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arXiv:2410.00838

Better Boosting of Communication Oracles, or Not

Nathaniel Harms, Artur Riazanov

Published Oct 1, 2024• 0 citations
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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).

DOI
10.48550/arxiv.2410.00838

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Suppose we have a two-party communication protocol for fff which allows the parties to make queries to an oracle computing ggg; 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 ggg. If qqq queries are made, the standard technique requires that we boost the error of each subroutine down to O(1/q)O(1/q)O(1/q), leading to communication complexity which grows as qlog⁡qq \log qqlogq. For which oracles ggg 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 kkk independent copies grows superlinear in kkk, 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).
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 naïve boosting technique be improved? We focus on the oracles which can be computed by constant-cost randomized protocols, and show that the naïve 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).
The unifying topic of this thesis is the relation of randomness to computation. We study two different facets of this: the hardness of sampling from probability distributions and the extent of usefulness of randomness in communication complexity. Hardness of Sampling. A circuit samples the distribution of its output on a uniformly random input. What distributions are samplable with small circuits? What about local circuits (NC0)? We make two contributions in this direction. The first is to show that sampling uniformly from the set of Hamming-weight-k n-bit string is hard for NC0-circuits (and even constant-width DNFs) as long as k=o(n). The second is to show that sampling a uniformly random permutation of [n] elements requires polylog(n) adaptive queries to a sequence of uniform input symbols from [n]---adaptive cell probes. These results refine our understanding of sampling by constant-depth de Morgan circuits (AC0). Randomness in Communication. The classical example separating deterministic and randomized communication is the Equality problem: Alice and Bob are to decide whether their n-bit inputs are equal. It requires transmitting n+1 in the deterministic model, but is solvable with a constant-cost protocol using public randomness. An active line of research found many non-trivial problems that are solvable with just constant-cost randomized communication [Harms-Wild-Zamaraev, STOC 2022; Hambardzumyan-Hatami-Hatami, Isr. J. Math.; Fang-Hambardzumyan-Harms-Hatami, STOC 2024; Fang-Göös-Harms-Hatami, STOC 2025]. All of those problems can be solved with polylog(n) queries to Equality oracle. Our first contribution is to exhibit a problem that is solvable with constant-cost randomized communication, but requires polynomially many calls to Equality oracle. This resolves an open problem in the survey [Hatami-Hatami, ACM SIGACT News]. Our second contribution is to investigate when the randomized communication complexity can be localized or condensed: for a matrix of randomized communication cost k, can we find its 2^k-by-2^k submatrix that retains communication cost at least linear in k$? In general this is impossible [Hambardzumyan-Hatami-Hatami, Discrete Appl. Math.]. We show, however, that in an important case it can be done: we condense randomized communication cost of sink-of-xor function---a recent counterexample to log-approximate-rank conjecture by [Chattopadhyay-Mande-Sherif, J. ACM], and thereby quantitatively improve the counterexample.
The canonical problem that gives an exponential separation between deterministic and randomized communication complexity in the classical two-party communication model is "Equality". In this work we show that even allowing access to an "Equality" oracle, deterministic protocols remain exponentially weaker than randomized ones. More precisely, we exhibit a total function on n bits with randomized one-sided communication complexity O(log n), but such that every deterministic protocol with access to "Equality" oracle needs Omega(n) cost to compute it. Additionally we exhibit a natural and strict infinite hierarchy within BPP, starting with the class P^{EQ} at its bottom.
We study an extension of the standard two-party communication model in which Alice and Bob hold probability distributions ppp and qqq over domains XXX and YYY, respectively. Their goal is to estimate Ex∼p, y∼q[f(x,y)] \mathbb{E}_{x \sim p,\, y \sim q}[f(x, y)] Ex∼p,y∼q​[f(x,y)] to within additive error ε\varepsilonε for a bounded function fff, known to both parties. We refer to this as the distributed estimation problem. Special cases of this problem arise in a variety of areas including sketching, databases and learning. Our goal is to understand how the required communication scales with the communication complexity of fff and the error parameter ε\varepsilonε. The random sampling approach -- estimating the mean by averaging fff over O(1/ε2)O(1/\varepsilon^2)O(1/ε2) random samples -- requires O(R(f)/ε2)O(R(f)/\varepsilon^2)O(R(f)/ε2) total communication, where R(f)R(f)R(f) is the randomized communication complexity of fff. We design a new debiasing protocol which improves the dependence on 1/ε1/\varepsilon1/ε to be linear instead of quadratic. Additionally we show better upper bounds for several special classes of functions, including the Equality and Greater-than functions. We introduce lower bound techniques based on spectral methods and discrepancy, and show the optimality of many of our protocols: the debiasing protocol is tight for general functions, and that our protocols for the equality and greater-than functions are also optimal. Furthermore, we show that among full-rank Boolean functions, Equality is essentially the easiest.
EFI pairs (Brakerski, Canetti, and Qian, ITCS 2023) and one-way puzzles (Khurana and Tomer, STOC 2024) are the leading candidates for the minimal assumption of quantum cryptography. The first are efficiently preparable quantum states, statistically far yet computationally indistinguishable; the second are classical puzzles, easy to sample and hard to solve. One-way puzzles imply EFI pairs, and whether the converse holds is open. We construct a single classical oracle relative to which one-way puzzles do not exist, even with an unbounded verifier, while an EFI pair survives every distinguisher that queries the oracle classically throughout and holds advice about it, making its one superposition query at the end. The oracle answers every question about the output probabilities of quantum samplers, which removes the puzzles, and hides a Haar-random half-dimensional subspace. To prove security we reduce it to communication complexity. An adversary whose knowledge of the subspace arrives as classical query answers can be simulated inside a two-party protocol against the party holding it, so it does no better than the best classical protocol for Vector-in-Subspace (Klartag and Regev, STOC 2011), whatever the oracle computes. That argument does not cover the superposition query, which we bound instead using tools from random matrix theory. The same attack gives a classical simulation of any quantum party in a classical-message protocol with no entanglement shared in advance, so relative to the oracle there is no proof of quantumness either. Quantum polynomial time therefore offers no advantage on any task with classical inputs and outputs, while the two quantum states stay indistinguishable. We state conjectures on removing the restriction on superposition queries.
In an influential paper, Linial and Shraibman (STOC '07) introduced the factorization norm as a powerful tool for proving lower bounds against randomized and quantum communication complexities. They showed that the logarithm of the approximate γ₂-factorization norm is a lower bound for these parameters and asked whether a stronger lower bound that replaces approximate γ₂ norm with the γ₂ norm holds. We answer the question of Linial and Shraibman in the negative by exhibiting a 2ⁿ×2ⁿ Boolean matrix with γ₂ norm 2^Ω(n) and randomized communication complexity O(log n). As a corollary, we recover the recent result of Chattopadhyay, Lovett, and Vinyals (CCC '19) that deterministic protocols with access to an Equality oracle are exponentially weaker than (one-sided error) randomized protocols. In fact, as a stronger consequence, our result implies an exponential separation between the power of unambiguous nondeterministic protocols with access to Equality oracle and (one-sided error) randomized protocols, which answers a question of Pitassi, Shirley, and Shraibman (ITSC '23). Our result also implies a conjecture of Sherif (Ph.D. thesis) that the γ₂ norm of the Integer Inner Product function (IIP) in dimension 3 or higher is exponential in its input size.