Steepest descent and conjugate gradient methods with variable preconditioning ilya lashuk the speaker and andrew knyazev department of mathematics and center for computational mathematics university of colorado at denver supported by national science foundation and copper mountain conference 2006. Then, we show that uncertainty sampling is preconditioned stochastic gradient descent on the zeroone loss in section 3. The regions of high curvature dominate the stepsize in gradient descent. An introduction to the conjugate gradient method without the. An introduction to the conjugate gradient method without. The gradient points directly uphill, and the negative gradient points directly downhill thus we can decrease f by moving in the direction of the negative gradient this is known as the method of steepest descent or gradient descent steepest descent proposes a new point where. Axb, where a is a positive definite symmetric matrix. The cga is only slightly more complicated to implement than the method of steepest descent but converges in a finite number of steps on quadratic problems. New threeterm conjugate gradient method with guaranteed global convergence. Stochastic gradient descent vs online gradient descent. Steepest descent and conjugate gradient methods with variable preconditioning article pdf available in siam journal on matrix analysis and applications 294 june 2006 with 522 reads. A new nonlinear conjugate gradient method and an associated implementation, based on an inexact line search, are proposed and analyzed.
A new conjugate gradient method with guaranteed descent and. Pluim 1 biomedical imaging group rotterdam, erasmus mc, rotterdam, the netherlands 2 lkeb, leiden university medical center, leiden, the netherlands 3 image sciences institute, university medical center utrecht. With exact line search, our method reduces to a nonlinear ver. Work to overcome this issue includes, for example, the work of huang, li and liu 24 where preconditioners based on gradient descent algorithms are designed and shown to work well for p. Jul 23, 2016 hi, ibraheem aldhamari, i checked your link, and gave psgd a trial on your problem. During the last decade, the data sizes have grown faster than the speed. Nonsymmetric preconditioning for conjugate gradient and. I created this video with the youtube video editor. Preconditioned stochastic gradient descent abstract. It differentiates itself from most methods by its inherent abilities of handling nonconvexity and gradient noise. Effective training of deep neural networks suffers from two main issues. In practice, the scales of problem can be so large that we need to sparsely represent a preconditioner to make its estimation affordable.
The basic idea is to combine the scaled memoryless bfgs method and the preconditioning technique in the frame of the conjugate gradient method. Psgd differentiates itself from most existing methods by its inherent abilities of handling nonconvexity and gradient noises. Our main result is that the conjugate gradient cg method with variable preconditioning in certain situations may not give improvement, compared to the steepest descent sd method for solving a linear sys. To build all programs concurrently, run make without any arguments make clean will remove compiled executables and created data files. Gradient descent is a firstorder iterative optimization algorithm for finding a local minimum of a. If one is using gradient descent to optimize over a vector space where each of the components is of a different. It is possible to precondition sgd to accelerate its convergence remarkably. Preconditioned steepest descentlike methods for symmetric. Lncs 6892 preconditioned stochastic gradient descent. Dec 14, 2015 stochastic gradient descent sgd still is the workhorse for many practical problems. Pdf constrained and preconditioned stochastic gradient. Psgd preconditioned stochastic gradient descent is a general secondorder optimization method. Also shows a simple matlab example of using conjugate gradient to solve a. Gradient descent also benefits from preconditioning, but this is not done as commonly.
Finally, we show that uncertainty sampling iterates in expectation move in a descent direction of zin section 3. Other topics include preconditioning and the nonlinear conjugate gradient. The preconditioned steepest descent psd iteration is a well known precursor of the optimal preconditioned conjugate gradient pcg algorithm for solving symmetric. Steepest descent and conjugate gradient methods with. Curve fitting and pde solving with gradient descent. Both asgd and rm employ a stochastic subsampling technique to accelerate the optimisation process. The first is that the parameter spaces of these models exhibit pathological curvature. Here, we insist that the directions pk be conjugate to each other.
In contrast to newton method, there is no need for matrix inversion. The eigenvalue problems are solved using the locally optimal block preconditioned conjugate gradient lobpcg method. The steepest descent algorithm is a classical iterative method for solving a linear system. On hager and zhangs conjugate gradient method with guaranteed descent. Problem outline 1 problem 2 stochastic average gradient sag 3 accelerating sgd using predictive variance reduction svrg 4 conclusion rie johnson, tong zhang presenter.
A new conjugate gradient method with guaranteed descent and an efficient line search. We propose a preconditioned accelerated stochastic gradient method suitable for large scale optimization. We then of n are being very large, say, n 106 or n 107. The idea of quadratic forms is introduced and used to derive the methods of steepest descent, conjugate directions.
Parallel preconditioned conjugate gradient algorithm on. Note that rk is the negative gradient of f at x xk, so the gradient descent method would be to move in the direction rk. Conjugate gradients an overview sciencedirect topics. Steepest descent, conjugate gradient, iterative method, innerouter iterations, variable preconditioning, random preconditioning, preconditioner, condition number. The convergence rate of inexact preconditioned steepest descent. Solution of a nonlinear system gradient descent can also be used to solve a system of nonlinear equations. Also, the theoretical convergence conditions of the optimisation method are taken into account. In both the original and the preconditioned conjugate gradient methods one only needs to set. Kao, ucla directional second derivative the directional second derivative along unit vector uis given by t hu. On the performance of preconditioned stochastic gradient descent. The conjugate gradient method is the most prominent iterative method for solving sparse systems of linear equations.
Below is an example that shows how to use the gradient descent to solve for three unknown variables, x1, x2, and x3. Largescale machine learning with stochastic gradient descent. Work to overcome this issue includes, for example, the work of huang, li and liu 24 where preconditioners based on gradient descent algorithms are designed and shown to work well for p up to. A new conjugate gradient method with guaranteed descent. This paper shows that uncertainty sampling is performing preconditioned stochastic gradient descent on the expected zeroone loss, which provides some explanation of the empirical finding that active learning with uncertainty sampling could yield lower zeroone loss than passive learning even with fewer labeled examples. On the momentum term in gradient descent learning algorithms pdf. However, it converges slow, and can be difficult to tune. Uncertainty sampling is preconditioned stochastic gradient. Preconditioned stochastic gradient descent optimisation for monomodal image registration stefan klein1,mariusstaring2, patrik andersson 3, and josien p.
On the performance of preconditioned stochastic gradient. Pdf statistically preconditioned accelerated gradient. Preconditioned stochastic gradient descent psgd psgd is a secondorder stochastic optimization method. But many attempts in this direction either aim at solving specialized. The analysis covers two classes of methods that are globally convergent on smooth, nonconvex functions. Dec 14, 2015 preconditioned stochastic gradient descent.
Stochastic gradient descent sgd still is the workhorse for many practical problems. Preconditioned gradient iterations for the eigenproblem of. Stochastic gradient descent with variance reduction. This generality is used in preconditioned gradient descent methods. Preconditioned stochastic gradient descent file exchange. This paper studies the performance of preconditioned stochastic gradient descent psgd, which can be regarded as an enhance stochastic newton method with the ability to handle gradient noise and nonconvexity at the same time. For example, one could use gradient descent or the conjugate gradient method. Our preconditioner, derived from the standard ssor, is an approximate inverse and can therefore be used in the pcg algorithm through a sparse matrixvector multiplication. Psgd does converge faster and better than gradient descent on your problem, although it is a simple convex problem with exact gradient. Unconstrained nonlinear optimization algorithms matlab.
Preconditioned stochastic gradient descent optimisation. An accelerated scaled memoryless bfgs preconditioned conjugate gradient algorithm for solving unconstrained optimization problems is presented. We have presented a parallel implementation, on gpu, of the preconditioned conjugate gradient algorithm for linear systems with symmetric, positive definite matrix. Preconditioned conjugate gradient preconditioner action. Pdf preconditioned stochastic gradient langevin dynamics. We consider the setting of distributed empirical risk minimization where multiple machines compute the gradients in parallel and a centralized server updates the model parameters. If a symmetric and positive denite matrix eis an approximateinverse of. Effectively, were just running gradient descent with gradients scaled by some positive semidefinite matrix.
It is possible to precondit ion sgd to accelerate its convergence remarkably. Preconditioned conjugate gradient pcg method is an equation solver that is suitable for parallelism. Aug 23, 2016 i created this video with the youtube video editor. Eecs 551 explored the gradient descent gd and preconditioned gradient descent pgd algorithms for solving leastsquares problems in detail. T or m ttt is called preconditioner in naive implementation, each iteration requires multiplies by t and tt and a. Accelerating sgd with preconditioning and adaptive. Pdf steepest descent and conjugate gradient methods with. Constrained and preconditioned stochastic gradient method article pdf available in ieee transactions on signal processing 6310. The convergence rate of inexact preconditioned steepest. Preconditioned spectral descent for deep learning david e.
We then give convergence analysis of the gradient method to show that it converges as the optimal richardson method. Symmetric successive overrelaxation preconditioner. Preconditioned stochastic gradient descent ieee journals. The result is conjugate gradient on the normal equations cgnr. The convergence rate of inexact preconditioned steepest descent algorithm for solving linear systems by hans munthekaas abstract. The gradient is rfu au b, so the steepest descent direction is given by the residual. As h in general is symmetric and thus has a set of real eigenvalues with. Pdf preconditioned descent algorithms for plaplacian. We have improved the implementation of psgd, unrevealed its relationship to equilibrated stochastic gradient descent esgd and feature.
Preconditioned conjugate gradient method jacobi preconditioner. Pdf we analyze the conjugate gradient cg method with variable preconditioning for solving a linear system with a real symmetric positive. But many attempts in this direction either aim at solving specialized problems, or result in significantly more complicated methods than sgd. Index termsstochastic gradient descent, preconditioner, non convex optimization.
We derive sufficient convergence conditions for the minimization of convex functions using a generic class of diagonal preconditioners and provide a formal convergence proof based on a framework originally used for online learning. Gradient methods use information about the slope of the function to dictate a direction of search where the minimum is thought to lie. This paper explores the convergence of nonlinear conjugate gradient methods without restarts, and with practical line searches. Online gradient descent, also known as sequential gradient descent or stochastic gradient descent, makes an update to the weight vector based on one data point at a time whereas, 2 describes that as subgradient descent, and gives a more general definition for stochastic gradient descent. Global convergence properties of conjugate gradient. Parallel preconditioned conjugate gradient algorithm on gpu. The proposed adaptive stochastic gradient descent asgd method is compared to a standard, nonadaptive robbinsmonro rm algorithm. Thus, every iteration of these steepest descent methods. The method combines both approaches by coupling nesterovs accelerated gradient descent. Tensorflow implementation of preconditioned stochastic. Recent methods address this problem by using adaptive preconditioning for stochastic gradient descent sgd.
Preconditioned stochastic gradient descent xilin li abstractstochastic gradient descent sgd still is the workhorse for many practical problems. These methods improve convergence by adapting to the local geometry of parameter space. In this paper we propose a preconditioned accelerated stochastic gradient descent pasgd method with a generic bounded preconditioner and analyze its convergence properties for convex cost functions. We have written the codes of exible pcg and psd by modifying the hypre standard pcg function. However, if the pdf of p is continuous, then the population zeroone loss is differentiable at most. The steepest descent algorithm is a classical iterative method for solving a linear system axb, where a is a positive definite symmetric matrix. More generally, if is a positive definite matrix, then. A preconditioned accelerated stochastic gradient descent. Preconditioned stochastic gradient descent the psgd is established as.
With this substitution, vectors p are always the same as vectors z, so there is no need to store vectors p. Jiawen yaostochastic gradient descent with variance reduction march 17, 2015 3 29. The conjugate gradient method can be applied to an arbitrary nbym matrix by applying it to normal equations a t a and righthand side vector a t b, since a t a is a symmetric positivesemidefinite matrix for any a. Boyd,lecture slides and notes for ee364b, convex optimization ii,lecturesonthe conjugategradientmethod. Carlson,1 edo collins, 2yaping hsieh, lawrence carin,3 volkan cevher2 1 department of statistics, columbia university 2 laboratory for information and inference systems lions, epfl 3 department of electrical and computer engineering, duke university abstract deep learning presents notorious. We choose the direction vector d 0 to be the steepest descent directionof the function fu. In order to reduce the number of communications required to reach a given accuracy, we propose a preconditioned accelerated gradient method where the preconditioning is done by solving a. Steepest descent and conjugate gradient methods with variable. The key ingredient to make a gradient iteration an efcient solver for the operator eigenvalue problem 1.
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