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Privacy-preserving empirical risk minimization model is crucial for the increasingly frequent setting of analyzing personal data, such as medical records, financial records, etc. Due to its advantage of a rigorous mathematical definition, differential privacy has been widely used in privacy protection and has received much attention in recent years of privacy protection. With the advantages of iterative algorithms in solving a variety of problems, like empirical risk minimization, there have been various works in the literature that target differentially private iteration algorithms, especially the adaptive iterative algorithm. However, the solution of the final model parameters is imprecise because of the vast privacy budget spending on the step size search. In this paper, we first proposed a novel adaptive differential privacy algorithm that does not require the privacy budget for step size determination. Then, through the theoretical analyses, we prove that our proposed algorithm satisfies differential privacy, and their solutions achieve sufficient accuracy by infinite steps. Furthermore, numerical analysis is performed based on real-world databases. The results indicate that our proposed algorithm outperforms existing algorithms for model fitting in terms of accuracy.

We propose a new iteration numerical algorithm to solve boundary integral equations of the first kind arising in the 2D scattering by soft obstacles. The main idea is to operate on each iteration step with an integral equation, which has a convolution kernel, by changing the full kernel with a special averaging procedure. The practical convergence of the algorithm is demonstrated by some examples for three different geometries. If M is the number of iterations then the computational cost of the algorithm is MNlog(N).