trained on $\{(x_i,y_i) | i\in I\}$ with latent variables fixed to $z_{I}^t$ in the last maximization of (\[eq:LSSVM\_objective\]).

We assume access to a loss function $\ell(w,x,y,z)$. For example, for the binary latent SVM where $y\in\{-1,1\}$, $\ell$ is the hinge loss: $\ell(w,x,y,z) = \max\{0,1-y\, w\cdot\phi(x,z)\}$.

We start by considering the Leave-One-Out (LOO) setting, $K=n$, and call the algorithm of @multifold LOO-MIL in this case. The update rule of LOO-MIL in iteration $t$ is to set $$\begin{aligned}
z_i^t=\operatorname*{argmin}_{z\in\mathcal{Z}}~\ell\left(w(S\backslash i,z_{S\backslash i}^{t-1}),x_i,y_i,z\right), ~~\forall i\in S.
\label{eq:update_rule}\end{aligned}$$ After updating the latent values for all training examples, the model $w$ is retrained by optimizing the resulting bound.

Now let us derive the bias function for a [G-MM]{}algorithm that mimics the behavior of LOO-MIL. Recall from Equation \[eq:boundFamily\_lssvm\] that each bound $b \in \mathcal{B}_t$ is associated with a joint latent configuration $z(b)=(z_1,\ldots,z_n)$. We use the following bias function: $$\begin{aligned}
g(b,w) = -\sum_{i\in S} \ell\left(w(S\backslash i,z_{S\backslash i}^{t-1}),x_i,y_i,z_i\right).
\label{eq:loomil}\end{aligned}$$ Note that picking a bound according to (\[eq:loomil\]) is equivalent to the LOO-MIL update rule of (\[eq:update\_rule\]) except that in (\[eq:loomil\]) only *valid* bounds are considered; that is bounds that make at least $\eta$-progress.

For the general multi-fold case ($K<n$), the bias function can be derived similarly.

**Remarks:** Let us 