# Differential privacy with strong composition under k-mechanisms with different (ε, δ)-DP bounds

The overall DP under the strong composition theorem for k-mechanisms is ($$\epsilon \sqrt{k log(1/\delta)}$$, k$$\delta$$) such that each individual mechanism has ($$\epsilon, \delta$$)-DP.

But what if say, 25% of the total k-mechanisms have $$(\epsilon_1, \delta_1)$$-DP, while the remaining 75% mechanisms have $$(\epsilon_2, \delta_2)$$-DP. What would the overall DP be under the rules of strong composition in this case?

This says that the mechanisms can be combined into an $$(\epsilon_g,\delta_g)$$ mechanism whenever $$\epsilon_g\ge 0$$ satisfies $$(1+\exp(\epsilon_1))^{-0.25k}(1+\exp(\epsilon_2))^{-0.75k}\left(\sum_{i=0}^{0.25k}\sum_{j=0}^{0.75k}\binom{0.25 k}i\binom{0.75 k}j\mathrm{max}\left(\exp(i\epsilon_1+j\epsilon_2)-\exp(\epsilon_g+(0.25k-i)\epsilon_1+(0.75k-j)\epsilon_2)\right)\right)\le 1-\frac{1-\delta_g}{(1-\delta_1)^{0.25k}(1-\delta_2)^{0.75k}}.$$