Start by clarifying the requirements and outlining the overall architecture, then implement the forward pass for scaled dot-product attention and the Transformer block, followed by the backward pass for attention using the provided softmax gradient. Emphasize modularity, numerical stability, and efficient tensor operations without relying on autograd.
Pro tip: Mention that you would cache intermediate values (e.g., softmax outputs, normalized inputs) during the forward pass to simplify and speed up the backward computation, and discuss the trade-off between memory and recomputation.
Confirm the input/output shapes, masking requirements, and the fact that autograd is not allowed. Sketch the overall data flow and identify the components to implement: attention, multi-head splitting, LayerNorm, feed-forward, and residual connections.
Compute queries, keys, and values, scale the dot products, apply causal masking (if needed), compute softmax, and produce the weighted sum. Handle multi-head splitting by reshaping and transposing tensors appropriately.
Using the provided softmax backward, derive gradients for queries, keys, and values. Propagate gradients through the scaling, masking, and multi-head concatenation steps, ensuring correct tensor shapes.
Implement LayerNorm (forward and backward), the feed-forward sublayer (two linear layers with activation), and residual connections. Integrate these with the attention module to form a complete Transformer block.
Write unit tests for each component, compare gradients with numerical approximations, and verify end-to-end functionality with a small example. Discuss potential optimizations and trade-offs.
AI-generated suggestions, not part of the candidate's original notes. May be inaccurate — verify before relying on them.
Felt more comfortable here than on the backward pass.
Start by formally defining the next-token cross-entropy loss for autoregressive training with teacher forcing, then explain the role of the causal mask in attention to prevent attending to future tokens, and finally show how perplexity is computed as the exponential of the average cross-entropy loss. Use clear notation and connect each component to the training objective.
Pro tip: Emphasize that the causal mask is applied to the attention scores before softmax, and mention that perplexity is a monotonic transformation of loss, so minimizing loss directly minimizes perplexity. Also, note that teacher forcing uses ground-truth tokens as inputs during training, which differs from inference.
Write the cross-entropy loss for a sequence of length T: L = - (1/T) * sum_{t=1}^{T} log p(x_t | x_{<t}), where p is the model's predicted probability for the true token at position t.
Clarify that during training, the model receives the ground-truth previous tokens as input (teacher forcing), so the conditional probabilities are computed in parallel for all positions.
Explain that in self-attention, a causal mask (upper triangular matrix of -inf) is added to the attention scores before softmax to ensure position t can only attend to positions ≤ t, preserving autoregressivity.
Show that perplexity = exp(L), where L is the average cross-entropy loss per token. If loss is reported as sum over tokens, divide by number of tokens first.
Summarize that minimizing cross-entropy loss with causal masking enables efficient parallel training while maintaining the autoregressive property, and perplexity provides an interpretable metric.
AI-generated suggestions, not part of the candidate's original notes. May be inaccurate — verify before relying on them.
They asked this as a follow-up and I think it was partly to see if I'd actually trust my own backward pass code.
Start by explaining the finite-difference gradient check method: perturb each parameter by a small epsilon, compute the loss, and compare the numerical gradient to the analytical gradient from backprop. Then discuss numerical stability, focusing on how log-sum-exp stabilizes softmax and cross-entropy by avoiding overflow/underflow, and mention other techniques like gradient clipping and careful epsilon selection.
Pro tip: Use a relative error metric (e.g., |a-b| / max(|a|+|b|, 1e-8)) and choose epsilon around 1e-4 to 1e-6; also, disable dropout and other stochastic layers during gradient checking to avoid noise.
Describe the central difference formula: (f(θ+ε) - f(θ-ε)) / (2ε) for each parameter, and how to compare it to the analytical gradient from backpropagation.
Choose a small network or function with few parameters, compute the analytical gradient via backprop, then compute numerical gradients for each parameter and calculate relative error.
Explain that naive softmax can overflow due to large exponentials; introduce the log-sum-exp trick: subtract the max logit before exponentiating, and show how it stabilizes both forward and backward passes.
Mention epsilon selection trade-offs (too small causes cancellation, too large causes truncation error), gradient clipping, and using double precision for gradient checks.
Emphasize that gradient checks are a debugging tool for custom layers, and that stable softmax is crucial for large-scale models like those at Tesla (e.g., vision or autonomous driving).
AI-generated suggestions, not part of the candidate's original notes. May be inaccurate — verify before relying on them.