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Understand The Distributions Used In Equivalence Testing
by Computational Technology for AllDecember 9th, 2024
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The aim is to assess whether the input distribution(s) possess specific properties or deviate significantly (i.e., “ε-far” for some ε 0) from meeting them. The goal is to do this while minimizing the number of queries made (also known as query complexity) to the distribution. The COND model and its variants have recently found applications in the areas like formal methods and machine learning.
Authors:
(1) Diptarka Chakraborty, National University of Singapore, Singapore
(2) Sourav Chakraborty, Indian Statistical Institute, Kolkata;
(3) Gunjan Kumar, National University of Singapore, Singapore;
(4) Kuldeep S. Meel, University of Toronto, Toronto.
Table of Links
4 An Efficient One-Round Adaptive Algorithm and 4.1 High-Level Overview
6 Acknowledgements and References
B An O(log log n)-query fully adaptive algorithm
2 Notations and Preliminaries
If the variation distance between two distributions is more than ε, then we say the distributions are ε-far (or just far, when it is clear from the context).
The Binomial distribution, with parameters n ∈ Z + and p ∈ [0, 1] denoted by Bin(n, p) is the distribution of the number of successes in n independent experiments, where each experiment yields a Boolean outcome, with success occurring with probability p and failure with probability 1 − p.
Definition 2.1 (COND Query Model). A conditional sampling oracle for a distribution D is defined as follows: the oracle takes as input a subset S ⊆ [n] and returns an element j ∈ S, such that the probability that j ∈ S is returned is equal to D(j)/D(S) if D(S) > 0 and 1/|S| if D(S) = 0. We denote such a conditional query by CONDD(S).
The formal definition of a k-round adaptive tester is given in [CG18]. For completeness, we present the formal definition of a one-round adaptive tester for equivalence in the COND Query Model.
Definition 2.2. Given conditional query access to distributions P and Q (over domain [n]), and given tolerance parameter ε as inputs, a one-round adaptive tester A makes conditional queries to the distributions in two rounds:
In our proof, we will extensively use concentration lemmas. In particular, we will use the following version of Chernoff bound.
This paper is available on arxiv under CC BY-NC-SA 4.0 DEED license.
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