Further, if $\exists \infty>M>m>0, \forall b \in \mathcal{F}, M I \succeq \nabla^2 b(w) \succeq m I$, where $I$ is the identity matrix, then $\nabla F(w^\dagger) = 0$; that is, [G-MM]{}converges to a local extremum or saddle point of $F$. \[thrm:GMM-converges-to-local-min\]

We postpone the proof of Theorem \[thrm:GMM-converges-to-local-min\] to the appendix.

Derived Optimization Algorithms {#sec:derived_algorithms}
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For simplicity and ease of exposition, we primarily focus on demonstrating [G-MM]{}on *latent variable models* where bound construction naturally corresponds to [imputing]{} latent variables in the model. In this section we derive [G-MM]{}algorithms for two models widely used in machine learning, namely, [$k$-means]{}clustering and [Latent Structural SVM]{}, both belonging to the category of latent variable models. It is worth mentioning that [G-MM]{}is fully capable of handling more general non-convex problems.

[**$\boldsymbol{k}$-means**]{}Clustering
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Let $\{x_1, \dots, x_n\}$ denote a set of sample points and $w=(\mu_1, \dots, 
\mu_k)$ denote a set of cluster centers. We use $z_i \in \{1, \dots, k\}$ to denote the index of the cluster assigned to the $i$-th sample point. The objective function in [$k$-means]{}clustering is defined as follows, $$\begin{aligned}
    F(w) = \sum_{i=1}^n \min_{z_i = 1}^k || x_i - \mu_{z_i} ||^2, 
    \;\; w=(\mu_1, \dots, \mu_k). 
\label{eq:kmeans_objective}\end{aligned}$$ **Bound construction:** We obtain a convex upper 