On the Power of Conditional Samples in Distribution Testing
Sourav Chakraborty, Eldar Fischer, Yonatan Goldhirsh and Arie Matsliah
Is the Lottery Fair?
Goldreich-Ron Test:
1. Run 𝑂( 𝑛)
draws
2. Reject if many
collisions found
Collision
Conditional Samples Test:
1. Pick a set H of 𝑂(1) balls
using the machine
2. Pick a set L of 𝑂(1) balls
uniformly at random
3. Test the machine with the
balls in 𝐿 ∪ 𝐻
Are balls picked uniformly at random?
Ratio Trees
Explicit Samplers
Summary of Results
Given a conditional samples oracle for a distribution 𝐷
1
3
3
5
2
5
Inner nodes:
2
3
1
2
Adaptive
Produce an oracles for 𝐷 close to 𝐷
which outputs samples and their probability
1
2
Pr 𝑙𝑒𝑓𝑡
Pr 𝑠𝑢𝑏𝑡𝑟𝑒𝑒
Leaves: universe elements
2
5
1
3
2
15
Uniformity
Identity
Label invariant
properties
General
properties
Non-adaptive
1
𝑝𝑜𝑙𝑦( )
𝜖
1
∗
𝑝𝑜𝑙𝑦(log 𝑛 , )
𝜖
1
𝑝𝑜𝑙𝑦(log 𝑛 , )
𝜖
Ω(√log log 𝑛)
Ω 𝑛
1
𝑝𝑜𝑙𝑦(log 𝑛 , )
𝜖
Ω(log log 𝑛)
Ω(log log 𝑛)
(following
uniformity)