A Comprehensive Survey on Local Differential Privacy

Table 6

Existing methods of private ERM under LDP. Bounds errors and complexity for optimization of convex functions in the local model as a function of the number T of rounds interaction (in the case of interactive setting), the number n of participants, and the dimension p of the parameter vector. α is the desired population excess risk (expected empirical error), optimization error is equivalent to expected population risk.

Method	Dimension (method)	Interactivity	Problem	Assumption on loss function	Assumption on variables and constrain set C	Bound error/complexity
Duchi et al. [31]	Low	Seq.	General convex optimization (minimax risk)	Generalized convex loss		Optimization error; lower bound
Smith et al. [53]	Low	Non.	General convex optimization	Generalized convex loss		Optimization error; sample complexity: (linear regression)
	Low	Seq.		Lipschitz convex loss		Round complexity: Optimization error:
Zheng et al. [88]	High (dimension reduction)	Non	Linear regression	Sparse linear regression		Optimization error:
	High (polynomial approximation)	Non		Smooth generalized linear		Sample complexity
Wang et al. [89]	Low	Non	Convex optimization with sample complexity	-Smooth		Sample complexity
Wang et al. [89]	Low (polynomial approximation)	Non	Convex optimization with sample complexity	-Smooth		Sample complexity
Wang et al. [89]	High (dimension reduction)	Non	Convex optimization	Smooth generalized linear		Optimization error
Wang et al. [90]	High (polynomial approximation)	Non	Convex optimization (ERM)	Lipschitz convex generalized linear loss		Sample complexity
Wang and Xu [91]	High (polynomial approximation)	Seq.	Sparse linear regression	Squared loss		Estimation error; lower: upper: