we set $v_0 = F(w_0)$ to ensure that the first bound *touches* $F$. In the next iterations, we set $v_t$ using a *progress coefficient* ${  \mbox{$\medmuskip=0mu\displaystyle\eta \in (0,1]$}}$ that ensures making, at least, $\eta d_t$ progress where ${  \mbox{$\medmuskip=0mu\displaystyled_t = b_t(w_t) - F(w_t)$}}$ is the gap between the bound and the true objective value at $w_t$.

Note that when ${  \mbox{$\medmuskip=0mu\displaystyle\eta=1$}}$ all valid bounds touch $F$ at $w_{t-1}$ (see (\[eq:valid\_bounds\_eta\])), corresponding to the MM requirement. Small $\eta$ values allow for gradual exploratory progress in each step while large $\eta$ values greedily select bounds that guarantee immediate progress.

Convergence of [G-MM]{}
=======================

We showed in (\[eq:bound\_sequence\_ineq\]) that, in any iteration $t$, the [G-MM]{}solution is no worse than the initial solution in terms of the objective value, $F(w_t) \leq F(w_0)$. In this section we also show that the gap between the [G-MM]{}bounds and the true objective converges to zero at the minimizers of the bounds (Theorem \[thrm:gmm\_bounds\_touch\_F\]). Moreover, we show, under mild conditions, [G-MM]{}converges to a solution (Theorem \[thrm:gmm\_sol\_converges\]) that is a local extremum or a saddle point of $F$ (Theorem \[thrm:GMM-converges-to-local-min\]). Other convergence properties of [G-MM]{}depend on the structure of the objective function $F$, the family of the bounds $\mathcal{F}$, and 