papers/hanzo-hmm/sections/06-multi-asset-generalization.tex

\section{Multi-Asset Generalization}

\label{sec:multiasset}

\subsection{Weighted Resource Pools}

For $n$ resource types with weight vector $\bm{w}$, the HMM invariant surface is:
\begin{equation}
\mathcal{S}_\kappa = \{(\bm{q}, \bm{p}) \in \Gamma : \mathcal{H}(\bm{q}, \bm{p}; \lambda, \bm{w}) = \kappa\}.
\end{equation}

The marginal price of resource $i$ in terms of resource $j$ is:
\begin{equation}
\pi_{ij} = \frac{\partial \mathcal{H}/\partial q_i}{\partial \mathcal{H}/\partial q_j} = \frac{w_i p_i + \lambda w_i (q_i - \bar{q}_i)/\bar{q}_i^2}{w_j p_j + \lambda w_j (q_j - \bar{q}_j)/\bar{q}_j^2}.
\end{equation}

\subsection{Resource Vectors and SLA Constraints}

AI compute jobs specify requirement vectors $\bm{r} = (r_{\text{gpu}}, r_{\text{vram}}, r_{\text{cpu}}, r_{\text{net}}, r_{\text{disk}})$ and SLA constraints $\bm{c} = (l_{\max}, z, \sigma)$ where $l_{\max}$ is maximum latency, $z \in \mathcal{Z}$ is geographic region, and $\sigma$ is privacy tier.

The multi-asset routing problem is:
\begin{align}
\min_{\Delta\bm{q}, \Delta\bm{p}} \quad & \sum_i \pi_i \Delta q_i + f(\|\Delta\bm{q}\|) \label{eq:routing} \\
\text{s.t.} \quad & \mathcal{H}(\bm{q} - \Delta\bm{q}, \bm{p} + \Delta\bm{p}) = \kappa, \nonumber \\
& \Delta q_i \ge r_i, \quad \forall i, \nonumber \\
& \text{SLA}(\Delta\bm{q}, \bm{c}) = \text{true}. \nonumber
\end{align}

\begin{proposition}
Problem (\ref{eq:routing}) is convex when $\mathcal{H}$ is convex in $(\bm{q}, \bm{p})$ and the SLA constraints define a convex feasible region.
\end{proposition}

\begin{proof}
The objective is linear in $\Delta\bm{q}$. The invariant constraint defines a convex level set (since $\mathcal{H}$ is a sum of products and quadratics, which is convex on $\mathbb{R}^n_{>0} \times \mathbb{R}^n_{>0}$ for $\lambda \ge 0$). Resource lower bounds and convex SLA constraints preserve convexity.
\end{proof}

We solve (\ref{eq:routing}) via interior-point methods with warm-starting from the previous trade, achieving median solve times of 12ms for $n = 5$ resource types.

\subsection{Quality-Weighted Reserves}

Worker $j$ supplying resource $i$ has quality score $q_j^{(i)} \in [0, 1]$ updated by PoAI attestations. The effective reserve is:
\begin{equation}
q_i^{\text{eff}} = \sum_{j \in \mathcal{W}_i} q_j^{(i)} \cdot q_{ij}^{\text{raw}},
\end{equation}
where $\mathcal{W}_i$ is the set of workers offering resource $i$ and $q_{ij}^{\text{raw}}$ is the raw reserve contributed by worker $j$.

Quality scores follow an exponential moving average:
\begin{equation}
q_j^{(i)}(t+1) = \alpha \cdot a_j(t) + (1 - \alpha) \cdot q_j^{(i)}(t),
\end{equation}
where $a_j(t) \in \{0, 1\}$ is the attestation result at time $t$ and $\alpha = 0.1$ is the learning rate.