Let $\mathrm{A}(x_0,y)$ be an algorithm with two inputs: an initial point $x_0$ and an integer parameter $y$, which specifies that $\mathrm{A}(.,.)$ executes at most $y$ iterations or steps. Given an integer $p\ge 1$, $p$ parallel processors search an appropriate value of $T$ for for $A(.)$. Each processor executes an infinite sequence of stages indexed by $i=0,1,2,\ldots$. At stage $i$, processor $j$ is assigned $T_{j,i}=h(j,i),$ where $h:\mathbb{N}\times\mathbb{N}\rightarrow\mathbb{R}^{+}$ is a prescribed function. Processor $j$ $(j=0,1,\ldots,p-1)$ then executes $\mathrm{A}(x_0,T_{j,i})$. The efficiency of the parallel framework is characterized by its $(p,α_p)$-approximation guarantee. Specifically, for every integer $T\ge T_0$, there exist a processor $j$ and a stage $i$ such that $T\le T_{j,i}\le T_{j,i}^*<α_p T,$ where $T_{j,i}^*=\sum_{t=0}^{i}T_{j,t}$ denotes the cumulative number of iterations executed by processor $j$ from the beginning to stage $i$. We prove that this framework achieves a $(p,α_p)$-approximation, and a tight lower bound for $α_p$ for all large $p$. We develop arithmetically simple stochastic gradient methods in which every division is of the form $x/2^t$ for some integer $t$, and integrate them into the proposed parallel framework.

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