[Paper Review] Query-to-Communication Lifting Using Low-Discrepancy Gadgets

Arkadev Chattopadhyay, Yuval Filmus+3 more|arXiv (Cornell University)|Apr 30, 2019

Complexity and Algorithms in Graphs4 citations

TL;DR

This paper presents a new lifting theorem that establishes a connection between query complexity and communication complexity for all low-discrepancy gadgets of logarithmic size, extending prior results that were limited to specific gadgets like the inner product. The key contribution is a general lifting framework using low-discrepancy gadgets, enabling randomized communication complexity lower bounds to be derived from query complexity lower bounds for a significantly broader class of functions.

ABSTRACT

Lifting theorems are theorems that relate the query complexity of a function $f:\\{0,1\\}^{n}\ o\\{0,1\\}$ to the communication complexity of the composed function $f \\circ g^{n}$, for some "gadget" $g:\\{0,1\\}^{b}\ imes\\{0,1\\}^{b}\ o\\{0,1\\}$. Such theorems allow transferring lower bounds from query complexity to the communication complexity, and have seen numerous applications in the recent years. In addition, such theorems can be viewed as a strong generalization of a direct-sum theorem for the gadget $g$. We prove a new lifting theorem that works for all gadgets $g$ that have logarithmic length and exponentially-small discrepancy, for both deterministic and randomized communication complexity. Thus, we significantly increase the range of gadgets for which such lifting theorems hold. Our result has two main motivations: First, allowing a larger variety of gadgets may support more applications. In particular, our work is the first to prove a randomized lifting theorem for logarithmic-size gadgets, thus improving some applications of the theorem. Second, our result can be seen as a strong generalization of a direct-sum theorem for functions with low discrepancy.

Motivation & Objective

• To extend lifting theorems beyond the special case of the inner product gadget to all low-discrepancy gadgets with logarithmic size.
• To establish a general lifting framework that connects query complexity of a function f to the communication complexity of f∘gⁿ for any low-discrepancy gadget g.
• To enable new applications of lifting theorems in communication complexity by broadening the range of applicable gadgets.
• To generalize direct-sum theorems by showing that low-discrepancy gadgets support strong lifting properties for both deterministic and randomized communication models.

Proposed method

• Introduce a deficiency-based analysis to track information loss during the simulation of a query protocol in a communication setting.
• Define a deficiency measure that combines entropy terms and free variable counts to bound the number of queries in a communication protocol.
• Use the gadget's low discrepancy to bound the probability of dangerous or leaking values during protocol simulation.
• Analyze each step of the simulation protocol to show that message transmission increases deficiency by O(log(1/p_M)), while query steps decrease it by Ω(b·|I|).
• Prove that the total deficiency is bounded and non-negative, leading to a query complexity upper bound of O((C + b)/b), where C is the communication cost.
• Leverage the gadget's exponential discrepancy and logarithmic size to ensure that the deficiency decrease per query dominates the increase from messages, enabling the lifting result.

Experimental results

Research questions

• RQ1 Can lifting theorems be extended beyond the inner product gadget to all low-discrepancy gadgets of logarithmic size?
• RQ2 What conditions on a gadget ensure that query complexity lower bounds lift to communication complexity lower bounds in the randomized setting?
• RQ3 How can a deficiency-based analysis be used to bound the number of queries in a communication simulation of a query protocol?
• RQ4 To what extent do low-discrepancy gadgets support strong direct-sum properties in communication complexity?
• RQ5 Can the lifting framework be generalized to work for both deterministic and randomized communication complexity with the same gadget constraints?

Key findings

• The paper proves a randomized lifting theorem for all low-discrepancy gadgets of logarithmic size, significantly generalizing prior results that were limited to the inner product function.
• The deficiency analysis shows that each message transmission increases deficiency by at most O(log(1/p_M) + 1), while each query step decreases it by Ω(b·|I|), ensuring bounded query count.
• The total number of queries in the communication protocol is bounded by O((C + b)/b), where C is the communication cost and b is the gadget size.
• The framework supports both deterministic and randomized communication complexity, extending the applicability of lifting theorems to a much broader class of functions.
• The result implies a strong generalization of direct-sum theorems, as it shows that low-discrepancy gadgets inherently support information-theoretic compression in communication protocols.
• The analysis holds under the condition that the gadget has exponentially small discrepancy and logarithmic size, which is satisfied by a wide range of natural functions.