Compute Matrix Prefix Products And Gradients
Company: OpenAI
Role: Machine Learning Engineer
Category: Machine Learning
Difficulty: hard
Interview Round: Onsite
You are given N square matrices A[0], A[1], ..., A[N-1], each of shape D by D. Define the inclusive prefix products:
Y[i] = A[0] @ A[1] @ ... @ A[i]
where @ denotes matrix multiplication.
Answer the following:
1. Implement a simple in-place forward computation of all prefix products and analyze its time and space complexity.
2. Explain why an in-place implementation is problematic for gradient computation in an automatic differentiation system.
3. Implement an out-of-place forward pass that preserves the values needed for backpropagation.
4. Given upstream gradients G[i] = dL/dY[i], derive and implement the backward pass that computes dL/dA[i].
5. Suppose you are given a generic Hillis-Steele inclusive scan function for an associative binary operation. Use it to compute the forward prefix products. Then explain how the backward pass can also be computed with scan-style operations.
Quick Answer: This question evaluates proficiency in matrix calculus, automatic differentiation, and algorithmic complexity, focusing on inclusive prefix matrix products, gradient propagation through sequences of matrix multiplications, and the use of scan-style parallel operations.