Question at position 11 Which of the following is True for Linear Regression with Regularization?Using L2 norm is LASSO RegressionUsing L1 norm is Ridge RegressionHigher level of penalty reduces model complexityThe loss function for Linear Regression with Regularization is always the same as the loss function for Linear Regression without Regularization

Question

BlackTom AI · Accepted Answer

Regularization adds a penalty term to the ordinary least squares objective; increasing the penalty parameter λ shrinks model coefficients and therefore reduces model complexity.

1) Relevant knowledge points and concepts
- Ordinary least squares (OLS) minimizes J = (1/2n) ||y − Xw||^2.
- Ridge regression uses the L2 norm penalty: J_ridge = (1/2n) ||y − Xw||^2 + (λ/2) ||w||_2^2.
- LASSO uses the L1 norm penalty: J_lasso = (1/2n) ||y − Xw||^2 + λ ||w||_1.
- L2 (Ridge) tends to shrink coefficients continuously; L1 (LASSO) can shrink some coefficients exactly to zero (sparsity).
- Larger λ increases regularization strength, discouraging large weights and reducing model complexity (variance).

2) Step-by-step reasoning
- Start from the OLS loss: J = (1/2n) (y − Xw)^T (y − Xw).
- Add L2 penalty for Ridge: J_ridge = (1/2n) (y − Xw)^T (y − Xw) + (λ/2) w^T w.
- Take derivative wrt w and set to zero:
   ∂J_ridge/∂w = −(1/n) X^T(y − Xw) + λ w = 0.
- Rearranged: (X^T X + nλ I) w = X^T y. (If λ defined per-sample, constants vary; common form uses (X^T X + λI)w = X^T y.)
- Solve: w = (X^T X + nλ I)^{-1} X^T y. The λI term increases diagonal eigenvalues of X^T X, leading to smaller (shrunk) weight magnitudes. As λ → ∞, w → 0 (maximal shrinkage), demonstrating reduction of model complexity.

3) Intermediate calculation detail (Ridge closed-form)
- OLS normal eqn: X^T X w = X^T y.
- With Ridge: X^T X w + nλ w = X^T y → (X^T X + nλ I) w = X^T y → w = (X^T X + nλ I)^{-1} X^T y.

4) Why the correct answer is correct
- "Higher level of penalty reduces model complexity" is true because increasing λ penalizes large coefficients more strongly, producing smaller parameter magnitudes, less variance, simpler decision functions, and in LASSO potentially sparser models.

5) Why the other options are incorrect
- "Using L2 norm is LASSO Regression" is incorrect: LASSO uses L1 norm, not L2.
- "Using L1 norm is Ridge Regression" is incorrect: Ridge uses L2 norm, not L1.
- "The loss function for Linear Regression with Regularization is always the same as the loss function for Linear Regression without Regularization" is incorrect: regularized loss includes an extra penalty term (λ ||w||_1 or λ ||w||_2^2), so it is not the same as the unregularized loss.

6) Conclusion
Therefore the correct statement is "Higher level of penalty reduces model complexity", which matches the provided answer.

类似问题

Which of the following statements is or are true regarding regularisation?

A project has a required return of 12.6 percent, an initial cash outflow of $42,100, and cash inflows of $16,500 in Year 1, $11,700 in Year 2, and $10,400 in Year 4. What is the net present value?

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