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Two Sigma·Data Scientist·Technical Phone Screen·Intermediate

Intermediate
May 2026

Summary

Two Sigma data scientist interview with a coding question that looked straightforward on the surface. One round, math-heavy, no ML libraries allowed which tripped me up at first since I kept reaching for numpy instinctively.

Questions Asked (1)

Q1

Implement simple linear regression from scratch using the ordinary least squares closed-form solution. No ML or stats libraries. Given a list of x and y values, return the fitted slope and intercept.

Algorithms & Data StructuresTechnical Trade-offs
Author's notes

My first instinct was to just call numpy's polyfit and I had to stop myself.

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AI HintsAI Generated

Suggested Approach

Derive the closed-form OLS formulas for slope and intercept, then implement them in pure Python using basic arithmetic. Emphasize numerical stability by using the computational form of the slope formula and discuss edge cases like zero variance in x.

Pro tip: Mention that you would use the computational formula for slope to avoid catastrophic cancellation, and explicitly handle the case where the denominator is zero (e.g., all x values identical) by raising an error or returning None.

1. Clarify requirements and edge cases

Confirm input format (lists of numbers), expected output (slope and intercept), and how to handle edge cases like empty lists, mismatched lengths, or zero variance in x.

2. Derive the closed-form solution

Recall that OLS minimizes sum of squared residuals, leading to slope = covariance(x,y)/variance(x) and intercept = mean(y) - slope*mean(x).

3. Implement using stable formulas

Compute means, then use the computational formula: slope = (n*sum(xy) - sum(x)*sum(y)) / (n*sum(x^2) - sum(x)^2) to reduce floating-point error.

4. Handle edge cases and validate

Check for zero denominator (e.g., all x equal) and raise ValueError; ensure inputs are non-empty and same length. Optionally, test with simple data.

5. Discuss trade-offs and extensions

Mention that this closed-form is efficient for small datasets but can be numerically unstable for large or ill-conditioned data, where gradient descent or QR decomposition might be preferred.

Key Points to Mention

  • Closed-form OLS solution: slope = Σ((x_i - x̄)(y_i - ȳ)) / Σ((x_i - x̄)^2), intercept = ȳ - slope * x̄
  • Computational formula for slope to avoid catastrophic cancellation: slope = (nΣxy - ΣxΣy) / (nΣx² - (Σx)²)
  • Edge case: zero variance in x (all x equal) leads to division by zero; must handle explicitly
  • Time complexity O(n) and space complexity O(1) beyond input storage
  • Numerical stability considerations: centering data or using alternative methods for large datasets
  • Assumptions of linear regression: linearity, independence, homoscedasticity, normality of residuals (mention briefly)

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