Start by clarifying the requirements: whether you need to implement the forward pass from scratch (e.g., using loops or tensor operations) or simply call PyTorch's built-in matmul. Then, outline a vectorized implementation using torch.matmul or the @ operator, and discuss how to handle edge cases like non-contiguous tensors or different dtypes. Finally, mention performance considerations such as GPU acceleration and autograd compatibility.
Pro tip: Emphasize that in a production setting, you would rely on PyTorch's optimized kernels (e.g., cuBLAS) rather than manual loops, but demonstrate understanding of the underlying computation by explaining the naive triple-loop approach and its inefficiencies.
Ask whether the implementation should be from scratch (e.g., using loops) or can use PyTorch's built-in functions. Also clarify if the function needs to support autograd, GPU, or batch dimensions.
Decide between a naive loop-based approach (for educational purposes) and a vectorized approach using torch.matmul or @. Explain the trade-offs in terms of performance and code simplicity.
Write the code: for the vectorized version, simply return A @ B or torch.matmul(A, B). For the naive version, use nested loops to compute each element of C. Ensure the function handles input validation (e.g., shape compatibility).
Mention that PyTorch's built-in matmul leverages optimized BLAS libraries and supports GPU acceleration. Address edge cases like non-contiguous inputs, different dtypes, and broadcasting if applicable.
Suggest testing with small random matrices against torch.matmul to ensure correctness. Also consider testing with non-square matrices and different devices.
AI-generated suggestions, not part of the candidate's original notes. May be inaccurate — verify before relying on them.
I knew the formulas going in: dA is dC @ B.T and dB is A.T @ dC.
First, clarify the shapes and the forward operation (e.g., A (M,K) times B (K,N) yields C (M,N)). Then derive the gradients using the chain rule: dA = dC @ B^T and dB = A^T @ dC, ensuring the shapes match. Finally, discuss implementation details like handling batch dimensions and memory efficiency.
Pro tip: Mention that you would verify the gradient shapes and consider using a small numerical gradient check to ensure correctness, especially in an interview setting.
Confirm the dimensions of A, B, and C (e.g., A: (M,K), B: (K,N), C: (M,N)) and the operation C = A @ B. This sets the stage for the backward pass.
For scalar loss L, dA = dC @ B^T and dB = A^T @ dC. Explain the derivation briefly, noting that each gradient sums over the appropriate dimension.
Check that dA has shape (M,K) and dB has shape (K,N). If there are batch dimensions, use batched matrix multiplication (e.g., torch.bmm or einsum).
Mention that you can compute dA and dB directly without materializing large intermediate tensors. For efficiency, consider using in-place operations or avoiding unnecessary transposes.
Suggest verifying the gradients using finite differences on a small example to ensure correctness, especially if the implementation is non-trivial.
AI-generated suggestions, not part of the candidate's original notes. May be inaccurate — verify before relying on them.
Start by explaining the sequential nature of the backward pass and how it creates a dependency chain that limits parallelism. Then, describe how you would restructure it using a parallel prefix scan (Hillis-Steele) to compute cumulative gradients in logarithmic time, reducing sequential dependencies. Finally, discuss the trade-offs in terms of increased work and memory usage, and how you would optimize for modern hardware.
Pro tip: Emphasize that while parallel prefix scan reduces depth, it increases total operations; therefore, a hybrid approach (e.g., blocked scan) often works best in practice. Also, mention that this technique is particularly beneficial for very deep networks or long sequences where sequential backprop becomes a bottleneck.
Analyze the backward pass to pinpoint the sequential chain of gradient computations, such as the cumulative sum of gradients in RNNs or residual networks.
Reformulate the backward pass as a prefix scan operation, where each step combines gradients using an associative operator, enabling parallel computation.
Implement the Hillis-Steele scan to compute the prefix sums in O(log n) depth, using parallel steps that double the stride each iteration.
Discuss the increased work (O(n log n) vs O(n)) and memory overhead, and propose optimizations like blocked scans or work-efficient algorithms (Blelloch) for better efficiency.
Consider hardware characteristics (e.g., GPU parallelism) and network architecture to determine when this approach yields speedups, and mention potential integration with existing frameworks.
AI-generated suggestions, not part of the candidate's original notes. May be inaccurate — verify before relying on them.