Approximate DP recovers non-private error rates exp(-r(n)) for ID and exp(-Ω(n)) for generation; pure DP degrades the exponent by exactly min{1,ε} with matching bounds.
Computational and sample-complexity barriers were analyzed by Arenas et al
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On the Price of Privacy for Language Identification and Generation
Approximate DP recovers non-private error rates exp(-r(n)) for ID and exp(-Ω(n)) for generation; pure DP degrades the exponent by exactly min{1,ε} with matching bounds.