Databricks·Data Scientist·Technical Phone Screen·Senior

Senior

Jun 2026

Summary

Interviewed for a Data Scientist role at Databricks and got hit with a pretty theoretical linear regression question that had more depth than I expected. The whole thing felt like a math exam disguised as a conversation.

Questions Asked (2)

You have two regressors x1 and x2. Model A is linear in x1 and x2 directly. Model B is linear in the transformed features (x1 + x2) and (x1 - x2). Are these two models equivalent? Walk through the math.

Technical Trade-offsData ModelingAlgorithms & Data Structures

Author's notes

I knew instinctively they were equivalent but fumbled the explanation for a moment.

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AI HintsAI Generated

Suggested Approach

Start by writing out the two models mathematically, then determine if one can be rewritten as the other through a linear transformation of the features. Show that the models are equivalent in terms of the function space they span, but the coefficients differ by a linear transformation. Conclude that they are equivalent for prediction but not for interpretation of individual coefficients.

Pro tip: Emphasize that while the models are mathematically equivalent in terms of predictions, the coefficients have different interpretations, which matters for explainability and feature importance. This shows you understand both the math and its practical implications.

1. Write the models explicitly

Model A: y = β0 + β1 x1 + β2 x2. Model B: y = γ0 + γ1 (x1 + x2) + γ2 (x1 - x2). Clearly define the coefficients.

2. Expand Model B

Expand Model B: y = γ0 + γ1 x1 + γ1 x2 + γ2 x1 - γ2 x2 = γ0 + (γ1 + γ2) x1 + (γ1 - γ2) x2.

3. Compare coefficients

Set the expanded Model B equal to Model A: β0 = γ0, β1 = γ1 + γ2, β2 = γ1 - γ2. This shows a one-to-one linear mapping between the coefficient sets.

4. Solve for the transformation

Solve for γ in terms of β: γ1 = (β1 + β2)/2, γ2 = (β1 - β2)/2. This confirms that any Model A can be represented as Model B and vice versa.

5. Conclude on equivalence

Conclude that the models are equivalent in terms of the function space they represent (same predictions for any data), but the coefficients have different interpretations. Mention that this holds for linear regression, but may not hold for regularized models or other algorithms.

Key Points to Mention

Linear transformation of features preserves the model's expressiveness in linear regression.
The models span the same column space, so they produce identical predictions.
Coefficient interpretations differ: β1 and β2 represent the effect of x1 and x2 individually, while γ1 and γ2 represent the effect of the sum and difference.
Equivalence holds for ordinary least squares, but regularization (e.g., Lasso, Ridge) can break equivalence because penalties are not invariant under linear transformations.
This concept relates to reparameterization and identifiability in linear models.
Practical implication: feature engineering like sum/difference can help with multicollinearity or interpretability without changing model fit.

AI-generated suggestions, not part of the candidate's original notes. May be inaccurate — verify before relying on them.

If you have more than 1,000 predictors and want to fit a linear model, what issues could come up and how would you handle them?

Technical Trade-offsData Modeling

Author's notes

This part I actually felt okay about.

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AI HintsAI Generated

Suggested Approach

Start by acknowledging the core statistical and computational challenges of high-dimensional data, such as multicollinearity, overfitting, and computational inefficiency. Then, structure your answer around a systematic workflow: diagnose the issues, apply appropriate regularization or dimensionality reduction techniques, and validate the model rigorously. Emphasize practical trade-offs and the importance of aligning the solution with business goals and interpretability needs.

Pro tip: Mention that with 1,000+ predictors, you would first check for data leakage and ensure proper cross-validation to avoid optimistic performance estimates. Also, highlight that in a Databricks environment, you can leverage distributed computing (e.g., Spark ML) to scale regularization techniques efficiently.

1. Identify Potential Issues

Discuss problems like multicollinearity, overfitting, high variance, computational cost, and interpretability challenges that arise with many predictors.

2. Diagnose the Data

Explain how to check for multicollinearity (e.g., VIF, correlation matrix), assess feature relevance, and evaluate the risk of overfitting using learning curves or cross-validation.

3. Apply Dimensionality Reduction or Regularization

Describe techniques such as PCA, feature selection (LASSO, recursive feature elimination), or regularization (Ridge, LASSO, Elastic Net) to mitigate issues while preserving predictive power.

4. Validate and Interpret

Emphasize the need for robust validation (e.g., k-fold cross-validation) and discuss how to interpret the final model, especially if using regularization that shrinks coefficients.

5. Consider Scalability and Tools

Mention scalable implementations (e.g., Spark MLlib on Databricks) and the trade-offs between distributed computing and single-node solutions for large-scale linear models.

Key Points to Mention

Multicollinearity and its impact on coefficient estimates and model stability.
Overfitting risk and the need for regularization (Ridge, LASSO, Elastic Net).
Dimensionality reduction techniques like PCA or feature selection methods.
Computational challenges and scalable solutions (e.g., Spark MLlib, distributed linear algebra).
Interpretability trade-offs: simpler models vs. black-box models, and the importance of business context.
Validation strategies: cross-validation, hold-out sets, and monitoring for data leakage.

AI-generated suggestions, not part of the candidate's original notes. May be inaccurate — verify before relying on them.