that satisfy (\[eq:GMMconstraint\]). We denote the set of valid bounds at iteration $t$ by $\mathcal{B}_t$. We consider two scenarios for constructing bounds in [G-MM]{}. The first scenario is when we have a bias function $g:\mathcal{B}_t \times \mathbb{R}^d \rightarrow \mathbb{R}$ over the set of valid bounds. The function $g$ takes in a bound $b \in \mathcal{B}_t$ and a current solution $w \in \mathbb{R}^d$ and returns a scalar indicating the goodness of the bound. In this scenario we select the bound with the largest bias value ${b}_t = \operatorname*{argmax}_{{b}\in \mathcal{B}_t} g({b}, w_{t-1})$.

In the second scenario we propose to choose one of the valid bounds from $\mathcal{B}_t$ at random. Thus, we have both a deterministic (the $1^{st}$ scenario) and a stochastic (the $2^{nd}$ scenario) bound construction mechanism. MM algorithms, such as EM and CCCP, fall into the first scenario.

In general MM algorithms are instances of [G-MM]{}that use a specific bias function $g(b,w) = -b(w)$. The bound that touches $F$ at $w$ maximizes this bias function. It also maximizes the amount of progress with respect to the previous bound at $w$. By choosing bounds that maximize progress, MM algorithms tend to rapidly converge to a nearby local minimum. For instance, at iteration 