the details of the bound selection strategy.

In the limit as $t \rightarrow \infty$, [G-MM]{}bounds and $F$ **touch** at the minimizer of the bounds, $\lim_{t \rightarrow \infty} d_t = 0$, where $d_t=b_t(w_t)-F(w_t)$. \[thrm:gmm\_bounds\_touch\_F\]

We have $b_t(w_t) \leq b_t(w_{t-1}) \leq v_{t-1}$, where $v_t$ is defined as in line \[alg:our\_framework:vt\] of Algorithm \[alg:our\_framework\]. The first inequality holds because $w_t$ minimizes $b_t$ and the second inequality follows from (\[eq:valid\_bounds\_eta\]). Summing over $t$ gives $$\begin{aligned}
        \sum_{t=1}^T b_t(w_t) \leq \sum_{t=1}^T v_{t-1} = v_0 + \sum_{t=1}^{T-1} b_t(w_t) - \eta d_t\\
        \Rightarrow \; \eta \sum_{t=1}^T d_t \leq v_0 - b_T(w_T) = F(w_0) - b_T(w_T).
        \label{eq:touching_inequality}
    \end{aligned}$$ Let $w^* \in \operatorname*{argmin}_w F(w)$ be an unknown *global* minimum of $F$. Using $F(w^*) \leq F(w_T) \leq b_T(w_T)$ and (\[eq:touching\_inequality\]), we have $$\begin{aligned}
        \eta \sum_{t=1}^T d_t \leq F(w_0) - F(w^*)  
        \; \Rightarrow \; \lim_{t \rightarrow \infty} d_t = 0.
        \label{eq:dt_converges_to_zero}
    \end{aligned}$$ If a global minimizer of $F$ does not exist, we can use the assumption that $F$ is bounded from below and replace $F(w^*)$ in (\[eq:dt\_converges\_to\_zero\]) with the value of the lower bound.

If $\mathcal{F}$ is a family of $m$-strongly convex functions where $m>0$, then $\exists w^\dagger$ [*s.t. *]{}$\lim_{t \rightarrow \infty} w_t = w^\dagger$; [G-MM]{}converges to a solution. \[thrm:gmm\_sol\_converges\]

Let $f$ 