Had seen this one floating around online so the forward pass felt manageable.
Start by clearly defining the forward pass for matrix multiplication, then derive the gradients for backpropagation using the chain rule. Implement both using PyTorch tensors and autograd, ensuring the backward pass correctly computes gradients for both inputs.
Pro tip: Emphasize that matrix multiplication is a linear operation, so its gradient is straightforward but must account for the transpose of the other matrix. Also, mention that using PyTorch's autograd can validate your manual implementation.
Implement the forward pass as C = A @ B, where A and B are input matrices. Ensure the dimensions are compatible for matrix multiplication.
Using the chain rule, derive the gradients: dA = dC @ B.T and dB = A.T @ dC, where dC is the gradient of the loss with respect to C.
Implement the backward function that takes dC and returns dA and dB using the derived formulas. Ensure the shapes match the original inputs.
Compare your manual gradients with PyTorch's autograd by creating tensors with requires_grad=True and checking that the gradients match.
Consider batched inputs, non-square matrices, and broadcasting. Ensure your implementation generalizes correctly.
AI-generated suggestions, not part of the candidate's original notes. May be inaccurate — verify before relying on them.
This one I had not seen anywhere and it showed.
Start by explaining the motivation for using parallel prefix scan in backpropagation, such as parallelizing sequential dependencies in RNNs or deep networks. Then outline how the Hillis-Steele algorithm works and how it can be adapted to compute gradients efficiently. Finally, discuss implementation details, trade-offs, and potential optimizations.
Pro tip: Emphasize that while parallel prefix scan can reduce time complexity from O(n) to O(log n) for certain operations, it increases work and memory usage; showing awareness of these trade-offs demonstrates depth. Also, mention that this approach is particularly beneficial for long sequences where parallelism outweighs overhead.
Analyze the backpropagation algorithm to find operations that are inherently sequential, such as gradient accumulation over time steps in RNNs or residual connections.
Express the sequential computation as a prefix scan (inclusive or exclusive) with an appropriate associative operator, such as matrix multiplication or addition.
Implement the parallel prefix scan using the Hillis-Steele approach, which iteratively doubles the stride to compute prefix results in O(log n) steps.
Replace the sequential loop with the parallel scan, ensuring that gradients are correctly computed and accumulated, and handle any necessary adjustments for reverse-mode differentiation.
Discuss the time and space complexity, parallelization overhead, and suitability for different hardware (e.g., GPUs). Mention potential optimizations like using shared memory or warp-level primitives.
AI-generated suggestions, not part of the candidate's original notes. May be inaccurate — verify before relying on them.