Lifting Theorems for Polynomial Discrepancy Functions

Temel Kavramlar

We prove a new lifting theorem that works for every two functions f and g such that the discrepancy of g is at most inverse polynomial in the input length of f. This significantly generalizes the known direct-sum theorem for discrepancy and extends the range of inner functions g for which lifting theorems hold.

Özet

The content discusses lifting theorems, which are theorems that bound the communication complexity of a composed function f ◦ g^n in terms of the query complexity of f and the communication complexity of g. Such theorems constitute a powerful generalization of direct-sum theorems for g, and have seen numerous applications in recent years.
The main contribution of the content is a new lifting theorem that works for every two functions f and g such that the discrepancy of g is at most inverse polynomial in the input length of f. This significantly generalizes the known direct-sum theorem for discrepancy, and extends the range of inner functions g for which lifting theorems hold.
The authors first provide background on lifting theorems and their relation to direct-sum theorems. They then state their main theorem, which shows that if the discrepancy of g is at least logarithmic in the input length of f, then the communication complexity of f ◦ g^n is lower bounded by the product of the query complexity of f and the discrepancy of g.
The authors then discuss the techniques used to prove this result, focusing on the main technical lemma. This lemma extends the main lemma of a previous work by [CFK+19], which was limited to the case where the discrepancy of g is at least linear in the input length of g. The authors overcome this limitation by introducing the notion of "recoverable" values, which allows them to handle the case where the discrepancy of g is smaller.
Finally, the authors discuss some related work and open questions, including the conjecture that lifting theorems should hold for every inner function g that has a sufficiently large information cost.

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Önemli Bilgiler Şuradan Elde Edildi

Lifting with Inner Functions of Polynomial Discrepancy

by Yahel Manor,... : arxiv.org 04-12-2024

https://arxiv.org/pdf/2404.07606.pdf

Daha Derin Sorular

What are some potential applications of the new lifting theorem beyond the ones mentioned in the content

The new lifting theorem has the potential for various applications beyond those mentioned in the content. One possible application could be in the field of machine learning, specifically in the optimization of neural networks. By understanding the communication complexity of composed functions and the query complexity of individual functions, the lifting theorem could potentially be used to optimize the training process of neural networks by improving the communication efficiency between different layers or components. Additionally, the theorem could have applications in cryptography, particularly in the development of secure communication protocols where minimizing communication complexity is crucial for maintaining security.

Can the requirement that the discrepancy of g is at least logarithmic in the input length of f be further relaxed, or is this a fundamental limitation of the current techniques

The requirement that the discrepancy of g must be at least logarithmic in the input length of f is a crucial aspect of the current techniques. Relaxing this requirement may pose challenges in maintaining the integrity and efficiency of the lifting theorem. The logarithmic discrepancy condition ensures that the inner function g provides enough information to effectively compose with the outer function f. Relaxing this condition could potentially lead to a loss of information or efficiency in the composition process, limiting the applicability and effectiveness of the lifting theorem. Therefore, it is a fundamental limitation that ensures the theorem's robustness and reliability in various scenarios.

How does the new lifting theorem relate to the conjecture that lifting theorems should hold for every inner function g that has a sufficiently large information cost

The new lifting theorem is closely related to the conjecture that lifting theorems should hold for every inner function g that has a sufficiently large information cost. The conjecture suggests that inner functions with a high information cost should lead to lifting theorems that effectively bound the communication complexity of composed functions. By proving the lifting theorem with a requirement on the discrepancy of g, the theorem aligns with the broader conjecture by demonstrating the importance of certain properties of inner functions in determining the communication complexity of composed functions. This connection highlights the significance of understanding the information cost and discrepancy of inner functions in the context of lifting theorems.