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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 arithmetic, converges in at most n steps, where n is the size of the matrix of the system (here n = 2).
XGBoost works as Newton-Raphson in function space unlike gradient boosting that works as gradient descent in function space, a second order Taylor approximation is used in the loss function to make the connection to Newton Raphson method. A generic unregularized XGBoost algorithm is:
Quadratic programming (QP) is the process of solving certain mathematical optimization problems involving quadratic functions.Specifically, one seeks to optimize (minimize or maximize) a multivariate quadratic function subject to linear constraints on the variables.
Much as the derivative of a function of a single variable represents the slope of the tangent to the graph of the function, [7] the directional derivative of a function in several variables represents the slope of the tangent hyperplane in the direction of the vector. The gradient is related to the differential by the formula = for any , where ...
To solve the system is to find the value of the unknown vector . [3] [5] A direct method for solving a system of linear equations is to take the inverse of the matrix , then calculate =. However, computing the inverse is computationally expensive.
HiGHS has an interior point method implementation for solving LP problems, based on techniques described by Schork and Gondzio (2020). [10] It is notable for solving the Newton system iteratively by a preconditioned conjugate gradient method, rather than directly, via an LDL* decomposition. The interior point solver's performance relative to ...
Whereas linear conjugate gradient seeks a solution to the linear equation =, the nonlinear conjugate gradient method is generally used to find the local minimum of a nonlinear function using its gradient alone. It works when the function is approximately quadratic near the minimum, which is the case when the function is twice differentiable at ...
Since is symmetric positive-definite, we can apply standard iterative methods like the gradient descent method or the conjugate gradient method to solve = in order to compute . The vector can be reconstructed by solving