An overview of the fractional-order gradient descent method and its applications

TL;DR

This paper reviews fractional calculus-based gradient descent methods, compares their convergence properties, and proposes a fractional continuous time approach that guarantees convergence to the extremum, with applications to complex chemical optimization problems.

Contribution

It introduces a fractional continuous time gradient descent method that ensures convergence to the extremum by modifying the time derivative, addressing limitations of previous fractional gradient methods.

Findings

Fractional continuous time method guarantees convergence for fractional order between 0 and 1.

Simulations on chemical optimization problems demonstrate the method's efficacy.

Previous fractional gradient methods are generally inefficient in reaching the extremum.

Abstract

Recent studies have shown that fractional calculus is an effective alternative mathematical tool in various scientific fields. However, some investigations indicate that results established in differential and integral calculus do not necessarily hold true in fractional calculus. In this work we will compare various methods presented in the literature to improve the Gradient Descent Method, in terms of convergence of the method, convergence to the extreme point, and convergence rate. In general, these methods that generalize the gradient descent algorithm by replacing the gradient with a fractional-order operator are inefficient in achieving convergence to the extremum point of the objective function. To avoid these difficulties, we proposed to choose the Fractional Continuous Time algorithm to generalize the gradient method. In this approach, the convergence of the method to the extreme point of the function is guaranteed by introducing the fractional order in the time derivative, rather than in of the gradient. In this case, the issue of finding the extreme point is resolved, while the issue of stability at the equilibrium point remains. Fractional Continuous Time method converges to extreme point of cost function when fractional-order is between 0 and 1. The simulations shown in this work suggests that a similar result can be found when $1 \leq \alpha \leq 2$. { This paper highlights the main advantages and disadvantages of generalizations of the gradient method using fractional derivatives, aiming to optimize convergence in complex problems. Some chemical problems, with n=11 and 24 optimization parameters, are employed as means of evaluating the efficacy of the propose algorithms. In general, previous studies are restricted to mathematical questions and simple illustrative examples.