Start by clearly defining the shapes of the input, weight, and bias matrices, then walk through the matrix multiplication and bias addition step by step, verifying shape compatibility at each stage. Emphasize the use of broadcasting for the bias addition and discuss how to implement this efficiently in NumPy.
Pro tip: Mention that you can implement the linear layer as `X @ W + b` where `X` is (batch_size, in_features), `W` is (in_features, out_features), and `b` is (out_features,), leveraging NumPy's broadcasting. Also, note that this is equivalent to `np.dot(X, W) + b` and that using `@` is preferred for clarity.
State that input X has shape (batch_size, in_features), weight W has shape (in_features, out_features), and bias b has shape (out_features,). Clarify that batch_size can be any positive integer.
Compute the linear transformation Z = X @ W. Verify that the inner dimensions match (in_features) and the result Z has shape (batch_size, out_features).
Add bias b to Z. Since b has shape (out_features,), broadcasting automatically expands it to (batch_size, out_features). The output Y has shape (batch_size, out_features).
Write the code: `def linear(X, W, b): return X @ W + b`. Optionally, include a check for shape compatibility and mention that this is a fully vectorized operation.
Mention that this can be extended to higher-dimensional inputs (e.g., (batch_size, seq_len, in_features)) by reshaping or using `np.tensordot`, and note that while NumPy is efficient for CPU, for large-scale training, frameworks like PyTorch or TensorFlow are preferred due to GPU support and autograd.
AI-generated suggestions, not part of the candidate's original notes. May be inaccurate — verify before relying on them.
They didn't go super deep on this with me but it was clearly on the table as a follow-up.
Start by defining a common abstraction for layers, such as a base class with forward and backward methods, then show how activation functions and attention can be implemented as subclasses. Emphasize modularity, composability, and performance considerations like vectorization and GPU support.
Pro tip: Mention that in production systems, you'd also consider serialization, device placement, and integration with autograd—showing you think beyond just the math.
Define a base Layer interface with methods like forward, backward, and parameters, ensuring all layer types can conform.
Create subclasses for activations (ReLU, Sigmoid, etc.) that apply element-wise operations and compute gradients.
Design attention as a layer that takes queries, keys, values, and returns weighted sums, handling masking and multi-head logic.
Allow layers to be stacked in a Sequential or Graph container, passing outputs of one as inputs to another.
Optimize with vectorized ops, support GPU, and integrate with autograd for automatic differentiation.
AI-generated suggestions, not part of the candidate's original notes. May be inaccurate — verify before relying on them.
Start by defining numerical stability in the context of your ML implementation, then systematically walk through the pipeline stages where instability can arise (data preprocessing, model initialization, forward/backward passes, loss computation, optimization). For each, describe the specific risk, its symptoms, and the mitigation techniques you applied or would apply. Conclude with how you validate stability through testing and monitoring.
Pro tip: Emphasize that numerical stability is not just about avoiding NaNs but also about preserving gradient fidelity and reproducibility; mention that you proactively log gradient norms and activation statistics during training to catch instability early.
Briefly explain what numerical stability means in ML (e.g., avoiding overflow/underflow, maintaining precision, ensuring convergence) and why it's critical for model performance and reliability.
Walk through key stages: data preprocessing (scaling, normalization), model architecture (activation functions, weight init), loss functions (log-sum-exp, softmax), and optimization (learning rates, gradient clipping).
For each risk area, give concrete examples of what can go wrong (e.g., exploding gradients in RNNs, underflow in log probabilities, division by zero in normalization) and how they manifest (NaNs, slow convergence, poor performance).
Detail the techniques you use to address each failure mode, such as gradient clipping, stable implementations of softmax (subtracting max), using double precision where needed, and careful initialization (Xavier, He).
Explain how you test for numerical stability (unit tests with extreme inputs, gradient checking) and monitor during training (logging gradient norms, loss spikes) to ensure ongoing stability.
AI-generated suggestions, not part of the candidate's original notes. May be inaccurate — verify before relying on them.