the object in the image. We define $\phi(x, y, z)$ to be a feature function with $6$ blocks; one block for each category. It extracts features from location $z$ of image $x$ and places them in the $y$-th block of the output and fills the rest with zero. We use the following multi-class classification rule: $$\begin{aligned}
    y(x) = \operatorname*{argmax}_{y, z} w \cdot \phi(x, y, z), \; w = (w_1, \ldots, w_6).\end{aligned}$$ In this experiment we use a setup similar to that in [@kumar12]: we use Histogram of Oriented Gradients (HOG) for the image feature $\phi$, and the 0-1 classification loss for $\Delta$. We set $\lambda=0.4$ in Equation \[eq:LSSVM\_objective\]. We report 5-fold cross-validation performance. Three initialization strategies are considered for the latent object locations: *image center*, *top-left corner*, and *random locations*. The first is a reasonable initialization since most objects are at the center in this dataset; the second initialization strategy is somewhat adversarial.

We try a stochastic as well as a deterministic bound construction method. For the stochastic method, in each iteration $t$ we uniformly sample a subset of examples $S_t$ from the training set, and update their latent variables using $z^t_i = \operatorname*{argmax}_{z_i} 
w_{t-1} \cdot \phi(x_i, y_i, z_i)$. Other 