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Modified Newton method | Wolfe Backtracking | Theory and Python Code | Optimization Algorithms #6
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In this one, I will show you what the modified newton algorithm is and how to use it with the backtracking search by Armijo rule. We will approach both methods from intuitive and animated perspectives. The difference between Damped and its modified newton method is that the Hessian may run into singularities at some iterations, and so we apply diagonal loading, or Tikhonov regularization at each iteration. As a reminder, Damped newton, just like newton’s method, makes a local quadratic approximation of the function based on information from the current point, and then jumps to the minimum of that approximation. Just imagine fitting a little quadratic surface in higher dimensions to your surface at the current point, and then going to the minimum of the approximation to find the next point. Finding the direction towards the minimum of the quadratic approximation is what you are doing. As a matter of fact, this animation shows you why in certain cases, Newton's method can converge to a saddle or a maximum. If the eigenvalues of the Hessian are non positive - in those cases the local quadratic approximation is an upside down paraboloid. Next, let’s talk about the line search we are going to use in this tutorial, which is based on Wolfe criterion. This is achieved by the Wolfe condition, which sufficiently decreases our function ! The Wolfe condition combines both the Armijo condition as well as an additional curvature condition to formulate the strong Wolfe condition. The curvature condition ensures a sufficient increase of the gradient. This is also a strong wolfe condition, which restricts slopes from getting too positive, hence excluding points far away from stationary points.
⏲Outline⏲
00:00 Introduction
00:55 Modified Newton Method
03:42 Wolfe Criterion
04:58 Python Implementation
18:15 Animation Module
33:42 Animating Iterations
38:20 Outro
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This lecture contains any optimization techniques.
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