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Better Bounds for the Distributed Experts Problem

David P. Woodruff, Samson Zhou

TL;DR

This paper gives a protocol that achieves regret roughly $R\gtrsim\frac{1}{\sqrt{T}\cdot\text{poly}\log(nsT)}$, using $\mathcal{O}\left(\frac{n}{R^2}+\frac{s}{R^2}\right)\cdot\max(s^{1-2/p,1)$ bits of communication, which improves on previous work.

Abstract

In this paper, we study the distributed experts problem, where $n$ experts are distributed across $s$ servers for $T$ timesteps. The loss of each expert at each time $t$ is the $\ell_p$ norm of the vector that consists of the losses of the expert at each of the $s$ servers at time $t$. The goal is to minimize the regret $R$, i.e., the loss of the distributed protocol compared to the loss of the best expert, amortized over the all $T$ times, while using the minimum amount of communication. We give a protocol that achieves regret roughly $R\gtrsim\frac{1}{\sqrt{T}\cdot\text{poly}\log(nsT)}$, using $\mathcal{O}\left(\frac{n}{R^2}+\frac{s}{R^2}\right)\cdot\max(s^{1-2/p},1)\cdot\text{poly}\log(nsT)$ bits of communication, which improves on previous work.

Better Bounds for the Distributed Experts Problem

TL;DR

This paper gives a protocol that achieves regret roughly , using bits of communication, which improves on previous work.

Abstract

In this paper, we study the distributed experts problem, where experts are distributed across servers for timesteps. The loss of each expert at each time is the norm of the vector that consists of the losses of the expert at each of the servers at time . The goal is to minimize the regret , i.e., the loss of the distributed protocol compared to the loss of the best expert, amortized over the all times, while using the minimum amount of communication. We give a protocol that achieves regret roughly , using bits of communication, which improves on previous work.
Paper Structure (19 sections, 18 theorems, 39 equations, 2 figures, 4 algorithms)

This paper contains 19 sections, 18 theorems, 39 equations, 2 figures, 4 algorithms.

Table of Contents

  1. Introduction
  2. Distributed Online Learning with Experts in the Coordinator Model
  3. Online learning with experts.
  4. Communication complexity.
  5. Our Contributions
  6. Algorithmic and technical novelties.
  7. Preliminaries
  8. Related Works
  9. Classic Learning with Experts
  10. Randomized weighted majority.
  11. Other randomized variants.
  12. Learning with Experts in Big Data Settings
  13. Learning with experts on data streams.
  14. Multi-armed bandits.
  15. Learning in streams.
  16. ...and 4 more sections

Key Result

Theorem 1.1

Let $b>a>0$ be fixed constants and suppose $\ell_i(j,t)\in[a,b]$ for all $t\in[T]$, $i\in[n]$ and $j\in[s]$. There exists an algorithm that achieves expected regret at most $\mathcal{O}\left(s^{1/p}\sqrt{\frac{\log n}{T}}\right)$ and with high probability, uses total communication at most $\mathcal{

Figures (2)

  • Figure 1: Our work is the first to study $\ell_p$ loss in the coordinator model; for the special case of $p=1$, we obtain better regret-communication tradeoffs for regret $R$.
  • Figure :

Theorems & Definitions (32)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4: Chernoff Bounds
  • Definition 1.5: Exponential random variable
  • Lemma 1.5
  • proof
  • Theorem 1.6
  • Lemma 3.1
  • proof
  • ...and 22 more