papers/hanzo-federated-agents/sections/14-deltasoup-detailed-filter-analysis.tex

\section{DeltaSoup: Detailed Filter Analysis}

\label{app:filter-analysis}

\begin{proposition}[Filter False Positive Rate]
\label{prop:fpr}
Under Assumption~\ref{ass:diversity}, the fraction of honest experiences
filtered by the MAD filter (Eq.~\ref{eq:filter}) with threshold $\tau = 2$
is bounded by:
\begin{equation}
    P(\text{honest filtered}) \leq \frac{d \cdot \mathrm{Var}[|\Delta|]}{4 \sigma_h^2}.
\end{equation}
For well-clustered libraries with $\mathrm{Var}[|\Delta|] \ll \sigma_h^2$,
this is negligible.
\end{proposition}

\begin{proof}
An honest experience $\hat{\bm{e}}_m$ is filtered when its mean normalized
deviation exceeds $\tau = 2$. By Markov's inequality:
$P(\bar{D}_m > 2) \leq \bar{D}_m / 2 \leq d \cdot \mathrm{Var}[|\Delta|] / (4 \sigma_h^2)$,
where we used $\mathbb{E}[\bar{D}_m] = d \cdot \mathrm{Var}[|\Delta|] / \sigma_h^2$
(the mean deviation of honest values from the coordinate-wise median is
proportional to the variance of $|\Delta|$ relative to $\sigma_h$).
\end{proof}