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4. Gradient descent. Gradient descent is one of the most popular algorithms in data science and by far the most common way to optimise neural networks. A function is minimised by iteratively moving a little bit in the direction of negative gradient. For the two-dimensional case, the step of iteration is given by the formula (**+²) = (*) – - € ▼ ƒ (xXn, Yn). Yn+1, In general, & does not have to be a constant, but in this question, for demonstrative purposes, we set = 0.1. Let f(x, y) = 3.5x² - 4xy +6.5y² and ro and yo be any real numbers. (a) For all x, y R compute Vf(x, y) and find a matrix A such that (-) = (-) - in terms of xo and yo and powers of A. A Write an expression for (5) Yn - ε V f (x, y). (b) Find the eigenvalues of A. (c) Find one eigenvector corresponding to each eigenvalue.

4. Gradient descent. Gradient descent is one of the most popular algorithms in data science and by far the most common way to optimise neural networks. A function is minimised by iteratively moving a little bit in the direction of negative gradient. For the two-dimensional case, the step of iteration is given by the formula (**+²) = (*) – - € ▼ ƒ (xXn, Yn). Yn+1, In general, & does not have to be a constant, but in this question, for demonstrative purposes, we set = 0.1. Let f(x, y) = 3.5x² - 4xy +6.5y² and ro and yo be any real numbers. (a) For all x, y R compute Vf(x, y) and find a matrix A such that (-) = (-) - in terms of xo and yo and powers of A. A Write an expression for (5) Yn - ε V f (x, y). (b) Find the eigenvalues of A. (c) Find one eigenvector corresponding to each eigenvalue.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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4. Gradient descent. Gradient descent is one of the most popular algorithms in data science and by far the most common way to optimise neural networks. A function is minimised by iteratively moving a little bit in the direction of negative gradient. For the two-dimensional case, the step of iteration is given by the formula Xn+1 Yn+1 n+¹) = (xn) - € V ƒ (Xxnx Y₁). E Yn In general, & does not have to be a constant, but in this question, for demonstrative purposes, we set = 0.1. Let f(x, y) = 3.5x² - 4xy +6.5y² and xo and yo be any real numbers. (a) For all x, y ER compute Vf(x, y) and find a matrix A such that Τη Yn A ¹ (*) = (*) in terms of xo and yo and powers of A. - ε ▼ f(x, y). Write an expression for (b) Find the eigenvalues of A. (c) Find one eigenvector corresponding to each eigenvalue. (d) Find matrices P and D such that D is diagonal and A = PDP-¹. (e) Find matrices Dn, P-1 and An. Find formulas for an and yn. (f) Suppose to = yo = 1. Find the smallest N EN such that |( YN < 0.05, ) IN-1 YN-1, N (g) Sketch the region R consisting of those (ro, yo) such that N ≥ 0, YN ≥ 0 and |(x)| ≤ > 0.05, where N is the number found in part (f). Write an equation for the boundary of R. Which points of the boundary belongs to R and which do not? ≤0.05.
Transcribed Image Text:4. Gradient descent. Gradient descent is one of the most popular algorithms in data science and by far the most common way to optimise neural networks. A function is minimised by iteratively moving a little bit in the direction of negative gradient. For the two-dimensional case, the step of iteration is given by the formula Xn+1 Yn+1 n+¹) = (xn) - € V ƒ (Xxnx Y₁). E Yn In general, & does not have to be a constant, but in this question, for demonstrative purposes, we set = 0.1. Let f(x, y) = 3.5x² - 4xy +6.5y² and xo and yo be any real numbers. (a) For all x, y ER compute Vf(x, y) and find a matrix A such that Τη Yn A ¹ (*) = (*) in terms of xo and yo and powers of A. - ε ▼ f(x, y). Write an expression for (b) Find the eigenvalues of A. (c) Find one eigenvector corresponding to each eigenvalue. (d) Find matrices P and D such that D is diagonal and A = PDP-¹. (e) Find matrices Dn, P-1 and An. Find formulas for an and yn. (f) Suppose to = yo = 1. Find the smallest N EN such that |( YN < 0.05, ) IN-1 YN-1, N (g) Sketch the region R consisting of those (ro, yo) such that N ≥ 0, YN ≥ 0 and |(x)| ≤ > 0.05, where N is the number found in part (f). Write an equation for the boundary of R. Which points of the boundary belongs to R and which do not? ≤0.05.
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