Linear Regression with Gradient Descent

In the context of gradient descent for simple linear regression, which of the following statements regarding the impact of the learning rate ($\alpha$) on coefficient updates is most accurate?

Consider a scenario where, during multiple linear regression using gradient descent, the gradients $\frac{\partial MSE}{\partial B}$ consistently point in nearly the same direction across multiple epochs. Which of the following optimization strategies would most effectively enhance convergence speed?

Suppose you are implementing multiple linear regression with gradient descent and observe that the Mean Squared Error (MSE) oscillates wildly between epochs. Which of the following is the most likely cause and its corresponding solution?

In the context of feature scaling for gradient descent in multiple linear regression, which statement best describes the potential consequences of omitting this step when features exhibit disparate scales?

When deploying gradient descent for a multiple linear regression model with a very large dataset, which of the following strategies would most effectively balance computational efficiency with convergence behavior?

Consider a scenario where you've implemented gradient descent for multiple linear regression, and the algorithm appears to converge to a suboptimal solution with a high residual error. Which of the following diagnostic steps would be most appropriate to investigate this issue?

In implementing gradient descent for multiple linear regression, what is the primary purpose of adding a bias (intercept) term to the feature matrix?

Assuming a multiple linear regression model, what adjustment to the gradient descent algorithm is required to incorporate L2 regularization (Ridge Regression)?

Consider an instance where the correlation between 'age' and 'experience' is high, and you're building a multiple linear regression model to predict 'income'. How does multicollinearity affect the interpretation and stability of the coefficients obtained through gradient descent?

Under what circumstances would normalizing the features (scaling them to a range between 0 and 1) be more appropriate than standardizing them (scaling to have a mean of 0 and a standard deviation of 1) before applying gradient descent in a multiple linear regression?

In the provided MultipleLinearRegression class, the fit method includes a tolerance parameter. What is the specific purpose of this parameter within the gradient descent algorithm?

In the sum_of_squared_errors method within the MultipleLinearRegression class, why is it important to calculate the sum of squared errors (SSE) instead of just the sum of errors?

In the provided code, the predict method in the MultipleLinearRegression class adds a bias column to the input feature matrix X if it's missing. Why is this step necessary for making accurate predictions?

The plot method in the MultipleLinearRegression class is designed to visualize the regression plane. What are the critical limitations of this method, and how do they restrict its applicability?

Examine the code snippet where the regression equation is dynamically constructed in the Evaluation of the model section. What is the purpose of this dynamic construction, and why is it preferred over a hardcoded equation?

When is it generally more appropriate to use Simple Linear Regression instead of Multiple Linear Regression?

In the update rule for the coefficients in Simple Linear Regression, $B_0 = B_0 - \alpha * \frac{\partial MSE}{\partial B_0}$ and $B_1 = B_1 - \alpha * \frac{\partial MSE}{\partial B_1}$, what does $\frac{\partial MSE}{\partial B_0}$ and $\frac{\partial MSE}{\partial B_1}$ represent?

In the context of gradient descent, what is the significance of initializing the coefficients ($\beta_0$ and $\beta_1$ in simple linear regression, or vector $B$ in multiple linear regression) to zero?

Suppose you implement a simple linear regression using gradient descent, but the error $E = y - \hat{y}$ consistently remains high, and the coefficients do not converge, even after a large number of epochs. What could be the most likely reason for this behavior?

In the provided formulas for updating coefficients in Simple Linear Regression, $B_0 = B_0 - \alpha * \frac{\partial MSE}{\partial B_0}$ and $B_1 = B_1 - \alpha * \frac{\partial MSE}{\partial B_1}$, explain the role and impact of the learning rate $\alpha$ on the training process.

Considering the formula for updating the coefficient vector B in Multiple Linear Regression, $B = B - \alpha * \frac{\partial MSE}{\partial B}$, where $\frac{\partial MSE}{\partial B} = \frac{-2}{n} * X^T E$, what does the term $X^T E$ represent, and why is it crucial in the gradient descent update?

Assume you have a multiple linear regression model, and after training with gradient descent, you observe that some coefficients are significantly larger than others. Which regularization technique can be applied, and how does it alter the gradient descent update rule to mitigate this issue?

How does the choice of batch size in mini-batch gradient descent affect the stability and speed of convergence in multiple linear regression?

How does the presence of outliers in the dataset affect the performance of linear regression models trained using gradient descent, and what strategies can be employed to mitigate these effects?

How can you diagnose whether your gradient descent implementation for linear regression contains bugs, and what are some common symptoms of such issues?

Suppose you are using scikit-learn's StandardScaler to preprocess your features before training a linear regression model with gradient descent. What are some critical considerations when applying the same scaler to both the training and testing datasets?

Considering the line of best fit equation, y = 0.0 -0.03 * x1 + 0.97 * x2, derived from the multiple linear regression model, how would you interpret the coefficients -0.03 and 0.97 in the context of the variables x1 (age) and x2 (experience) when predicting income?

How could you validate that the assumptions of linear regression (linearity, independence of errors, homoscedasticity, normality of errors) are reasonably met when using gradient descent to train your model?

What are the key distinctions between Batch Gradient Descent, Stochastic Gradient Descent (SGD), and Mini-Batch Gradient Descent, particularly concerning their convergence properties and computational costs?