An overview of the fractional-order gradient descent method and its applications
Laboratory of Mathematical-Chemistry, Institute of Chemistry, Universidade Federal de Alfenas (UNIFAL)（阿尔芬纳斯联邦大学化学研究所数学化学实验室）；Institute of Chemistry and Center for Computational Engineering & Science, Universidade Estadual de Campinas (UNICAMP)（坎皮纳斯州立大学化学学院与计算科学与工程中心）；Institute of Exact Science, Universidade Federal de Alfenas (UNIFAL)（阿尔芬纳斯联邦大学精确科学学院）
Higor V. M. Ferreira, Camila A. Tavares, Nelson H. T. Lemes, José P. C. dos Santos
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 $1 \leq α\leq 2$. { 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.
In this comment, we raise serious concerns over the derivation of the rate of convergence of fractional steepest descent algorithm in Fractional Adaptive Learning (FAL) approach presented in `Fractional Extreme Value Adaptive Training Method: Fractional Steepest Descent Approach' [IEEE Trans. Neural Netw. Learn. Syst., vol. 26, no. 4, pp. 653--662, April 2015]. We substantiate that the estimate of the rate of convergence is grandiloquent. We also draw attention towards a critical flaw in the design of the algorithm stymieing its applicability for broad adaptive learning problems. Our claims ar…
Motivated by gradient methods in optimization theory, we give methods based on $ψ$-fractional derivatives of order $α$ in order to solve unconstrained optimization problems. The convergence of these methods is analyzed in detail. This paper also presents an Adams-Bashforth-Moulton (ABM) method for the estimation of solutions to equations involving $ψ$-fractional derivatives. Numerical examples using the ABM method show that the fractional order $α$ and weight $ψ$ are tunable parameters, which can be helpful for improving the performance of gradient descent methods.
This paper focuses on the convergence problem of the emerging fractional order gradient descent method, and proposes three solutions to overcome the problem. In fact, the general fractional gradient method cannot converge to the real extreme point of the target function, which critically hampers the application of this method. Because of the long memory characteristics of fractional derivative, fixed memory principle is a prior choice. Apart from the truncation of memory length, two new methods are developed to reach the convergence. The one is the truncation of the infinite series, and the ot…
Zhejiang University（浙江大学）；Huzhou Institute of Zhejiang University（浙江大学湖州研究院）
The vanilla fractional order gradient descent may oscillatively converge to a region around the global minimum instead of converging to the exact minimum point, or even diverge, in the case where the objective function is strongly convex. To address this problem, a novel adaptive fractional order gradient descent (AFOGD) method and a novel adaptive fractional order accelerated gradient descent (AFOAGD) method are proposed in this paper. Inspired by the quadratic constraints and Lyapunov stability analysis from robust control theory, we establish a linear matrix inequality to analyse the conver…
We consider a class of structured fractional minimization problems, in which the numerator part of the objective is the sum of a differentiable convex function and a convex non-smooth function, while the denominator part is a convex or concave function. This problem is difficult to solve since it is non-convex. By exploiting the structure of the problem, we propose two Coordinate Descent (CD) methods for solving this problem. The proposed methods iteratively solve a one-dimensional subproblem \textit{globally}, and they are guaranteed to converge to coordinate-wise stationary points. In the ca…
Linear programming (LP) relaxation is a standard technique for solving hard combinatorial optimization (CO) problems. Here we present a gradient descent algorithm which exploits the special structure of some LP relaxations induced by CO problems. The algorithm can be run in parallel mode and was implemented as CUDA C/C++ program to be executed on GPU. We exemplify efficiency of the algorithm by solving a fractional 2-matching problem. Our results demonstrate that a fractional 2-matching problem with 100,000 nodes is solved by our algorithm on a modern GPU on a scale of a second while solving t…
We propose a novel Caputo fractional derivative-based optimization algorithm. Upon defining the Caputo fractional gradient with respect to the Cartesian coordinate, we present a generic Caputo fractional gradient descent (CFGD) method. We prove that the CFGD yields the steepest descent direction of a locally smoothed objective function. The generic CFGD requires three parameters to be specified, and a choice of the parameters yields a version of CFGD. We propose three versions -- non-adaptive, adaptive terminal and adaptive order. By focusing on quadratic objective functions, we provide a conv…
Nazarbayev University（纳扎尔巴耶夫大学）；Korea Advanced Institute of Science and Technology (KAIST)（韩国高等科学技术研究院（KAIST））；The University of Western Australia（西澳大学）；Karachi Institute of Economics and Technology（卡拉奇经济与技术学院）
Fractional learning algorithms are trending in signal processing and adaptive filtering recently. However, it is unclear whether the proclaimed superiority over conventional algorithms is well-grounded or is a myth as their performance has never been extensively analyzed. In this article, a rigorous analysis of fractional variants of the least mean squares and steepest descent algorithms is performed. Some critical schematic kinks in fractional learning algorithms are identified. Their origins and consequences on the performance of the learning algorithms are discussed and swift ready-witted r…
ENS（巴黎高等师范学院）；Univ. PSL（巴黎文理研究大学）；Univ. Grenoble Alpes（格勒诺布尔阿尔卑斯大学）；CNRS（法国国家科学研究中心）；Grenoble INP（格勒诺布尔国立理工学院）；Inria（法国国家信息与自动化研究所）；Criteo AI Lab（Criteo人工智能实验室）
In this paper, we analyze the local convergence rate of optimistic mirror descent methods in stochastic variational inequalities, a class of optimization problems with important applications to learning theory and machine learning. Our analysis reveals an intricate relation between the algorithm's rate of convergence and the local geometry induced by the method's underlying Bregman function. We quantify this relation by means of the Legendre exponent, a notion that we introduce to measure the growth rate of the Bregman divergence relative to the ambient norm near a solution. We show that this …
Substantial progress has been made recently on developing provably accurate and efficient algorithms for low-rank matrix factorization via nonconvex optimization. While conventional wisdom often takes a dim view of nonconvex optimization algorithms due to their susceptibility to spurious local minima, simple iterative methods such as gradient descent have been remarkably successful in practice. The theoretical footings, however, had been largely lacking until recently. In this tutorial-style overview, we highlight the important role of statistical models in enabling efficient nonconvex optimiz…
This book intends to deepen the study of the fractional calculus, giving special emphasis to variable-order operators. It is organized in two parts, as follows. In the first part, we review the basic concepts of fractional calculus (Chapter 1) and of the fractional calculus of variations (Chapter 2). In Chapter 1, we start with a brief overview about fractional calculus and an introduction to the theory of some special functions in fractional calculus. Then, we recall several fractional operators (integrals and derivatives) definitions and some properties of the considered fractional derivativ…
In this paper, we consider a broad class of nonsmooth and nonconvex fractional programs, where the numerator can be written as the sum of a continuously differentiable convex function whose gradient is Lipschitz continuous and a proper lower semicontinuous (possibly nonconvex) function, and the denominator is weakly convex over the constraint set. This model problem includes the composite optimization problems studied extensively lately, and encompasses many important modern fractional optimization problems arising from diverse areas such as the recently proposed scale invariant sparse signal …
Recent studies on diffusion-based sampling methods have shown that Langevin Monte Carlo (LMC) algorithms can be beneficial for non-convex optimization, and rigorous theoretical guarantees have been proven for both asymptotic and finite-time regimes. Algorithmically, LMC-based algorithms resemble the well-known gradient descent (GD) algorithm, where the GD recursion is perturbed by an additive Gaussian noise whose variance has a particular form. Fractional Langevin Monte Carlo (FLMC) is a recently proposed extension of LMC, where the Gaussian noise is replaced by a heavy-tailed α-stable noise. …
This paper reviews the gradient sampling methodology for solving nonsmooth, nonconvex optimization problems. An intuitively straightforward gradient sampling algorithm is stated and its convergence properties are summarized. Throughout this discussion, we emphasize the simplicity of gradient sampling as an extension of the steepest descent method for minimizing smooth objectives. We then provide overviews of various enhancements that have been proposed to improve practical performance, as well as of several extensions that have been made in the literature, such as to solve constrained problems…
