Mean-Biased Processes for Balanced Allocations

TL;DR

This work introduces Mean-Biased processes for balanced allocations, unifying a broad family of strategies that bias allocations toward underloaded bins via probability or weight bias. By coupling Mean-Thinning, Twinning, and related Relative-Threshold variants into a single framework, the authors prove a logarithmic gap $ ext{Gap}(m)=O( ext{log }n)$ w.h.p. in heavily loaded regimes, with matching lower bounds for several processes. The analysis hinges on a triplet of potential functions (linear, quadratic, exponential) and a new mean-quantile stabilization phenomenon that ensures progress even when the system is far from equilibrium. The results extend known bounds for classic models, reveal improved sample-efficiency over One- and Two-Choice in several cases, and open the door to further improvements and extensions in noisy or dynamic settings with practical implications for load balancing and distributed systems.

Abstract

We introduce a new class of balanced allocation processes which bias towards underloaded bins (those with load below the mean load) either by skewing the probability by which a bin is chosen for an allocation (probability bias), or alternatively, by adding more balls to an underloaded bin (weight bias). A prototypical process satisfying the probability bias condition is Mean-Thinning: At each round, we sample one bin and if it is underloaded, we allocate one ball; otherwise, we allocate one ball to a second bin sample. Versions of this process have been in use since at least 1986. An example of a process, introduced by us, which satisfies the weight bias condition is Twinning: At each round, we only sample one bin. If the bin is underloaded, then we allocate two balls; otherwise, we allocate only one ball. Our main result is that for any process with a probability or weight bias, with high probability the gap between maximum and minimum load is logarithmic in the number of bins. This result holds for any number of allocated balls (heavily loaded case), covers many natural processes that relax the Two-Choice process, and we also prove it is tight for many such processes, including Mean-Thinning and Twinning. Our analysis employs a delicate interplay between linear, quadratic and exponential potential functions. It also hinges on a phenomenon we call "mean quantile stabilization", which holds in greater generality than our framework and may be of independent interest.

Figures (7)

• Figure 1.1: (left) The three stages $A, B, C$ for an allocation in round $t$ using Two-Choice, which assumes that no other ball is being allocated to the two sampled bins (so the two bins are being "held" until both loads $x_{i_1}^t$ and $x_{i_2}^t$ are reported and the allocation is complete). (right) The (at most) two stages $A,B$ for an allocation using Two-Thinning process with threshold $f^t$. Here, none of the bins needs to be "held" and there are at most two stages.
• Figure 3.1: Two possibilities for Twinning: $(i)$ allocate one ball to an overloaded bin, or $(ii)$ allocate two balls to an underloaded bin.
• Figure 3.2: Three different possibilities for Mean-Thinning: $(i)$ first bin is underloaded, so the ball is allocated there, $(ii)$ first bin is overloaded and the ball is allocated to an underloaded bin, or $(iii)$ the ball is allocated to an overloaded bin.
• Figure 4.1: Overview of the recovery phase and stabilization phase. In the recovery phase starting with $V^t = \operatorname{poly}(n)$ (and so $\mathop{\mathrm{Gap}}\nolimits(t) = \mathcal{O}(n \log n)$ and $\Lambda^t \leqslant e^{\mathcal{O}(\log n)}$), after $n^3 \log^4 n$ rounds w.h.p. we find a round $s_0$ with $\Lambda^{s_0} \leqslant cn$. We then switch to the stabilization phase, where in \ref{['lem:stabilization']} we prove that starting with $\Lambda^{\tau} \leqslant 2cn$ in the next $\@nameuse{constantstab_time} n \log n$ rounds there is a round $t$ with $\Lambda^{t} \leqslant cn$, which allows us to infer that $\mathop{\mathrm{Gap}}\nolimits(m)=\mathcal{O}(\log n)$.
• Figure 4.2: Summary of the key steps in the proof for \ref{['thm:main_technical']}. By $\widetilde{G}_{t_0}^{t_1}$ we denote the number of rounds $t \in [t_0, t_1]$ with $\Delta^t \leqslant Cn$ and by $G_{t_0}^{t_1}$ the number of rounds $t \in [t_0, t_1]$ with $\delta^t \in [\varepsilon, 1 - \varepsilon]$, also $r=\varepsilon^2$ and $\widetilde{r}=4\varepsilon^2$.

Theorems & Definitions (92)

• Theorem 3.1: Main Theorem
• Theorem 3.2
• Lemma 3.3
• proof
• Lemma 3.4
• Remark 3.5
• Lemma 3.6
• proof
• Lemma 3.7
• Lemma 3.8