Conjugate direction methods can be regarded as being between the method of steepest descent (first-order method that uses gradient) and Newton’s method (second-order method that uses Hessian as well). Motivation: ! steepest descent is slow. Goal: Accelerate it! ! Newton method is fast… BUT: we need to calculate the inverse of the Hessian Typically, you'd use gradient ascent to maximize a likelihood function, and gradient descent to minimize a cost function. Both gradient descent and ascent are practically the same. Let me give you an concrete example using a simple gradient-based optimization friendly algorithm with a concav/convex likelihood/cost function: logistic regression. In gradient descent, we compute the update for the parameter vector as $\boldsymbol \theta \leftarrow \boldsymbol \theta - \eta \nabla_{\!\boldsymbol \theta\,} f(\boldsymbol \theta)$. Steepest descent is typically defined as gradient descent in which the learning rate $\eta$ is chosen such that it yields maximal gain along the negative gradient ONJUGATE GRADIENT METHOD. We use conjugate gradient method to solve the system of linear equations given in the form of Ax = b; (11) where A is a positive definite matrix with n n sizes.As a result of operation of this method we obtain a sequence of vectors starting from a vector initial x. 0. x. 0!x. 1!! x. n; Conjugate gradient method in Python With the conjugate_gradient function, we got the same value (-4, 5) and wall time 281 μs, which is a lot faster than the steepest descent. Visualizing steepest 13. Conjugate Gradients on the Normal Equations 41 14. The Nonlinear Conjugate Gradient Method 42 14.1. Outline of the Nonlinear Conjugate Gradient Method 42 14.2. General Line Search 43 14.3. Preconditioning 47 A Notes 48 B Canned Algorithms 49 B1. Steepest Descent 49 B2. Conjugate Gradients 50 B3. Preconditioned Conjugate Gradients 51 i Conjugate Gradients vs Steepest Descents Next: Comparison of Preconditioning Methods Up: Solution of second order Previous: Direct Minimisation: Conjugate Gradients Contents In figure 4.1 the performance of a steepest descents minimiser is compared to that of a conjugate gradients minimiser when applied to the determination of for -quartz. Gradient descent is the method that iteratively searches for a minimizer by looking in the gradient direction. Conjugate gradient is similar, but the search directions are also required to be orthogonal to each other in the sense that $\boldsymbol{p}_i^T\boldsymbol{A}\boldsymbol{p_j} = 0 \; \; \forall i,j$. $\begingroup$ The wikipedia article actually does a pretty good job of illustrating the difference between conjugate gradient method and the gradient descent method and even approaches conjugate gradient from the perspective of both a direct solve and an iterative solve. So we are looking at different methods for solving system of linear equations. Equivalently solving the matrix equation A x = b. When A is n⨉n and x and b are vectors. In Gauss–Seidel method we decompose A as A=L* + U where L* is the diagonal and
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