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Parallel Best Arm Identification in Heterogeneous Environments

Nikolai Karpov, Qin Zhang

TL;DR

The paper analyzes parallel best arm identification in heterogeneous collaborative learning, proving that heterogeneity inherently worsens the time-round tradeoff relative to homogeneous CL. It introduces a generalized round-elimination framework with interleaved local mean constructions and distribution classes to establish near-tight lower bounds, and presents a non-adaptive, successive-elimination algorithm achieving matching performance on the upper bound side. The main contributions include a $K$-agent/round tradeoff with lower bound $H n^{\Omega(1/R)}/K$ and a corresponding $R$-round algorithm achieving $0.99$ success for budgets above $c_T H n^{1/R}/K$, demonstrating the intrinsic difficulty imposed by heterogeneity. These results have implications for federated and multi-agent learning systems where non-IID data across agents is common, guiding the design of communication-round budgets and aggregation strategies.

Abstract

In this paper, we study the tradeoffs between the time and the number of communication rounds of the best arm identification problem in the heterogeneous collaborative learning model, where multiple agents interact with possibly different environments and they want to learn in parallel an objective function in the aggregated environment. By proving almost tight upper and lower bounds, we show that collaborative learning in the heterogeneous setting is inherently more difficult than that in the homogeneous setting in terms of the time-round tradeoff.

Parallel Best Arm Identification in Heterogeneous Environments

TL;DR

The paper analyzes parallel best arm identification in heterogeneous collaborative learning, proving that heterogeneity inherently worsens the time-round tradeoff relative to homogeneous CL. It introduces a generalized round-elimination framework with interleaved local mean constructions and distribution classes to establish near-tight lower bounds, and presents a non-adaptive, successive-elimination algorithm achieving matching performance on the upper bound side. The main contributions include a -agent/round tradeoff with lower bound and a corresponding -round algorithm achieving success for budgets above , demonstrating the intrinsic difficulty imposed by heterogeneity. These results have implications for federated and multi-agent learning systems where non-IID data across agents is common, guiding the design of communication-round budgets and aggregation strategies.

Abstract

In this paper, we study the tradeoffs between the time and the number of communication rounds of the best arm identification problem in the heterogeneous collaborative learning model, where multiple agents interact with possibly different environments and they want to learn in parallel an objective function in the aggregated environment. By proving almost tight upper and lower bounds, we show that collaborative learning in the heterogeneous setting is inherently more difficult than that in the homogeneous setting in terms of the time-round tradeoff.
Paper Structure (9 sections, 27 theorems, 45 equations, 1 algorithm)

This paper contains 9 sections, 27 theorems, 45 equations, 1 algorithm.

Table of Contents

  1. Introduction
  2. Related Work
  3. Technical Overview of The Main Result
  4. The Impossibility Result
  5. The Hard Input Distribution (When $K = 2$) and Its Properties.
  6. Classes of Hard Distributions
  7. The Lower Bound for $K = 2$
  8. General $K$
  9. The Algorithm

Key Result

Theorem 1

For any $1 \le R \le \frac{\log n}{24\log\log n}$ and any $T < H n^{\Omega\left(\frac{1}{R}\right)}/K$, any $R$-round $T$-time $K$-agent algorithm that solves $n$-arm BAI in the heterogeneous CL model has a success probability less than $0.99$.

Theorems & Definitions (28)

  • Theorem 1: Main Theorem
  • Theorem 2
  • Lemma 3
  • Lemma 4: Chernoff-Hoeffding Inequality
  • Lemma 5
  • Lemma 6
  • Lemma \ref{lem:key-pi}$'$
  • Lemma \ref{lem:key-pi}$'$
  • Lemma \ref{lem:key-pi}$'$
  • Lemma \ref{lem:key-pi}$'$
  • ...and 18 more