261 views
0 votes
0 votes
Suppose we have a regularized linear regression model: \[ \text{argmin}_{\mathbf{w}} \left||\mathbf{Y} - \mathbf{Xw} \right||^2 + k \|\mathbf{w}\|_p^p. \] What is the effect of increasing \( p \) on bias and variance (for \( p \geq 1 \)) if the weights are all larger than $1$?

(a)] Increases bias, increases variance

(b)] Increases bias, decreases variance

(c)] Decreases bias, increases variance

(d)] Decreases bias, decreases variance

(e)] Not enough information to tell

1 Answer

0 votes
0 votes

Regularization and Bias-Variance Trade-off:

Regularization aims to control the complexity of a model to reduce overfitting and improve generalization. Increasing regularization generally increases bias (underfitting) but decreases variance (overfitting).

Effect of Increasing \( p \):

  • Penalty on Larger Weights:The \( \|\mathbf{w}\|_p^p \) term is a regularization term that penalizes large weights. As \( p \) increases, the penalty on weights larger than $1$ becomes more severe. This leads to smaller weights overall, making the model simpler.
  • Impact on Bias: Simpler models tend to have higher bias because they might not fully capture the underlying patterns in the data.
  • Impact on Variance: Simpler models also tend to have lower variance because they are less sensitive to noise in the training data.

Key Point: The fact that weights are initially larger than $1$ ensures that increasing \( p \) does indeed lead to smaller weights.

Therefore, increasing \( p \) in this model increases bias but decreases variance.

Related questions

178
views
1 answers
0 votes
rajveer43 asked Jan 13
178 views
Suppose we have a regularized linear regression model: \[ \text{argmin}_{\mathbf{w}} \left||\mathbf{Y} - \mathbf{Xw} \right||^2 + \lambda \ ... , increases variance(d)] Decreases bias, decreases variance(e)] Not enough information to tell
359
views
1 answers
0 votes
rajveer43 asked Jan 13
359 views
After applying a regularization penalty in linear regression, you find that some of the coefficients of $w$ are zeroed out. Which of the following penalties might have been used?(a) ... (c) L2 norm(d) either (A) or (B)(e) any of the above
584
views
1 answers
0 votes
rajveer43 asked Jan 14
584 views
Suppose you have a three-class problem where class label \( y \in \{0, 1, 2\} \), and each training example \( \mathbf{X} \) has 3 binary attributes \( X_1, ... an example using the Naive Bayes classifier?(a) 5b) 9(c) 11(d) 13(e) 23
179
views
0 answers
0 votes
rajveer43 asked Jan 13
179 views
Using the same data as above \( \mathbf{X} = [-3, 5, 4] \) and \( \mathbf{Y} = [-10, 20, 20] \), assuming a ridge penalty \( \lambda = 50 \), what ratio versus the MLE ... \mathbf{w}}_{\text{ridge}} \) will be?(a)] 2b)] 1(c)] 0.666(d)] 0.5