Design And Application Of Fractional Order Gradient Descent Based On Momentum Information

The definition of fractional calculus contains historical information about functions and has the characteristic of memory.Its combination with many engineering applications has shown significant advantages.The combination of fractional calculus and deep learning optimization algorithms has attracted extensive research.Momentum information can improve optimization algorithms’ performance and the convergence speed and accuracy of deep learning model optimization.Some scholars have proposed fractional order gradient optimization algorithms and achieved significant results.However,exploring fractional order gradient optimization algorithms based on momentum information still faces many difficulties and challenges.Based on this,this paper researches momentum information and fractional order gradient optimization algorithms.Firstly,a fractional order gradient descent method based on momentum information is proposed and its advantages were demonstrated through experiments.Then,fractional order gradient descent methods based on adaptive momentum are further proposed and demonstrated to be superior to other algorithms in the industry.Finally,fractional order gradient optimization algorithms proposed in this paper have been successfully applied to the inverted pendulum and robot control tasks and have achieved significant results.The specific summary is as follows:(1)Based on the definition of Caputo fractional derivative,a fractional order stochastic gradient descent method(FOSGD)is proposed,and its convergence is proved.Furthermore,a fractional order stochastic gradient descent method with momentum(FOSGDM)is proposed,proving its convergence.The relationship between the proposed fractional order gradient optimization algorithm and the corresponding integer order optimization algorithm is analyzed.According to the inherent meaning of fractional order and the observed phenomenon of FOSGDM,two fractional order adjustment algorithms,linear adjustment and exponential adjustment,are further proposed.Experiments on classification tasks for CIFAR-10 datasets on Res Net and Dense Net have verified that FOSGD and FOSGDM perform better than corresponding integer order optimization algorithms SGD and SGDM.(2)Based on the fractional order stochastic gradient descent method with momentum information,we introduce second-order momentum information to propose adaptive momentum fractional gradient optimization algorithms Adam FO and Adam FOW and provide theoretical proof of convergence.Through classification task experiments on the CIFAR-10 dataset on Res Net and Dense Net,and classification task experiments on the Image Net-1K dataset on Vi T-S,it is verified that Adam FO and Adam FOW perform better compared to the current optimal algorithm.(3)The proposed fractional order gradient optimization algorithm is applied to two reinforcement learning tasks: inverted pendulum control and robotic arm grasping.It is verified that the proposed fractional order gradient optimization algorithm can achieve significant results in practical systems,as well as the practicality of the proposed algorithm and the high reusability advantage of being able to train different network models.