To protect both local gradients and estimated parameters in distributed learning, this paper introduces a masked diffusion (MD) strategy, leading to two algorithms: the MD stochastic gradient (MD-SG) and the MD primal-dual stochastic gradient (MPD-SG). The two algorithms distinguish themselves from existing privacy diffusion methods by incorporating two mechanisms: non-zero mean protection noise and a random matrix step-size. The first mechanism ensures the confidentiality of the transmitted values, while the second protects the gradient information. We analyze the mean-square stability and privacy of the proposed methods under standard assumptions. The results indicate that the MPD-SG algorithm, with a sufficiently small parameter γ, can achieve better steady-state performance than the MD-SG algorithm in heterogeneous data scenarios. Finally, simulations illustrate the effectiveness of the proposed algorithms and support the theoretical analysis.

In this article, we consider the data privacy issue of distributed learning over adaptive networks under zero-mean protection noise. First, using a nonzero-mean protection noise, a new privacy-preserving diffusion adaptive least-mean-squares algorithm is devised, named NZPD-LMS. Different from the existing differential privacy noise, the nonzero-mean protection noise is designed with two noises with zero-mean and nonzero-mean, allowing the zero-mean noise to retain differential privacy properties, and the nonzero-mean noise to prevent the use of a sliding average over time to obtain transmission values. Then, based on mean-square analysis, we evaluate stability conditions and steady-state error bounds for the NZPD-LMS algorithm, as well as how each algorithmic parameter affects steady-state error. Finally, several simulations are conducted to illustrate the theoretical findings and effectiveness of the proposed approach.

Affine projection (AP) algorithm has been widely used in many applications because of the fast convergence ability for colored inputs. To reduce high steady-state errors resulting from weight-noise correlations, this letter presents an interval-extraction mechanism for AP algorithm, named IEAP. In the procedure, we recommend using an interval between each data point of the output vector in the AP update. Finally, by means of theoretical mean square deviations analysis and computer simulations, the interval-extraction method is demonstrated to be effective.

This study develops a pseudo-energy-conservation relation to analyze the complementary mean-square performance of a complex-valued normalized least mean-square (CNLMS) algorithm. Based on this relation, we derive a recursion describing the transient complementary mean-square deviation (CMSD) behavior of the CNLMS. Closed-form expressions to predict the steady-state CMSD and complementary excess mean-square error conjugate (CEMSEc) are obtained. Because this method inherits the advantages of the energy-conservation method, our results do not restrict the distribution of input signals. Numerical simulations validate our theoretical analysis results.

In this paper, we present a robust variable step-size decorrelation normalized least-mean-square (RVSSDNLMS) algorithm. A new constrained minimization problem is developed by minimizing the l2norm of the a decorrelated posteriori error signal with a constraint on the filter coefficients in the l2norm sense. Solving this minimization problem gives birth to the efficient RVSSDNLMS algorithm. The convergence performance and computational complexity of RVSSDNLMS algorithm are analyzed. Finally, simulations show that the proposed RVSSDNLMS considerably outperforms the normalized least-mean-square (NLMS), robust variable step-size NLMS, and pseudoaffine projection algorithms in terms of convergence rate and steady-state error in Gaussian noise and impulsive noise environments.