How does calculating only a few of eigenvalues out of thousands work with conjugate-gradient method?
I read that if you have a large square matrix, say more than 1000x1000 and you want to get its eigenvalues but all you need is just, say, 10 of them, then there is the so-called conjugate gradient method that can save you a significant amount of time of calculating exactly the number of eigenvalues you want instead of all of them. Can someone point me to existing numerical libraries (does BLAS or LAPACK have it) and references? EDIT: The matrix can be 10^6 x 10^6.
[Numerical Analysis] How good is the tridiagonal preconditioner for the conjugated gradient?
I am trying to use this preconditioner (made by simply taking the 3 diagonals of my matrix) to fasten the convergence of the conjugated gradient, but it keeps bahving way worse than without any, so I'm wondering if it is known that it behaves nicely in very few cases or it normally behaves good and I probably did some mistakes writing my code (though I checked more times than my sanity can handle). Maybe it depends on the fact that I am working with Toeplitz matrices. Thanks in advance!
Tridiagonal preconditioner for conjugated gradient
I am trying to use this preconditioner (made by simply taking the 3 diagonals of my matrix) to fasten the convergence of the conjugated gradient, but it keeps bahving way worse than without any, so I'm wondering if it is known that it behaves nicely in very few cases or it normally behaves good and I probably did some mistakes writing my code (though I checked more times than my sanity can handle). Thanks in advance!
I am preparing for my final exams, and in my preparations for a Numerical Methods exam on Monday, I was trying to review the Conjugate Gradient Method, and my book is not really clear on how to attempt a problem. How am I supposed to go about solving a problem? Any advice would be greatly appreciated. Thank you!
[R] Overfitting in Neural Nets: Backpropagation, Conjugate Gradient, and Early Stopping, NIPS2000
I've been applying deep neural networks to medical images. One of the main obstacles in this domain is the lack of enough training data which often leads to poor generalization. While trying hyper parameter tuning to overcome the overfitting including depth/width of ths network, dropout ratio, weight decay term, etc, I found simple early stopping was quite effective. Below is the link to the very interesting paper which explains what happens during the back prop and the role of the early stopping: https://papers.nips.cc/pape1895-overfitting-in-neural-nets-backpropagation-conjugate-gradient-and-early-stopping
Can we use Conjugate Gradients method to maximize a function instead of minimizing?
So I am trying to Conjugate Gradients method to find the vector x which maximizes a function L(x): Rn -> R1. I tried to cheat the algorithm by changing the signal of the function, thinking it would minimize -L(x). But it showed to be immune to it because of the way it computes the alpha factor. I sincerely appreciate any help.
Question about the behavior of conjugate gradient descent optimization
So I'm playing around with sparse autoencoders, and I'm trying to train a simple example with conjugate gradient descent. I just witnessed some behavior I can't explain and I'm hoping someone here can help me understand what's going on. The neural network I'm training is small, and meant to solve the XOR problem. It has two inputs plus a bias on the input layer, two hidden units (plus a bias), and a single output. This creates 3*2 + 3 = 9 total weights to be trained. I have confidence that my gradient calculations are correct, because they pass the gradient estimation check described here, and are used to generate edge detectors for natural images with the backpropagation algorithm as described here. It should be a short couple of steps to train this network to solve XOR with conjugate gradient descent using my already-coded gradient calculation plus an erf() function that calculates overall network error. I'm using the Polak-Ribiere method to generate the Beta coefficient. My erf() function is more of less exactly as described at the UFLDL site. Finally, the problem: My CGD algorithm seems to be sensitive to the magnitude of the weights that I initialize the network with. When I initialize the weights with uniform random numbers in the range of [-0.1 0.1], the algorithm reliably converges on a bad local minima (all inputs result in an output of 0.5). If I hange the weight initialization to uniform random numbers of the range [-0.3 0.3], then the network converges to a state that solves XOR. What's the principle at work here? Is this kind of weight sensitivity something specific to CGD? Thanks!
I am trying to implement conjugate gradient descent numerically but I'm stuck. The update rule in normal gradient descent algorithm is: [; x_{n+1} = x_n + \alpha \nabla f(x_n) ;] My problem is that I don't have the explicit defention of function f or its gradient, but I do have a table where I can look up the value of its gradient at any point. I'm also using some fixed [; \alpha ;] here. Now that I'm trying to implement the conjugate gradient descent I'm stuck and don't know what I should do. As mentioned on the Wikipedia article this algorithm can be used to solve a system of linear equations Ax = b. But I don't have a matrix A or vector b. I just have a numeric value of the gradient of a function at a point. So, when in the algorithm I have to update the coefficients alpha and beta I don't know how I should proceed because I don't have the matrix A. I also have the same problem in normal gradient descent method to have a dynamic alpha. What is it that I don't understand here?
There seems to an explanation for gradient descent, but not one for conjugate gradient descent which seems to be similar yet different in crucial ways.
Born today : February 2nd - Cornelius Lanczos, Mathematician, Physicist, "developed a number of techniques for mathematical calculations ... Lanczos algorithm for finding eigenvalues, Lanczos approximation for the gamma function, conjugate gradient method for solving systems of linear equations"
Born today : February 2nd - Cornelius Lanczos, Mathematician, Physicist, "developed a number of techniques for mathematical calculations ... Lanczos algorithm for finding eigenvalues, Lanczos approximation for the gamma function, conjugate gradient method for solving systems of linear equations"
China rockets to top of supercomputing charts with homegrown tech: TaihuLight 1st on Top500 list. 2nd on Graph500 list, which ranks supercomputers by ability to handle data-intensive applications. 3rd in High Performance Conjugate Gradient, focuses on dealing with complicated engineering problems
Is negative step size an issue in conjugate gradient descent?
I have followed this tutorial and implemented conjugate gradient descent. It works fairly good for my problems. I started plotting the step size '[; \alpha ;]`, and noticed for some initial conditions it is negative. Although it solves the problem it is negative. Is this an issue? I mean mathematically, does this violate any assumption for stability of the algorithm or some thing like that? If I use this condition [; \alpha = max(0, \alpha) ;] it gets stuck in local minima and can not converge to the solution.
NN with a genetic algorithm used for an initial phase for a starting point in a conjugate gradient algorithm. I have a few questions, does someone have some time to help?
I'm a Structural Engineering grad student and I'm doing a research paper for a Structural Reliability class base on this paper. I understand how the neural network works but I'm having some trouble with the optimization using the genetic algorithm and the conjugate gradient algorithm. This portion of the paper falls between equations (7) and (8). Would someone mind ELI5 this process using first iteration of these algorithms with actual numbers, as used in this paper? These algorithms are beyond the scope of this class and I've been trying to find examples but have not had any luck.
The conjugate gradient method is built upon the idea of reducing the number of jumps and make sure the algorithm never selects the same direction twice. To achieve this, one needs the following choices for the size of the jumps and search directions: The conjugate gradient method is a conjugate direction method ! Selects the successive direction vectors as a conjugate version of the successive gradients obtained as the method progresses. ! The conjugate directions are not specified beforehand, but rather are determined sequentially at each step of the iteration. We can summarize Conjugate Gradient in a line as: at each step , go to the minimizer of ( ) = 1 2 ⊤ − ⊤ in the subspace 0 + ( ; 0). At step , move to the -minimizer in 0 + ( ; 0). Lemma 4.3. The following hold. 1. = argmin ∈ 0+ ( ; 0) ( ). 2.The residual The Conjugate Gradient Method is the most prominent iterative method for solving sparse systems of linear equations. Unfortunately, many textbook treatments of the topic are written with neither illustrations nor intuition, and their victims can be found to this day babbling senselessly in the corners of dusty libraries. For this reason, a deep, geometric understanding of the method has been It looks like the conjugate gradient method is meant to solve systems of linear equations of the for $$ A\mathbf{x} = \mathbf{b} $$ Where $A$ is an n-by-n matrix that is symmetric, positive-definite and real. On the other hand, when I read about gradient descent I see the example of the Rosenbrock function, which is 在Conjugate Gradient中，我们希望在这样一组正交基底的方向上更新，每次都在该方向上一步到位，然后再在与其正交的空间里面搜索。原则上这样n步就能找到凸优化问题的最小值。 二、符号定义. Residual是目标 与现在 的差， ；同时注意到，它也是凸函数的负导数 The conjugate gradient is a numerical method meaning that the we get out is not an exact solution. Consequently, the vector is only close but not exactly one. You can find the file on my github repository . Conjugate gradient method A comparison of the convergence of gradient descent with optimal step size (in green) and conjugate vector (in red) for minimizing a quadratic function associated with a given linear system. Conjugate gradient, assuming exact arithmetics, converges in at most nsteps where n Conjugate Gradient Method • direct and indirect methods • positive definite linear systems • Krylov sequence • spectral analysis of Krylov sequence • preconditioning EE364b, Stanford University. Three classes of methods for linear equations methods to solve linear system Ax = b, A ∈ Rn×n • dense direct (factor-solve methods) – runtime depends only on size; independent of data The conjugate gradient method is an algorithm for the numerical solution of particular systems of linear equations, namely those whose matrix is symmetric and positive-definite. The conjugate gradient method is often implemented as an iterative algorithm , applicable to sparse systems that are too large to be handled by a direct implementation or other direct methods such as the Cholesky decomposition.
About Press Copyright Contact us Creators Advertise Developers Terms Privacy Policy & Safety How YouTube works Test new features Press Copyright Contact us Creators ... This video will explain the working of the Conjugate Gradient (Fletcher Reeves) Method for solving the Unconstrained Optimization problems.Steepest Descent M... Lecture course 236330, Introduction to Optimization, by Michael Zibulevsky, TechnionMotivation 0:0Scalar product, definition 4:47 (slide on 8:53), and exampl... This is helpfull to every students of civil engineering from Tribhuvan University private colleges and I.O.E. pulchowk campus T.U.link of I.O.E old question... Video lecture on the Conjugate Gradient Method A brief overview of steepest descent and how it leads the an optimization technique called the Conjugate Gradient Method. Also shows a simple Matlab example ... In mathematics, the conjugate gradient method is an algorithm for the numerical solution of particular systems of linear equations, namely those whose matrix... Conjugate gradient is a more advanced algorithm than steepest descent for obtaining a minimum energy configuration of a molecular system. The step history is used to accelerate the convergence to ... Design and Optimization of Energy Systems by Prof. C. Balaji , Department of Mechanical Engineering, IIT Madras. For more details on NPTEL visit http://nptel... Video lecture on the Conjugate Gradient Method.A brief overview of steepest descent and how it leads the an optimization technique called the Conjugate Gradi...
