ML Take‑home: Bias–Variance, Regularization, Leakage, and From‑scratch Logistic Regression

Context

You are given user event logs in a Pandas dataframe df with columns:

• user_id: unique user identifier
• event_time: timestamp of the event
• event_type: categorical event name (e.g., view, click, add_to_cart, purchase)
• purchase: indicator (0/1) if the event is a purchase

Your goal is to build a leakage‑free binary classifier that predicts whether a user will purchase within the next 7 days, then evaluate AUC on a held‑out set.

Tasks

1. Bias–variance decomposition
   • State the bias and variance terms and interpret them in the bias–variance decomposition.
2. Regularization and sparsity
   • Which regularization technique(s) can shrink linear‑model coefficients exactly to zero, and why?
3. Detecting data leakage
   • Name two practical approaches for detecting data leakage in a supervised learning pipeline.
4. Modeling: Logistic regression from scratch
   • Using df(user_id, event_time, event_type, purchase), build a binary classifier to predict whether a user will purchase within the next 7 days.
   • Use a temporally correct split and report AUC on a held‑out set.
   • Implement logistic regression with gradient descent using only numpy for the model (pandas allowed for data prep). Provide basic convergence diagnostics.

Implementation requirements

• Ensure no temporal leakage: features must use data up to an anchor time; labels look forward 7 days after the anchor.
• Clean, vectorized Python; no sklearn for the model or metrics (implement AUC yourself).