Reduce overfitting and explain the mean squared error loss

A machine learning phone screen asks several short conceptual questions in quick succession. Two of them are reconstructed below: how to reduce overfitting, and the mean squared error loss. Answer each as you would aloud, concisely, but give the reasoning behind every claim rather than a list of terms. Unless told otherwise, assume supervised learning with a held-out validation set. ### Clarifying Questions - What kind of model is in play (a linear model, gradient-boosted trees, a deep network)? The most effective overfitting remedies differ between them. - How much labeled data is there relative to the number of features or parameters, and can more be collected? - For the loss question, is the target continuous, and are there outliers or heavy-tailed errors? ### Part 1 — Reducing overfitting A model scores much better on its training data than on held-out data. How do you confirm that the gap is overfitting, and what would you do to reduce it? For each remedy, explain why it works and what it costs. ```hint Diagnose before fixing Decide what evidence shows that the gap comes from the model fitting noise, rather than from leakage or from a difference between the training and evaluation data. ``` #### What This Part Should Cover - Diagnosis from training and validation curves, and ruling out leakage or distribution shift - Remedies grouped by mechanism, with the reason each one reduces variance - The cost of each remedy, and how its strength is tuned on held-out data ### Part 2 — The mean squared error loss Define the mean squared error (MSE) loss for a regression model. What does minimizing it make the model predict, why might you choose it, and when is it a poor choice? Compare it with at least one alternative loss. ```hint The best constant Ask which single number minimizes the average squared error over a set of targets, and what happens to that number when one target is extreme. ``` #### What This Part Should Cover - The formula and its gradient with respect to a prediction - What MSE minimization estimates, and the noise assumption under which it is maximum likelihood - Sensitivity to outliers, and the losses used instead when that matters ### What a Strong Answer Covers - Concise answers that still give a mechanism for each claim, such as why a remedy lowers variance or why squaring changes how errors are weighted - Advice matched to the setting raised in the clarifying questions: the model family, the amount of labeled data, and outliers in the target - Where each claim stops holding: when a remedy only adds bias, and which noise assumptions make MSE the principled choice - How a regularization penalty combines with the squared-error objective, linking the two parts ### Follow-up Questions - Add an L2 penalty to linear regression trained with MSE. What is the closed-form solution, and what does the penalty do to it? - Decompose the expected squared error on a new point into bias, variance and noise. Which term does each Part 1 remedy target? - Why is MSE on a sigmoid output usually avoided for binary classification? - Your validation set has only a few hundred examples. How does that change the way you detect and tune against overfitting?