The proposed algorithm is in the fashion of classical majorizationminimization algorithms. Systems engineering a dynamical systems perspective on non. A parallelizable augmented lagrangian method applied to. A robust control perspective on optimization of strongly.
A variant algorithm of the stochastic heavyball method is proposed which will be incrementally processed by both the stochastic heavyball method. Accelerated gossip via stochastic heavy ball method deepai. Convex optimization lecture notes for ee 227bt draft, fall. Denote by d the diameter of the underlying convex set p, i. Convergence of heavy ball method for non convex optimization. In this paper, we provide extensive theoretical analysis of signbased methods for nonconvex optimisation under transparent assumptions. Leastsquares, linear and quadratic programs, semidefinite programming, minimax, extremal volume, and other problems. After the nonconvergence issue of adam has been raised in reddi et al. Firstorder methods for such problems can be either deterministic or stochastic. Siam journal on imaging sciences society for industrial. Optimality conditions, duality theory, theorems of alternative, and applications.
Convergence of heavyball method for non convex optimization. In addition, we show how to apply the approach on a wide family of algorithms, which includes the fast gradient method and the heavy ball method, and. Universal gradient methods for convex optimization problems. Nonetheless, the mechanisms by which momentum speedup optimization algorithms are still not very wellunderstood. When the objective function has lipschitzcontinuous gradient, we show that the cesaro average of the iterates converges to. Mar 27, 2018 in a numerical feasibility problem, the inertial alternating projection method significantly outperforms its noninertial variants. Global optimization with nonconvex constraints reduction to one dimension 511 multivariate index method 5 convergence conditions 523 8. Concentrates on recognizing and solving convex optimization problems that arise in engineering. Such problems arise, for example, in the splitvariable deterministic. For convex problems, primal and dual solutions are equivalent. In this paper, we consider a heavyball method for the constrained stochastic optimization problem by focusing to the situation that the constraint set is specified as the intersection of possibly finitely many constraint sets.
Dec 23, 2014 global convergence of the heavyball method for con vex optimization 3 1. The optimization problem induced from classical machine learning methods is often a convex and smooth one, for which gradient descent is guaranteed to solve it ef. A variant algorithm of the stochastic heavy ball method is proposed which will be incrementally processed by both the stochastic heavy ball method and random constraint. On the behavior of lagrange multipliers in convex and non. The heavyball method also called gradient descent with momentum is commonly used in optimization. The 4th conference on optimization methods and software, december 1620, 2017, havana, cuba. Issues in nonconvex optimization mit opencourseware. The method requires time o 74 log1 to nd an stationary point, meaning a point xsuch that krfxk.
The heavyball method seems also appealing in the nonconvex setting. Notes on firstorder methods for minimizing smooth functions. How to design e cient algorithms with convergence guarantees have been an active area of research due to their practical importances. Pdf global convergence of the heavyball method for.
Why should nonconvexity be a problem in optimization. Examples of non convex problems include combinatorial optimization problems, where some if not all variables are constrained to be boolean, or integers. We note that a generalization of the heavy ball method to constrained convex optimization was previously considered in 3. Nonconvex optimization problems arises in numerous advanced machine learning, statistical learning and structural estimation settings 19,3.
Liang, fadili, and peyre 2016 in nonconvex optimization. Modi cations of the heavyball method appear in the convex setting for. In particular, we focus on two special cases of shb. The class of problems we consider is that of nonsmooth and nonconvex optimization problems. This paper establishes global convergence and provides global bounds of the convergence rate of the heavyball method for convex optimization problems. A convex optimization problem is an optimization problem in which the objective function is a convex function and the feasible set is a convex set.
The heavy ball method also called gradient descent with momentum is commonly used in optimization. The sign of the stochastic gradient is a biased approximation to the true gradient, making it more challenging to analyse compared to standard sgd. There are a lot of ways to solve class of math optimization problems which is called convex optimization problems with various tradeoffs. Strekalovsky russianacademyofsciences, siberianbranch, instituteforsystemdynamicsandcontroltheory.
Optimization problems of sorts arise in all quantitative disciplines from computer science and engineering to operations research and economics, and the development of solution methods. The nonconvex part f2 is at each iteration approximated by means of a majorizing convex surrogate function. Stochastic proximal quasinewton methods for non convex composite optimization. The basic idea of the heavyball method is that, rather than using only. We obtain several new algorithms for nonconvex optimization problems. Stochastic heavy ball toulouse school of economics. Another fruitful concept from convex optimization is that of composite objective functions involving linear operators. Firstorder convex optimization methods for signal and image. On lower and upper bounds in smooth and strongly convex. Once we have an unconstrained problem, we can solve it by newtons method. Yes, non convex optimization is at least nphard can encode most problems as non convex optimization problems example.
Mathematical optimization alternatively spelt optimisation or mathematical programming is the selection of a best element with regard to some criterion from some set of available alternatives. Bilevel problems like 1 have already been considered in the literature. Variance amplification of accelerated firstorder algorithms for. The idea is to use a barrier function that sets a barrier against leaving the feasible region. In this paper we show how the stochastic heavy ball method shb a popular method for solving stochastic convex and nonconvex optimization problems operates as a randomized gossip algorithm. Jan 16, 2020 in this paper, we consider a heavy ball method for the constrained stochastic optimization problem by focusing to the situation that the constraint set is specified as the intersection of possibly finitely many constraint sets.
The heavyball method applied to the sum of distance functions to proxregular sets. While many ipms for general non convex optimization problems have been developed, there is little analysis of the sequences they generate. However, to get a convex optimization problem, the set of feasible solution, that is the shape of plan of the islandcave must be convex, too. In this paper, we consider a heavyball method for the constrained stochastic optimization problem by focusing to the situation that the constraint set.
Introduction in the eld of mathematical optimization one is interested in e ciently solving a minimization problem of the form min x2x fx. Here are some examples that are of particular relevance to problems in regression, machine learning and classi. Analysis and design of optimization algorithms via integral. Accelerated gossip via stochastic heavy ball method papers. Largescale numerical optimization 0 20 40 60 80 100 1010 105 10 0 10 5 k f x k. Convergence of gradient descent and heavy ball function values on a strongly convex. Youdidntneed to learn it at least when it wasten years ago. The sequence of iterates is attracted by a local or global minimum, stays in its neighborhood and. Local convergence of the heavyball method and ipiano for non.
From an argument based on a lyapunov function, this work shows that heavy ball converges to version of november 27, 2017. Convex optimization problem minimize f0x subject to fix. Inertial proximal algorithm for nonconvex optimization. It can be seen as a nonsmooth split version of the heavyball method from polyak.
We consider the smooth nonconvex optimization problem. The problems solved in practice, especially in machine learningstatistics, are mostlyconvex. Global convergence of the heavyball method for convex optimization. In a numerical feasibility problem, the inertial alternating projection method significantly outperforms its noninertial variants. In what follows, we discuss some existing results related to our. In the remaining part of chapter 1 we describe methods for solving convex optimization problems with a strong emphasis on. The second contribution is to tailor the heavyball method to network optimization problems. Convergence plots for the feasibility problem in section 5. Heavyball method in nonconvex optimization problems. In order to build upon these components, it is helpful to understand not just how they work. It generates and solves a sequence of convex optimization problems.
Next we will talk about a special class of optimization problems where objective functions are a sum of a convex smooth di erentiable function and a convex nondi erentiable function. Before going to the math where do we use nonconvex optimization. Unless stated otherwise, we assume that the cost functions f. These methods are rarely used directly, more often they occur as the building blocks for distributed, composite, or non convex optimization. Stochastic heavyball method for constrained stochastic. June 9, 2017 abstract a local convergence result for abstract descent methods is proved. However, due to the non existence of a central path shanno and vanderbei, 2000 it is unknown how to extend the homogenous algorithm to non convex optimization. We then derive the convergence rates of the norm of gradient for the nonconvex optimization problem, and analyze the generalization performance. What are some recent advances in nonconvex optimization.
A stochastic derivative free optimization method with momentum. Nonconvex optimization problems i smooth nonconvex problems can be solved via generic nonlinear numerical optimization algorithms sd, cg, bfgs. The convergence result of the heavyball method and ipiano translate directly to these new methods in the non convex. The result applies to constrained optimization and banach spaces. They do not need to know in advance the actual level of smoothness of the objective function. We end our study with limit theorems on several rescaled algorithms. In this paper we show how the stochastic heavy ball method shba popular method for solving stochastic convex and nonconvex optimization problems operates as a randomized gossip algorithm. January 29, 2018 abstract a local convergence result for an abstract descent method. I, e denotes the indices of the equality constraints, and i denotes the indices of the inequality constraints. Our aim in the present paper is to solve a bilevel or hierarchical optimization problem of the form. In particular, the abstract theory in this paper applies to the inertial forward backward splitting method. The core algorithms of convex optimization are gradient descent gd and the accelerated gradient method agm. The algorithm ipiano combines forwardbackward splitting with an inertial force.
Let a2r mn, y2r, 0, and jjxjj 0 is the 0 pseudonorm see example 4. Section 3 nonconvex projected gradient descent this section will introduce the simple and intuitive projected gradient descent method in the context of nonconvex optimization. We summarized the method in algorithm 1, with theoretical guarantees for nonconvex, convex and strongly convex functions under generic sampling directions d. In 59, the method was reinterpreted and analysed in the non smooth convex setting. Convergence of heavyball method for nonconvex optimization. A function mapping some subset of into is convex if its domain is convex and for all and all in its domain, the following condition holds. Global convergence of the heavyball method for convex optimization 3 1. I hard to generalize to constraints, or nondi erentiable functions i linesearch procedure can be time intensive i a reasonable idea is to develop algorithms for special classes of. Global convergence of the heavyball method for convex. Accelerated gossip via stochastic heavy ball method.
Pdf the method of projections for finding the common. Stochastic proximal quasinewton methods for nonconvex. Though, originally designed for smooth optimization problems, the bfgs method shows good performance on non smooth problems as well. Random notes about nonconvex optimization burlachenkok. Contributions to the complexity analysis of optimization. One needs a good estimate of the condition number of the hessian dnn are not quadratics. Several popular optimization problems with stronglyconvex objective functions include ridge regression 7, 2regularized logistic regression 9, and smooth support vector machine 10. The convergence result of the heavyball method and ipiano translate directly to these new methods in the non convex setting. However, mdps require precise specification of model parameters, and often the cost of a policy can be highly sensitive to the estimated parameters. We contribute improvements to a lagrangian dual solution approach applied to largescale optimization problems whose objective functions are convex, continuously differentiable and possibly nonlinear, while the non relaxed constraint set is compact but not necessarily convex.
On iteratively reweighted algorithms for nonsmooth non. Variants of this method will be used in later sections to solve problems such as sparserecoveryandrobustlearning. Abstractin this paper we show how the stochastic heavy ball method shba popular method for solving stochastic convex and nonconvex optimization problems operates as a randomized gossip algorithm. On the other hand, modern machine learning methods, like deep neural networks, often require solving a non smooth and non convex problem. January 29, 2018 abstract a local convergence result for an abstract descent method is proved. Compressed optimisation for nonconvex problems methods.
Abstract we present an accelerated gradient method for nonconvex optimization problems with lip schitz continuous rst and second derivatives. Pdf local convergence of the heavyball method and ipiano for. The sequence of iterates is attracted by a local or global minimum, stays in its neighborhood and converges. Theoretical guarantees of the original bfgs method for one or two dimen. If the optimal solution occurs at the boundary of the feasible region, the procedure moves from the interior to the. In the early 1980s, nemirovski and yudin proved that no firstorder method can converge at a rate faster than o1k2 on convex optimization problems with. This is a note about various nonconvex optimization algorithms, it doesnt cover approximately nothing from convex optimization.
Logarithmic regret algorithms for online convex optimization. Focus on manyvariable problems but will illustrate in 2d. The heavy ball method is usually at tributed topolyak1964. Local convergence of the heavyball method and ipiano for. In this paper, we revisit the convergence of the heavy ball method, and present improved convergence complexity results in the convex setting. Nonergodic convergence analysis of heavyball algorithms. The heavyball method was studied in the nonconvex setting in. We show that essentially all these algorithms behave as gossip algorithms when used to solve carefully structure problems. Markov decision processes mdp are a widely used model for dynamic decisionmaking problems. Heavyball algorithms always escape saddle points ijcai. Journal of optimization theory and applications 177. Modern methods for nonconvex optimization problems alexander s.
In this paper we study an algorithm for solving a minimization problem composed of a differentiable possibly nonconvex and a convex possibly nondifferentiable function. June 29, 2016 abstract a local convergence result for abstract descent methods is proved. Robust mdps ameliorate this issue by allowing one to specify uncertainty sets around the parameters, which leads to a non convex optimization problem. Non convex problems tend to work better in practice, but until now theory was only available for convex relaxation methods. A multistep inertial forwardbackward splitting method for. Keywords inertial forwardbackward splitting nonconvex feasibility proxregularity gradient of moreau envelopes heavyball method alternating projection averaged projection ipiano. Local nonconvex optimization gradient descent difficult to define a proper step size newton method newton method solves the slowness problem by rescaling the gradients in each direction with the inverse of the corresponding eigenvalues of the hessian can result in moving in the wrong direction negative eigenvalues.
Pdf the heavy ball with friction dynamical system for. In 30, it was generalized to smooth nonconvex functions and in 25 to a class of structured nonsmooth nonconvex optimization problems. In such problems,a collection of decisionmakers collaborate to. Siam journal on imaging sciences society for industrial and. Moreover, it reveals an equivalence between ipiano and inertial averagedalternating proximal minimization and projection methods. Accelerating convergence of largescale optimization algorithms. Conceptionally the algorithms are known from the convex setting or from their noninertial versions, however there are no guarantees for the inertial versions in the nonconvex setting.
Pdf accelerated gossip via stochastic heavy ball method. Universal gradient methods for convex optimization problems yu. We then examine the situation of convex and strongly convex potentials and derive some non asymptotic results about the stochastic heavy ball method. Pdf a local convergence result for abstract descent methods is proved. The left panel of figure2shows the iterates of gradient descent bouncing from wall to wall. You had a constrained minimization problem, which may be hard to solve dual problem may be easier to solve simpler constrains when you solve the dual problem, it also gives the solution for the. Unlike the ordinary gradient method, the subgradient method is not a descent.
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