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Multitask Online Learning: Listen to the Neighborhood Buzz

Juliette Achddou, Nicolò Cesa-Bianchi, Pierre Laforgue

TL;DR

This work addresses multitask online learning over decentralized networks by introducing MT-CO2OL, a fully decentralized algorithm built on MT-FTRL that propagates neighborhood information through fetch-and-send operations. The core contribution is a family of graph-aware regret bounds that interpolate between single-task and multitask performance, scaling with local task variance $\sigma_j^2$ and neighborhood size $N_j$ as $R_T(U) = \widetilde{O}\left(\sum_{j=1}^N \max_{i\in\mathcal{N}_j} w_{ij} \sqrt{1+\sigma_j^2(N_j-1)} \sqrt{\sum_{i\in\mathcal{N}_j} T_i}\right)$ in adversarial activations, with improved bounds under stochastic activations. A private variant, DPMT-CO2OL, achieves $\varepsilon$-DP with only polylogarithmic overhead in $T$, enabling privacy-preserving multitask collaboration on graphs. The paper also establishes lower bounds that demonstrate the tightness of the presented results on regular and structured graphs and provides experiments on synthetic networks validating the theoretical predictions. Overall, the framework offers a principled approach to personalized, communication-constrained learning with privacy guarantees in networked systems.

Abstract

We study multitask online learning in a setting where agents can only exchange information with their neighbors on an arbitrary communication network. We introduce $\texttt{MT-CO}_2\texttt{OL}$, a decentralized algorithm for this setting whose regret depends on the interplay between the task similarities and the network structure. Our analysis shows that the regret of $\texttt{MT-CO}_2\texttt{OL}$ is never worse (up to constants) than the bound obtained when agents do not share information. On the other hand, our bounds significantly improve when neighboring agents operate on similar tasks. In addition, we prove that our algorithm can be made differentially private with a negligible impact on the regret. Finally, we provide experimental support for our theory.

Multitask Online Learning: Listen to the Neighborhood Buzz

TL;DR

This work addresses multitask online learning over decentralized networks by introducing MT-CO2OL, a fully decentralized algorithm built on MT-FTRL that propagates neighborhood information through fetch-and-send operations. The core contribution is a family of graph-aware regret bounds that interpolate between single-task and multitask performance, scaling with local task variance and neighborhood size as in adversarial activations, with improved bounds under stochastic activations. A private variant, DPMT-CO2OL, achieves -DP with only polylogarithmic overhead in , enabling privacy-preserving multitask collaboration on graphs. The paper also establishes lower bounds that demonstrate the tightness of the presented results on regular and structured graphs and provides experiments on synthetic networks validating the theoretical predictions. Overall, the framework offers a principled approach to personalized, communication-constrained learning with privacy guarantees in networked systems.

Abstract

We study multitask online learning in a setting where agents can only exchange information with their neighbors on an arbitrary communication network. We introduce , a decentralized algorithm for this setting whose regret depends on the interplay between the task similarities and the network structure. Our analysis shows that the regret of is never worse (up to constants) than the bound obtained when agents do not share information. On the other hand, our bounds significantly improve when neighboring agents operate on similar tasks. In addition, we prove that our algorithm can be made differentially private with a negligible impact on the regret. Finally, we provide experimental support for our theory.
Paper Structure (25 sections, 19 theorems, 88 equations, 4 figures, 5 algorithms)

This paper contains 25 sections, 19 theorems, 88 equations, 4 figures, 5 algorithms.

Table of Contents

  1. INTRODUCTION
  2. Related works
  3. PROBLEM SETTING
  4. Setting.
  5. Preliminaries.
  6. ALGORITHM AND ANALYSIS
  7. Adversarial Activations
  8. Stochastic Activations
  9. Lower Bounds
  10. A PRIVATE VARIANT
  11. EXPERIMENTS
  12. CONCLUSION
  13. Technical Proofs (Results from Section \ref{['sec:comm-graph']})
  14. Proof of Theorem \ref{['thm:any-G']}
  15. Proof of Corollary \ref{['cor:reggraphs_cliques']}
  16. ...and 10 more sections

Key Result

Theorem 1

Let $G$ be a clique. The regret of MT-FTRL with $\beta_{t-1}=\sqrt{t}$ satisfies for all $U \in \mathcal{U}$ where $\sigma^2 = \sigma^2(U) = \frac{1}{N-1} \sum_{i=1}^N \|U_{i:} - \frac{1}{N} \sum_{j=1}^N U_{j:}\|_2^2$ is the comparator variance.We use $g \stackrel{\widetilde{\mathcal{O}}}{=} f$ to denote $g = \widetilde{\mathcal{O}}(f)$, where $\widetilde{\mathcal{O}}$ hides logarithmic factors in

Figures (4)

  • Figure 1: Multitask regret over time of MT-CO2OL on a random communication graph with stochastic activations against two baselines: i-FTRL ($N$ independent instances of FTRL) and ST-FTRL (the multi-agent single-task algorithm of cesa2020cooperative).
  • Figure 2: One Iteration of MT-CO2OL. Active agent $2$ fetches their neighbor's models and predicts with their weighted average $x_t$ (left). Then $2$ pays $\ell_t(x_t)$, observes $g_t \in \partial\ell_t(x_t)$, and sends back $w_{2j}\,g_t$. Finally, each local $\textnormal{AlgoClique}\xspace$ instance updates the models for the agents in the virtual clique centered on the agent running the instance (right).
  • Figure 3: Multitask regret at horizon $T=150\,000$.
  • Figure :

Theorems & Definitions (36)

  • Remark 1: Comparison to cesa2020cooperative
  • Theorem 1: cesa2022multitask
  • Definition 1
  • Lemma 2
  • proof
  • Theorem 3
  • Corollary 3
  • Theorem 4
  • Remark 2: Extension to unknown $q_i$
  • Theorem 5
  • ...and 26 more