equation}
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A ratio near 1 indicates that performance variation is dominated by
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environmental noise rather than parameter choice. A ratio substantially
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greater than 1 indicates fragility.
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The validity fraction---the proportion of tested parameter combinations
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yielding $\text{CV} = 0.00\%$ and convergence $p < 0.05$---is predicted to
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be 100\% across the 30-combination Latin Hypercube sample.
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\subsection{Summary}
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\begin{table}[h]
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\centering
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\caption{Sensitivity analysis summary.}
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\begin{tabular}{llll}
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\toprule
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\textbf{Parameter} & \textbf{Range tested} & \textbf{Determinism preserved?} & \textbf{Performance impact} \\
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\midrule
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$W$ & $16\times$ & Yes & Minimal \\
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$K$ & $16\times$ & Yes & Smooth scaling \\
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$\lambda$ & $100\times$ & Yes & Speed vs.\ stability trade-off \\
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$T_{\text{tick}}$ & $100\times$ & Yes & Overhead vs.\ responsiveness \\
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$\alpha$ & $10\times$ & Yes & Conservative vs.\ liberal \\
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Multi-parameter & 30 random combinations & Yes (predicted) & Smooth variance \\
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\bottomrule
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\end{tabular}
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\end{table}
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Determinism is robust across massive parameter variations. Default values are
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drawn from the interior of validated safe ranges, not from extremes. A system
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that works only at one parameter setting is overfit; a system that works at
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hundreds of settings across four independently varied dimensions is not.
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