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Openai·AI Engineer·Technical Phone Screen·Senior

SeniorPrefer not to say
May 2026Remote

Summary

OpenAI AI Engineer technical screen, one round focused entirely on numerical computing for ML. The problem was deceptively hard once you got past the basic softmax definition, and the streaming extension really separated people who actually understand floating point from people who just memorized formulas.

Questions Asked (2)

Q1

Given a vector of logits, compute the entropy of the softmax distribution in a numerically stable way, avoiding overflow in the exponentiation and underflow in the log.

Algorithms & Data StructuresTechnical Trade-offs
Author's notes

I knew the log-sum-exp trick going in, so the first part felt okay.

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AI HintsAI Generated

Suggested Approach

First, explain the numerical stability issues with naive softmax entropy computation: overflow when exponentiating large logits and underflow when taking log of small probabilities. Then, present the log-sum-exp trick to compute log probabilities directly from logits, and derive the entropy formula using these log probabilities. Finally, discuss implementation details and potential pitfalls.

Pro tip: Mention that in practice, frameworks like PyTorch and TensorFlow have built-in numerically stable functions (e.g., log_softmax) that you should use, but understanding the underlying math is crucial for debugging and custom implementations.

1. Identify numerical issues

Explain that directly exponentiating logits can overflow (for large values) and taking log of softmax probabilities can underflow (for small probabilities).

2. Introduce log-sum-exp trick

Describe how subtracting the maximum logit before exponentiation prevents overflow, and how log-sum-exp computes the log of the sum of exponentials stably.

3. Derive stable entropy formula

Show that entropy H = -sum(p_i * log(p_i)) can be computed as H = logsumexp(logits) - sum(softmax(logits) * logits), using log probabilities.

4. Implement and validate

Outline code steps: compute max logit, shifted logits, logsumexp, log probabilities, then entropy. Suggest testing with extreme values to ensure stability.

Key Points to Mention

  • Log-sum-exp trick: logsumexp(x) = max(x) + log(sum(exp(x - max(x))))
  • Softmax probabilities: p_i = exp(x_i - logsumexp(x))
  • Entropy formula: H = -sum(p_i * log(p_i)) = logsumexp(x) - sum(p_i * x_i)
  • Avoid underflow by working in log space: log(p_i) = x_i - logsumexp(x)
  • Use built-in functions like log_softmax for stability in practice
  • Edge cases: uniform distribution (max entropy), one-hot (min entropy), and large logits

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

Q2

Extend the entropy computation to a streaming setting where logits arrive one at a time. Maintain the entropy incrementally without reprocessing previous logits, and handle the case where a new logit exceeds the current running maximum.

Algorithms & Data StructuresTechnical Trade-offsSystem Design
Author's notes

This is where I got properly humbled.

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AI HintsAI Generated

Suggested Approach

Explain that entropy can be maintained incrementally by tracking the running maximum logit and the sum of exponentials of shifted logits. When a new logit arrives, update the sum and adjust for a new maximum if necessary, then compute entropy using the current sum and maximum. Emphasize numerical stability and the need to avoid reprocessing previous logits.

Pro tip: Highlight that the shift by the maximum is crucial for numerical stability, and that when the maximum changes, the sum must be rescaled by exp(old_max - new_max). This shows you understand both the math and practical implementation concerns.

1. Define the incremental state

Maintain the current maximum logit (m) and the sum of exponentials of shifted logits (S = sum(exp(x_i - m))). Also track the count of logits (n) to compute the average.

2. Update with a new logit

When a new logit x arrives, if x > m, rescale S by exp(m - x) and set m = x; then add exp(x - m) to S. If x <= m, simply add exp(x - m) to S. Increment n.

3. Compute entropy incrementally

The entropy (in nats) is log(S) + m - (sum of x_i * exp(x_i - m))/S. To compute this incrementally, also maintain the weighted sum T = sum(x_i * exp(x_i - m)), updating it similarly when a new logit arrives and rescaling when m changes.

4. Handle numerical stability and edge cases

Ensure that exp(m - x) does not underflow when m is much larger than x. Discuss the case of negative infinity or very large logits, and how to avoid overflow/underflow by using the shift.

5. Analyze complexity and trade-offs

The update is O(1) per logit, with O(1) memory. Discuss the trade-off between maintaining additional state (T) versus recomputing entropy from scratch, and the impact on precision.

Key Points to Mention

  • Numerical stability via shifting by the maximum logit
  • Rescaling the sum of exponentials when the maximum changes
  • Maintaining the weighted sum for entropy computation
  • O(1) time and space per update
  • Handling of edge cases like -inf logits or extreme values
  • Comparison with batch computation and potential precision issues

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