Linear regression aims to fit the parameters based on the training set 6T X= D = {(x(i),y(i)), i = 1,2,..., m} so that the hypothesis function he (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 repeatedly to minimize the least squares cost function J(0). =
Linear regression aims to fit the parameters based on the training set 6T X= D = {(x(i),y(i)), i = 1,2,..., m} so that the hypothesis function he (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 repeatedly to minimize the least squares cost function J(0). =
MATLAB: An Introduction with Applications
6th Edition
ISBN:9781119256830
Author:Amos Gilat
Publisher:Amos Gilat
Chapter1: Starting With Matlab
Section: Chapter Questions
Problem 1P
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![**Understanding Linear Regression and Stochastic Gradient Descent**
Linear regression aims to fit the parameters \(\hat{\boldsymbol{\theta}}\) based on the training set:
\[ D = \{ (\vec{x}^{(i)}, y^{(i)}), i = 1, 2, \ldots, m \} \]
The goal is to ensure that the hypothesis function:
\[ h_{\theta}(\vec{x}) = \hat{\boldsymbol{\theta}}^T \cdot \vec{x} = \theta_0 + \theta_1 x_1 + \theta_2 x_2 + \ldots + \theta_n x_n \]
can accurately predict the output \( y \) for a new input vector \(\vec{x}\).
For effective parameter fitting, we derive the stochastic gradient descent update rule. This rule helps in updating \(\hat{\boldsymbol{\theta}}\) iteratively to minimize the least squares cost function \( J(\hat{\boldsymbol{\theta}}) \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ff9cf94d2-c54d-41a5-b095-0e143a04d70d%2Ff93b338c-a512-457d-9720-e2c657bbf150%2Fqn5xkf_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Understanding Linear Regression and Stochastic Gradient Descent**
Linear regression aims to fit the parameters \(\hat{\boldsymbol{\theta}}\) based on the training set:
\[ D = \{ (\vec{x}^{(i)}, y^{(i)}), i = 1, 2, \ldots, m \} \]
The goal is to ensure that the hypothesis function:
\[ h_{\theta}(\vec{x}) = \hat{\boldsymbol{\theta}}^T \cdot \vec{x} = \theta_0 + \theta_1 x_1 + \theta_2 x_2 + \ldots + \theta_n x_n \]
can accurately predict the output \( y \) for a new input vector \(\vec{x}\).
For effective parameter fitting, we derive the stochastic gradient descent update rule. This rule helps in updating \(\hat{\boldsymbol{\theta}}\) iteratively to minimize the least squares cost function \( J(\hat{\boldsymbol{\theta}}) \).
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