Suppose N = 4 and D = 3. Start with the least-squares objective function, as show nd derive the gradient descent update equation for (a) wo and (b) w2. Show all ste 2 1(w) = ||Xw
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- Solve in R programming language: 5) Use R to calculate and simulate with the exponential distribution as follows. (a) For an exponential random variable X with λ = 4, simulate 1000 independent exponential random variables by using the R function rexp(n, λ). Calculate the mean and variance of this sample. (b) Compare the empirical results from part (a) with the distribution mean 1/λ and the distribution standard deviation 1/λ.1. In a two-class problem, the likelihood ratio is p(x|C₁) p(x|C₂) Write the discriminant function in terms of the likelihood ratio.Generate the graph of f(xk) vs k where k is the iteration number and xk is the current estimate of x at iteration k. This graph should convey the decreasing nature of function values.
- 1. Let f(x₁, x₂) = x₁ cos(x₂) e¯*₁. What is the gradient Vf(x) of f?Given the function, f(A, B, C, D) = E m(0,1,4,5,10,11,14), minimize it using the Karnaugh map method. %3!For logistic regression, the gradient of the cost function is given by J(0) = (i) E (he (x) – y')x;). Write down mathematical expression(s) for the correct m gradient descent update for logistic regression with a learning rate of a. (In the expression, he(x^) should be replaced by the sigmoid function.)
- 2. Let D be a distribution over R where the mean is 5 and variance is 9. Suppose x1, ... , x10 are indepen- dent draws from D. Plot the possible positions of these random variables on the real line. 3. Let w E Rd be the variable, and let x E Rd and y E R be given. Calculate the gradient of the following functions with respect to w: • F(w) = (y – w · x)100; • F(w) = vrwsi 1 y+w.x' • F(w) = log(1+ yw · x); F(w) = e(w-x)².Program the Gaussian elimination method with no partial pivoting for solving a linear system of the form Ax=b, where b is a single column vector. Your function should take in a coefficient matrix A, and a single input vector b. Your function should return the solution vector x. Your code should also return the appropriate error message. The first line of your function should look like: function x = gaussElimination (A,b)Use the golden section search method to find the local maximum (left of the y-axis) and local minimum (right of the y-axis) of the following function: f(x) = x³ + 4x? – 9x + 5 Use a stopping criterion ɛa = 0.0001%. For each local optimum, list the x-location, function value, and number of iterations required.
- Consider the same house rent prediction problem where you are supposed to predict price of a house based on just its area. Suppose you have n samples with their respective areas, x(1), x(2), ... , x(n), their true house rents y(1), y(2),..., y(n). Let's say, you train a linear regres- sor that predicts f(x()) = 00 + 01x(e). The parameters 6o and 0, are scalars and are learned by minimizing mean-squared-error loss with L2-regularization through gradient descent with a learning rate a and the regularization strength constant A. Answer the following questions. 1. Express the loss function(L) in terms of x), y@), n, 0, 01, A. 2. Compute L 3. Compute 4. Write update rules for 6, and O1Assume that we have been running gradient descent for 50 iterations with a =0.02. It has been observed that J(0) decreases slowly and is still decreasing after 50 iterations. What does a stand for? Based on the observation, comment on whether a has been chosen appropriately.You run gradient descent for 15 iterations with alpha = 0.3, and compute J(theta) after each iteration. You find that the value of J(theta) increases over time. What would you do to correct this issue?