Increasing epsilon decreases noise

[echo66]

Greetings!

I'm trying to understand your paper and implementation. I've noticed that the more you increase epsilon, the less noise will be generated. In order to understand if that is the expected behavior, I looked into your paper and PrivBayes paper (and, also, a Java implementation) and everyone seems to say that the scale of the noise is given by:

4 * (n_cols - k) / (n_rows * epsilon)

But the definition of differential privacy implies that if epsilon gets closer to 0, there won't be any difference for the query output between the original and synthetic datasets. Am I getting something wrong?

Thanks in advance!

[haoyueping]

Thanks for your question! I will update the documentation for epsilon soon.

There are several concepts in Differential Privacy (DP):

1. an original dataset D1
2. a neighboring dataset D2
3. a randomized algorithm A
4. an output O
5. a small value epsilon

DP requires that

Under the context of DataSynthesizer, D1 is the input dataset, O is the synthetic dataset.

When epsilon=0, Pr(A(D1)=O) = Pr(A(D2)=O). There is no difference between D1 and D2 for A, then A is fully randomized. So reducing epsilon injects more noises.

[eunbeejang]

@haoyueping

Hello, I'm also trying to understand your work. If you say increased epsilon translates to reducing noise, then what epsilon value will be equivalent to having no DP?

[haoyueping]

Hi @eunbeejang ,

The noise required by DP is nearly 0 when the epsilon value is infinity.

But in the implementation of DataSythesizer, the DP is turned off if epsilon=0

[amdjedbens]

But in the implementation of DataSythesizer, the DP is turned off if epsilon=0.

So with that being said, and in simpler terms, within your practical implementation of DataSynthetizer, you don't strictly adhere to the definition of differential privacy.
When epsilon is higher, less noise is added, and vice versa. However, when epsilon=0, it means no noise is added at all.