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Better Boosting of Communication Oracles, or Not

Nathaniel Harms, Artur Riazanov

TL;DR

It is shown that the naive boosting strategy can be improved for the Equality oracle but not the 1-Hamming Distance oracle, and a new proof that Equality is not complete for the class of constant-cost randomized communication.

Abstract

Suppose we have a two-party communication protocol for $f$ which allows the parties to make queries to an oracle computing $g$; for example, they may query an Equality oracle. To translate this protocol into a randomized protocol, we must replace the oracle with a randomized subroutine for solving $g$. If $q$ queries are made, the standard technique requires that we boost the error of each subroutine down to $O(1/q)$, leading to communication complexity which grows as $q \log q$. For which oracles $g$ can this naive boosting technique be improved? We focus on the oracles which can be computed by constant-cost randomized protocols, and show that the naive boosting strategy can be improved for the Equality oracle but not the 1-Hamming Distance oracle. Two surprising consequences are (1) a new example of a problem where the cost of computing $k$ independent copies grows superlinear in $k$, drastically simplifying the only previous example due to Blais & Brody (CCC 2019); and (2) a new proof that Equality is not complete for the class of constant-cost randomized communication (Harms, Wild, & Zamaraev, STOC 2022; Hambardzumyan, Hatami, & Hatami, Israel Journal of Mathematics 2022).

Better Boosting of Communication Oracles, or Not

TL;DR

It is shown that the naive boosting strategy can be improved for the Equality oracle but not the 1-Hamming Distance oracle, and a new proof that Equality is not complete for the class of constant-cost randomized communication.

Abstract

Suppose we have a two-party communication protocol for which allows the parties to make queries to an oracle computing ; for example, they may query an Equality oracle. To translate this protocol into a randomized protocol, we must replace the oracle with a randomized subroutine for solving . If queries are made, the standard technique requires that we boost the error of each subroutine down to , leading to communication complexity which grows as . For which oracles can this naive boosting technique be improved? We focus on the oracles which can be computed by constant-cost randomized protocols, and show that the naive boosting strategy can be improved for the Equality oracle but not the 1-Hamming Distance oracle. Two surprising consequences are (1) a new example of a problem where the cost of computing independent copies grows superlinear in , drastically simplifying the only previous example due to Blais & Brody (CCC 2019); and (2) a new proof that Equality is not complete for the class of constant-cost randomized communication (Harms, Wild, & Zamaraev, STOC 2022; Hambardzumyan, Hatami, & Hatami, Israel Journal of Mathematics 2022).
Paper Structure (8 sections, 9 theorems, 22 equations, 1 figure, 2 algorithms)

This paper contains 8 sections, 9 theorems, 22 equations, 1 figure, 2 algorithms.

Table of Contents

  1. Introduction
  2. Definitions: Communication Problems and Oracles
  3. Better Boosting of Equality Protocols
  4. No Better Boosting for Hamming Distance, and Consequences
  5. Randomized Reduction Lemma
  6. Direct Sum Theorem for 1 Hamming Distance
  7. Lower Bound on Computing 1-Hamming Distance with Equality Queries
  8. Noisy-Tree Fails for Other Oracles

Key Result

Theorem 1.4

eq:intro-informal-badboosting can be improved for the Equality oracle, but it is tight for the 1-Hamming Distance oracle.

Figures (1)

  • Figure 1: The picture represents the runtime of \ref{['alg:noisy-tree']}. The thick green path is $P'_{i,j}$ for some $i$ and $j$. The walk corresponding to the runtime of \ref{['alg:noisy-tree']} is represented with thin arrows: green arrows represent good rounds, solid red arrows represent bad rounds where protocol makes a mistake, and dashed red arrows represent bad rounds where protocol backtracks.

Theorems & Definitions (29)

  • Example 1.1
  • Example 1.2
  • Theorem 1.4: Informal; see \ref{['thm:intro-eq-betterboosting', 'thm:no-boosting-for-hd1']}
  • Conjecture 1.5
  • Definition 2.1: Query Set
  • Definition 2.2: Communication with oracles
  • Remark 2.3
  • Theorem 3.1
  • proof
  • Claim 3.2
  • ...and 19 more