Problem 2: Convergence of Gradient Descent in Over-Parameterized Neural Networks Statement: Consider an over-parameterized neural network (i.e., a network with more parameters than necessary to fit the training data) trained using gradient descent on a squared loss function. Prove that, under appropriate initialization and with a sufficiently small learning rate, gradient descent converges to a global minimum of the loss function. Key Points for the Proof: • • • Define the over-parameterization regime and its implications for the loss landscape. Analyze the dynamics of gradient descent in the high-dimensional parameter space. Use tools from optimization theory to show that all local minima are global minima in this setting. Ensure that the initialization is within the basin of attraction for convergence to a global minimum.

Given a collection of data points {(xi, y₁)}₁ find the best least squares approxima- ax² + bx³. tion of the form y =

Ex[lain the Two Stage Least Squares Estimator?

An article presents the results of an experiment in which the surface roughness (in μm) was measured for 15 D2 steel specimens and compared with the roughness predicted by a neural network model. The results are presented in the following table. TRUE Value (x) Predicted Value (y) of 6 0.45 0.400 0.82 0.7 0.54 0.52 0.41 0.39 0.77 0.74 2:30:15 0.79 0.78 0.25 0.27 pped 0.62 0.6 0.91 0.87 0.52 Book 0.51 1.02 0.91 Hint 0.6 0.71 Ask 0.58 0.5 erences 0.87 0.91 1.06 1.04 ۵ To check the accuracy of the prediction method, the linear model y=Bo+B1x+ & is fit. If the prediction method is accurate, the value of Bo will be 0 and the value of ẞ1 will be 1. Note: This problem has a reduced data set for ease of performing the calculations required. This differs from the data set given for this problem in the text.

2. Let consider given points: (1,1), (3,2), (4,3), (5,6). (a) Find the best least square fit by a linear function (linear regression). Compute the sum of square errors. (b) Find a polynomial of degree 3 that interpolates the points. (c) Find the best least squares exponential fit y = piet. Compute the sum of square errors and compare with result from (a).

Graph the data and observe the relationship between x and y, is it linear or nonlinear? Predict which of least squares approximation P₁(x) (linear) or P₂(x) (2nd order polynomial) is better? or they are similar? (Do not need to find P₁ or P₂, no calculations are involved in this problem) Xi Yi 0 1.0 0.15 1.004 0.31 0.5 1.031 1.117 0.6 1.223 0.75 1.422

all 3 - true or false

Can someone please explain to me ASAP??!!

Show that the first least squares assumption, E(ui | Xi) = 0, implies thatE(Yi | Xi) = β0 + β1Xi.

True or false II If false, explain briefly. a) Some of the residuals from a least squares linear model will be positive and some will be negative. b) Least Squares means that some of the squares of the residuals are minimized. c) We write y n to denote the predicted values and y to  denote the observed values.