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Two Sigma·Software Engineer·Technical Phone Screen·Senior

Senior
Apr 2026

Summary

Two Sigma Research Engineer interview with a coding round that leaned heavily on applied math and stats rather than pure algorithms. The linear regression question had two parts and the second one tripped me up more than I expected.

Questions Asked (1)

Q1

Implement linear regression through the origin (no intercept) from scratch. First, solve it in the standard batch setting where you have the full dataset available. Then extend it to a streaming setting where data arrives one point at a time and you need to update your estimate incrementally without reprocessing everything.

Algorithms & Data StructuresTechnical Trade-offs
Author's notes

The batch part was fine, just derive the closed-form slope estimator and compute it directly.

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AI HintsAI Generated

Suggested Approach

Start by deriving the closed-form solution for linear regression through the origin, then explain how to compute it incrementally using sufficient statistics. Emphasize the equivalence between batch and streaming updates, and discuss practical considerations like numerical stability and memory efficiency.

Pro tip: Mention that the streaming update is exact and equivalent to batch, and highlight the importance of using Welford's algorithm or similar for numerical stability when updating sums of squares.

1. Derive the batch solution

For the model y = wx, minimize the sum of squared errors. The optimal w is (Σ x_i y_i) / (Σ x_i^2).

2. Identify sufficient statistics

Recognize that only two quantities are needed: S_xy = Σ x_i y_i and S_xx = Σ x_i^2. These can be updated incrementally.

3. Design the streaming update

When a new point (x, y) arrives, update S_xy += x*y and S_xx += x^2. Then recompute w = S_xy / S_xx.

4. Address numerical stability

For large datasets, naive summation can cause overflow or loss of precision. Use compensated summation (Kahan) or Welford's algorithm for S_xx.

5. Discuss trade-offs and extensions

Compare batch vs streaming in terms of memory, computation, and adaptability. Mention that streaming allows real-time updates but may be sensitive to outliers.

Key Points to Mention

  • Closed-form solution: w = Σ(x_i y_i) / Σ(x_i^2)
  • Sufficient statistics: S_xy and S_xx
  • Incremental update: S_xy += x*y, S_xx += x^2
  • Numerical stability techniques (e.g., Welford's algorithm)
  • Memory efficiency: O(1) space for streaming
  • Exact equivalence between batch and streaming results

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