Problem 2: Convergence of Gradient Descent in Over-Parameterized NeuralNetworksStatement: Consider an over-parameterized neural network (i.e., a network with more parametersthan 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, gradientdescent 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 thissetting.Ensure that the initialization is within the basin of attraction for convergence to a globalminimum.

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Answered: Problem 2: Convergence of Gradient…

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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In linear regression, the most popular and widely used method to estimate the parameters of an approximate linear relationship between a response variable y and a p-dimensional vector of predictors x (1,..., p) is by minimizing the residual sum of squares. This minimum is obtained by the method of least squares and it always exists but may not be unique doesn't always exist always exists and is unique
Q3 (4 pts): Determine whether the Mean Value Theorem applies on the given interval. If so, find the point(s) that are guaranteed to exist by the Mean Value Theorem. f(x) = 3x² + 2x+5 [−1,1]
(i) Find the new weights using gradient descent for the data presented in Table 6 for a linear regression classifier Table 6 1 3 4 y 3 4 Initial weights: slope = 0.92 and intercept = 2.4, learning rate=D0.02. (ii) State the condition for convergence of gradient descent algorithm
2 Softmax classifier gradient. For softmax classifier, derive the gradient of the log likelihood. Concretely, assume a classification problem with c classes • Samples are (x(1), y(1)),..., (x(m), y(m)), where x(1) Rn, y) = {1,...,c}, j = 1,...,m • Parameters are 0 = {w₁b₁}=1,...c . Probablistic model is where Pr r(yG) = i | xG), 0) = softmax; (x(j)) softmax, (x) = Σk=1 Derive the log-likelihood £, and its gradient w.r.t. the parameters, VwL and V₁L, for i = 1,..., c. W Note: We can group w, and b, into a single vector by augmenting the data vectors with an additional dimension of constant 1. Let x wx. = 1₁ [M]. then a, (x) = w/x+b₁ = ' ew/x+b; ew/x+bk This unifies VL and V, L into VL. W₂ W₁
find the critical point(s) and determine if it corresponds to a local max, local min, or saddle point
Linear regression aims to fit the parameters based on the training set D = {(x(i),y(i)), i = 1,2,..., m} so that the hypothesis function ho (x) 6T.x = 00+ 01x₁ + 0₂x₂+......+ Onxn can better predict the output y of a new input vector x. Please derive the stochastic gradient descent update rule which can update o repeatedly to minimize the least squares cost function (). =
Consider 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 regressor that predicts ƒ (x(i)) : = 0 + 0₁x¹). The parameters and ₁ are scalars and are learned by minimiz- ing mean-squared-error loss through gradient descent with a learning rate a. Answer the following questions. 1. Express the loss function(L) in terms of x(i),y(i), n, 00, 0₁. 2. Compute L მძი ƏL 3. Compute 20₁ 4. Write update rules for 0 and 0₁
We use the residual deviance to test: a) the model with common p (null hypothesis) against the logistic regression model (alternative hypothesis) b) the model with common p (null hypothesis) against the maximal model (alternative hypothesis) c) the logistic regression model (null hypothesis) against the maximal model (alternative hypothesis)
Write out the design matrix 1. Polynomial regression : 2. Data transformation: X and find its rank for the following Y₁ = Bot fixi + B₂X²³² + ... + B₂-, X; log7; = B₁ + B₁ (-log D₁) + E; P-1 > +E; " i=1.- `n models. n>p. i=1...n
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2. An experiment was conducted to explore the effects of fatigue on the performance of a simple manual task. A random sample of 15 male, college students was obtained. We will assume that this sample forms a random sample from all of the male, college students that were available to the investigators. These students were trained to make a continuous horizontal right-to-left arm movement from a microswitch to a barrier, so that the barrier was knocked over at exactly the same time as the clock sweep second hand arrived at the 6 o'clock position. The absolute value of the difference between the time, in milliseconds, that it took to knock over the barrier and the time for the sweep second hand to reach the 6 o'clock position was recorded. Note that the student's goal is to make this time difference as close to zero as possible. Each participant performed the task five times under rested conditions (the student was not fatigued) and fatigued conditions (the student was fatigued). The sums…
(only if possible, preferably in minitab, but you can do it manually)

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