latent variable.

**Bound optimization:** Optimization of a bound ${b}\in 
\mathcal{F}$ corresponds to a convex quadratic program and can be solved using different techniques, including gradient based methods (SGD) and the cutting-plane method [@joachims09]. We use the cutting-plane method in our experiments.

### Bias Function for Multi-fold MIL {#sec:bias_for_mfmil}

The multi-fold MIL algorithm [@multifold] was introduced for training latent SVMs for weakly supervised object localization, to deal with stickiness issues in training with CCCP. It modifies how latent variables are updated during training. @multifold divide the training set into $K$ folds, and updates the latent variables in each fold using a model trained on the other $K-1$ folds. This algorithm does not have a formal convergence guarantee. By defining a suitable bias function, we can derive a [G-MM]{} algorithm that mimics the behavior of multi-fold MIL, and yet, is convergent.

Consider training an [LS-SVM]{}. Let $S=(1,\ldots,n)$ be an ordered sequence of training sample indices. Also, let ${  \mbox{$\medmuskip=0mu\displaystylez_i \in \mathcal{Z}$}}$ denote the latent variable associated to training example $(x_i, y_i)$. Let $I$ denote a subsequence of $S$. Also, let $z_{I}^t$ denote the fixed latent variable values of training examples in $I$ in iteration $t$, and let $w(I, z_{I}^t)$ be the model 