We introduce and study the communication complexity of computing the inner product of two vectors, where the input is restricted w.r.t. a norm N on the space Rn. Here, Alice and Bob hold two vectors v, u such that ∥v∥N ≤ 1 and ∥u∥N∗ ≤ 1, where N∗ is the dual norm. The goal is to compute their inner product <v, u> up to an ε additive term. The problem is denoted by IPN, and generalizes important previously studied problems, such as: (1) Computing the expectation Ex∼D[f(x)] when Alice holds D and Bob holds f is equivalent to IPℓ1. (2) Computing vTAv where Alice has a symmetric matrix with bounded operator norm (denoted S∞) and Bob has a vector v where ∥v∥2 = 1. This problem is complete for quantum communication complexity and is equivalent to IPS∞. We systematically study IPN, showing the following results, near tight in most cases: 1. For any symmetric norm N, given ∥v∥N ≤ 1 and ∥u∥N∗ ≤ 1 there is a randomized protocol using Õ(ε−6 log n) bits of communication that returns a value in <u, v> ± ϵ with probability 23 -we will denote this by Rε1/3(IPN) ≤ Õ(ε−6 log n). In a special case where N = ℓp and N∗ = ℓq for p−1 + q−1 = 1, we obtain an improved bound Rε1/3(IPℓp) ≤ O(ε−2 log n), nearly matching the lower bound Rε1/3(IPℓp) ≥ Ω(min(n, ε−2)). 2. One way communication complexity −→Rε,δ(IPℓp) ≤ O(ε−max(2,p) · log nε ), and a nearly matching lower bound −→Rε1/3(IPℓp) ≥ Ω(ε−max(2,p)) for ε−max(2,p) ≪ n. 3. One way communication complexity −→Rε,δ(N) for a symmetric norm N is governed by the distortion of the embedding ℓk∞ into N. Specifically, while a small distortion embedding easily implies a lower bound Ω(k), we show that, conversely, non-existence of such an embedding implies protocol with communication kO(log log k) log2 n. 4. For arbitrary origin symmetric convex polytope P, we show Rε1/3(IPN) ≤ O(ε−2 log xc(P)), where N is the unique norm for which P is a unit ball, and xc(P) is the extension complexity of P (i.e. the smallest number of inequalities describing some polytope P′ s.t. P is projection of P′).
|Title of host publication||14th Innovations in Theoretical Computer Science Conference, ITCS 2023|
|Editors||Yael Tauman Kalai|
|Publisher||Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing|
|State||Published - 1 Jan 2023|
|Event||14th Innovations in Theoretical Computer Science Conference, ITCS 2023 - Cambridge, United States|
Duration: 10 Jan 2023 → 13 Jan 2023
|Name||Leibniz International Proceedings in Informatics, LIPIcs|
|Conference||14th Innovations in Theoretical Computer Science Conference, ITCS 2023|
|Period||10/01/23 → 13/01/23|
Bibliographical notePublisher Copyright:
© Alexandr Andoni, Jarosław Błasiok, and Arnold Filtser; licensed under Creative Commons License CC-BY 4.0.
- communication complexity
- symmetric norms