# $(\epsilon, \delta)$-differential privacy: main motivation of $\delta$

I am wondering why (not how) we relax $$\epsilon$$-differential privacy to $$(\epsilon, \delta)$$-differential privacy. Is the main motivation to reduce the variance of the noise added to the query with a slight sacrifice from the strength of privacy?

• Have you read this answer, crypto.stackexchange.com/a/61200/82720. I think there is trade-off between epsilon and delta. Commented Jul 31, 2020 at 22:39
• @CloudCho thanks! more than the answer, the comments in the link provided addresses to my question. Commented Aug 1, 2020 at 0:36

I help implement and ship anonymization strategies based on differential privacy in a large tech company. In my experience, the $$\delta$$ is mainly used for two reasons.
• Partition selection: when computing histograms on an unbounded space, you can threshold the results (or do smarter things) and, at the cost of a non-zero $$\delta$$, avoid having to specify the full list of partitions in advance>.
• Gaussian noise: since Gaussian is based on the $$L_2$$ sensitivity, it is very convenient to use when adding noise to a lot of metrics at once; if a single user can influence $$k$$ metrics, the noise needs to be scaled by $$\sqrt{k}$$ instead of $$k$$ with Laplace noise$$^1$$. But Gaussian noise doesn't give you pure $$\varepsilon$$-DP, you have to have a non-zero $$\delta$$.
Gaussian noise is particularly used for machine learning use cases. In such contexts, you also often want to use results on amplification by sampling, which also require a non-zero $$\delta$$.