Research Of Online Saddle Point Optimization Algorithm Based On Regularization And Escaping From Saddle Point

With the advent of the big data era,distributed optimization has been widely used to process massive data.The most commonly used is the distributed stochastic gradient descent algorithm,but it is vulnerable to different types of Byzantine attacks.In order to solve the problem of deliberate attacks and the optimal value of the objective function in the distributed dimensional Byzantine environment,the introduction of Byzantine fault-tolerant technology enables stable distributed systems to function normally under a certain amount of Byzantine.The main work of this article is divided into two parts:The first part,based on the gradient update rule a new Byzantine attack by saddle point is proposed.It also analyzes that when the objective function falls into the saddle point,the proposed AD ABOUND can escape the saddle point faster comparing with other optimization methods on the classification experiment.Secondly,an aggregation rule Saddle(·)is proposed to filter Byzantine agents.Theoretical analysis shows that it is a dimensional Byzantine fault-tolerant technology.Therefore,in a distributed dimensional Byzantine environment,the ADABOUND combined with the Saddle(-)aggregation rule can effectively resist saddle point attacks.Finally,the advantages and disadvantages of AD ABOUND,adaptive and non-adaptive methods are compared and analyzed from the error rate and error of the classification experiment results.The results show that AD ABOUND combined with the Saddle(·)aggregation rule is less affected by the saddle point attack in a distributed dimensional Byzantine environment.The second part builds a game model of optimization problem with Byzantine attack and Byzantine fault tolerance coexisting in the distributed system,and studies the online saddle point optimization algorithm to solve the game's Nash equilibrium point,which is the optimal decision value.Firstly,the FTL algorithm is proposed to update the optimal decision,and a regularization method is added to stabilize the time-varying decision.Secondly,SP-Regret measures the difference between the cumulative loss functions and the saddle point value of the total loss function,and proves that SP-Regret can be sublinear.Further,the algorithm is extended to network with Byzantine faults and 3f +1 agents.The FTL algorithm with regularization based on online saddle point optimization in a Byzantine environment is obtained.Finally,the numerical simulation results show that the decision variables on both sides can fluctuate within a certain range after adding regularization.Figure[22]table[1]reference[41]...