Assume that your hypothesis function for linear regression is of the form f(x) = w0 + w1x and that the current values of w0 and w1 are 1 and 2 respectively. Further assume that you are using a learning rate (alpha) of 0.001What is the new w0 value associated with the point (1, 12), after one gradient update?

Consider a function f(x)=x3−4x2+7𝑓(𝑥)=𝑥3−4𝑥2+7. What is the updated value of x𝑥 after 2nd iteration of the gradient descent update, if the learning rate is 0.10.1 and the initial value of x𝑥 is 5?

This problem involves updating the parameters of a multivariate linear regression model using gradient descent. The model is defined by the hypothesis function \( h_{\theta}(x) = \theta_0 + \theta_1 x_1 + \theta_2 x_2 \), and the cost function is the mean squared error. Given the initial parameters \( \theta = [\theta_0, \theta_1, \theta_2] = [0, 0.5, 1] \) and a learning rate \( \alpha = 0.8 \) for the first iteration and \( \alpha = 0.4 \) for the second iteration, we need to perform the updates for two iterations. The update rule for gradient descent is: \[ \theta_j := \theta_j - \alpha \frac{1}{m} \sum_{i=1}^{m} (h_{\theta}(x^{(i)}) - y^{(i)}) \cdot x_j^{(i)} \] where \( m \) is the number of training examples, \( x^{(i)} \) is the input features of the \( i \)-th training example, \( y^{(i)} \) is the actual output of the \( i \)-th training example, and \( x_j^{(i)} \) is the \( j \)-th feature of the \( i \)-th training example. Let's calculate the updates for each \( \theta_j \) for the first iteration: First, we need to compute the hypothesis for each instance: - For instance 1: \( h_{\theta}(x^{(1)}) = 0 + 0.5 \cdot (-1) + 1 \cdot 0.5 = 0 - 0.5 + 0.5 = 0 \) - For instance 2: \( h_{\theta}(x^{(2)}) = 0 + 0.5 \cdot (-0.5) + 1 \cdot 1 = 0 - 0.25 + 1 = 0.75 \) - For instance 3: \( h_{\theta}(x^{(3)}) = 0 + 0.5 \cdot 2 + 1 \cdot 0.5 = 0 + 1 + 0.5 = 1.5 \) Now, we calculate the gradient for each \( \theta_j \): - For \( \theta_0 \): \( \frac{1}{3} \sum_{i=1}^{3} (h_{\theta}(x^{(i)}) - y^{(i)}) \cdot x_0^{(i)} \), where \( x_0^{(i)} = 1 \) for all \( i \) (since \( x_0 \) is the bias term). - Gradient for \( \theta_0 \): \( \frac{1}{3} [(0 - 0) \cdot 1 + (0.75 - 1) \cdot 1 + (1.5 - 1) \cdot 1] = \frac{1}{3} [0 - 0.25 + 0.5] = \frac{1}{3} \cdot 0.25 = \frac{1}{12} \) - For \( \theta_1 \): - Gradient for \( \theta_1 \): \( \frac{1}{3} [(0 - 0) \cdot (-1) + (0.75 - 1) \cdot (-0.5) + (1.5 - 1) \cdot 2] = \frac{1}{3} [0 + 0.125 + 1] = \frac{1}{3} \cdot 1.125 = \frac{3.375}{12} \) - For \( \theta_2 \): - Gradient for \( \theta_2 \): \( \frac{1}{3} [(0 - 0) \cdot 0.5 + (0.75 - 1) \cdot 1 + (1.5 - 1) \cdot 0.5] = \frac{1}{3} [0 - 0.25 + 0.25] = 0 \) Now we update the parameters using the learning rate \( \alpha = 0.8 \): - \( \theta_0 := \theta_0 - 0.8 \cdot \frac{1}{12} = 0 - 0.8 \cdot \frac{1}{12} = 0 - \frac{1}{15} = -\frac{1}{15} \) - \( \theta_1 := \theta_1

Given a learning rate of 0.01 and a gradient of 0.05, what is the update step for the weights?

In a deep learning model, the learning rate is set to 0.001. If the gradient of the loss function is 0.2, what is the update value for the weights?

For our Gradient Descent algorithm, the cost function = Σ(Y−(mX+1))2Σ(𝑌−(𝑚𝑋+1))2  and our learning rate = 0.01.We are interested in approximating a value for the parameter m using three points. Y is the true y-coordinate of each point and X is the true x-coordinate.We initialize m with 0 and the new m is calculated as the old m - (0.083m - 124) * 0.01.(a) What is the first step size?