Reason About Train and Test Loss Under Regularization and More Data

Explain the bias–variance tradeoff using training and test loss. Address these cases:

1. Training error is much lower than test error.
2. Both training and test loss are large, including for OLS.
3. Data are fixed while regularization strength changes.
4. The model class is fixed while the amount of training data changes.

Constraints & Assumptions

Distinguish expected learning-curve tendencies from guarantees for one finite dataset. Assume loss definitions and data distributions are comparable unless diagnosing a mismatch. When discussing regularization, separate prediction loss from the penalized optimization objective.

Clarifying Questions

How large are the losses relative to a baseline and measurement noise? Are train and test drawn from the same distribution? Is optimization converged? Does “fixed model” mean a fixed model class that is retrained, or literally fixed fitted parameters?