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Scalable Decentralized Algorithms for Online Personalized Mean Estimation

Franco Galante, Giovanni Neglia, Emilio Leonardi

TL;DR

This study proposes a framework where agents self-organize into a graph, allowing each agent to communicate with only a selected number of peers r, and proposes two collaborative mean estimation algorithms: one employs a consensus-based approach, while the other uses a message-passing scheme.

Abstract

In numerous settings, agents lack sufficient data to directly learn a model. Collaborating with other agents may help, but it introduces a bias-variance trade-off, when local data distributions differ. A key challenge is for each agent to identify clients with similar distributions while learning the model, a problem that remains largely unresolved. This study focuses on a simplified version of the overarching problem, where each agent collects samples from a real-valued distribution over time to estimate its mean. Existing algorithms face impractical space and time complexities (quadratic in the number of agents A). To address scalability challenges, we propose a framework where agents self-organize into a graph, allowing each agent to communicate with only a selected number of peers r. We introduce two collaborative mean estimation algorithms: one draws inspiration from belief propagation, while the other employs a consensus-based approach, with complexity of O( r |A| log |A|) and O(r |A|), respectively. We establish conditions under which both algorithms yield asymptotically optimal estimates and offer a theoretical characterization of their performance.

Scalable Decentralized Algorithms for Online Personalized Mean Estimation

TL;DR

This study proposes a framework where agents self-organize into a graph, allowing each agent to communicate with only a selected number of peers r, and proposes two collaborative mean estimation algorithms: one employs a consensus-based approach, while the other uses a message-passing scheme.

Abstract

In numerous settings, agents lack sufficient data to directly learn a model. Collaborating with other agents may help, but it introduces a bias-variance trade-off, when local data distributions differ. A key challenge is for each agent to identify clients with similar distributions while learning the model, a problem that remains largely unresolved. This study focuses on a simplified version of the overarching problem, where each agent collects samples from a real-valued distribution over time to estimate its mean. Existing algorithms face impractical space and time complexities (quadratic in the number of agents A). To address scalability challenges, we propose a framework where agents self-organize into a graph, allowing each agent to communicate with only a selected number of peers r. We introduce two collaborative mean estimation algorithms: one draws inspiration from belief propagation, while the other employs a consensus-based approach, with complexity of O( r |A| log |A|) and O(r |A|), respectively. We establish conditions under which both algorithms yield asymptotically optimal estimates and offer a theoretical characterization of their performance.
Paper Structure (38 sections, 28 theorems, 162 equations, 12 figures, 3 tables, 3 algorithms)

This paper contains 38 sections, 28 theorems, 162 equations, 12 figures, 3 tables, 3 algorithms.

Table of Contents

  1. Introduction
  2. Related Work
  3. Model and Background
  4. 1) Identifying Similarity Classes.
  5. 2) Estimating the Mean.
  6. ColME's theoretical guarantees.
  7. ColME's limitations.
  8. Scalable Algorithms over a Graph $\mathcal{G}$
  9. Consensus-based Algorithm: C-ColME
  10. Message-passing Algorithm: B-ColME
  11. Choice of the Graph and other Parameters
  12. Algorithms' Comparison
  13. Numerical Experiments
  14. Conclusions
  15. Appendix A - ColME's Guarantees
  16. ...and 23 more sections

Key Result

Theorem 1

[Proof in Appendix appendix:th1] Considering an arbitrarily chosen agent $a$ in $\mathcal{A}$, for any $\delta \in (0,1)$, employing either B-ColME or C-ColME we have: with $\zeta_a^D=n_{\gamma}^{\star}\left(\frac{\Delta_a}{4}\right) + 1$, $\Delta_a=\underset{a' \in \mathcal{A} \setminus \mathcal{C}_a }{\min}\Delta_{a,a'}$, $\gamma= \frac{\delta}{4 r | \mathcal{CC}_a|}$. $n^{\star}_{\gamma}(x)$

Figures (12)

  • Figure 1: Fraction of agents with estimate deviates by more than $\epsilon$ from the true value, i.e., $| \{ a \in \mathcal{A} : |\hat{\mu}_a^t - \mu_a| > \varepsilon \} | / | \mathcal{A} |$ (top) and fraction of wrong links (bottom) for B-ColME (a) and C-ColME (b), over 20 realizations with 95% confidence intervals.
  • Figure 2: Comparison of our algorithms and two versions of ColME, over 10 realizations.
  • Figure 3: Accuracy of a local model (Local), a decentralized FL over a static graph (FL-SG), and our approach over a dynamic graph (FL-DG). We also show the fraction of links between classes (wrong links) over time for FL-DG.
  • Figure 4: Fraction of the wrong links (log scale) over time, as a function of the dimension $K$ of the mean values, $\bm{x} \in \mathbb{R}^K$.
  • Figure 5: Simplified illustration of the functioning of the B-ColME algorithm from the point of view of node $a$ (all the quantities are already aggregated in the messages $m_{h,i}^{t, a' \rightarrow a}$, so that to exclude the '' self'' info sent by $a$).
  • ...and 7 more figures

Theorems & Definitions (48)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Corollary 5
  • Proposition 6
  • Theorem 7
  • Theorem 8
  • Theorem 9: Incorrect neighborhood estimation
  • proof
  • ...and 38 more