I got through the normal equations fine but stumbled a bit explaining exogeneity vs homoscedasticity in plain terms.
Start by setting up the OLS problem as minimizing the sum of squared residuals, then derive the normal equations using matrix calculus. Solve for the coefficients to get the closed-form solution, and finally discuss the assumptions required for this solution to be valid and unbiased.
Pro tip: Emphasize the geometric interpretation of OLS as an orthogonal projection onto the column space of the design matrix, which provides intuition and connects to the normal equations. Also, mention that while the closed-form solution is elegant, in practice numerical stability and computational efficiency often favor iterative methods like gradient descent for large-scale problems.
State the OLS problem: minimize the sum of squared residuals, expressed as ||y - Xβ||^2. Explain that this is equivalent to minimizing the squared Euclidean norm of the error vector.
Take the gradient of the objective with respect to β, set it to zero, and solve. Show that the gradient is -2X^T(y - Xβ), leading to X^T X β = X^T y.
Assuming X^T X is invertible, the closed-form solution is β = (X^T X)^{-1} X^T y. Discuss the role of the Moore-Penrose pseudoinverse when X^T X is singular.
List the Gauss-Markov assumptions: linearity, exogeneity (E[ε|X] = 0), homoscedasticity (constant variance), no autocorrelation, and full rank of X. Mention that normality is needed for inference but not for unbiasedness.
Explain that under these assumptions, OLS is BLUE (Best Linear Unbiased Estimator). Note that violations affect properties, and that the closed-form solution may be computationally expensive for large datasets.
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Talked through the penalty terms and sparsity with L1.
Start by writing the standard linear regression objective (OLS) and then show how L1 (Lasso) and L2 (Ridge) add penalty terms to it. Explain the practical differences in the solutions: L1 promotes sparsity (feature selection) while L2 shrinks coefficients smoothly and handles multicollinearity. Conclude with when to use each, emphasizing trade-offs in bias-variance and computational aspects.
Pro tip: Mention that L1 regularization can be solved efficiently via coordinate descent or LARS, while L2 has a closed-form solution, and that elastic net combines both—this shows depth and practical awareness.
State the ordinary least squares (OLS) objective: minimize ||y - Xw||^2. This sets the baseline for comparison.
Explain that L1 adds λ * sum(|w_i|) and L2 adds λ * sum(w_i^2) to the OLS objective, with λ controlling regularization strength.
Describe how L1 yields sparse solutions (some weights exactly zero) due to the diamond-shaped constraint, while L2 yields small but non-zero weights due to the circular constraint.
Cover use cases: L1 for feature selection and interpretability, L2 for handling multicollinearity and improving generalization, and elastic net for combining benefits.
Highlight bias-variance trade-off, computational considerations (closed-form for L2, iterative for L1), and the importance of tuning λ via cross-validation.
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Start by defining bias and variance in the context of linear regression, then explain the trade-off and how it affects model performance. Use the mean squared error decomposition to illustrate the balance, and discuss practical implications for model selection and regularization.
Pro tip: Relate the trade-off to the bias-variance decomposition of expected prediction error, and mention how techniques like ridge regression or lasso can help control variance at the cost of bias. This shows you understand both theory and application.
Explain bias as the error from erroneous assumptions in the learning algorithm (e.g., assuming linearity when the true relationship is nonlinear), and variance as the error from sensitivity to small fluctuations in the training set.
Describe how increasing model complexity (e.g., adding polynomial terms) decreases bias but increases variance, and vice versa. The goal is to find the sweet spot that minimizes total error.
Present the decomposition of expected prediction error into bias squared, variance, and irreducible error. This mathematically shows the trade-off.
Discuss how in linear regression, bias and variance depend on the number of features, regularization, and model assumptions. For example, ordinary least squares has low bias but high variance if features are many or correlated.
Mention techniques to manage the trade-off, such as cross-validation for model selection, regularization (ridge, lasso), and dimensionality reduction.
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This is where the interview got more interesting.
Start by defining self-attention as a mechanism that allows each token to weigh the importance of all other tokens in the sequence. Then, walk through the query-key-value computation step by step, using a concrete example to illustrate the process. Finally, connect this to the broader transformer architecture and discuss trade-offs like computational complexity.
Pro tip: Emphasize the intuition behind queries, keys, and values as a soft dictionary lookup, and mention how this enables parallelization and long-range dependencies. Also, briefly touch on scaling and efficiency considerations, as these are critical in real-world ML systems.
Explain that self-attention lets each token attend to all others, dynamically aggregating context. Use an analogy like a soft dictionary lookup where queries match keys to retrieve values.
Describe how each token's embedding is linearly projected into Q, K, and V matrices. Clarify that these projections are learned and shared across positions.
Detail the dot product between queries and keys, scaling by sqrt(d_k), and applying softmax to obtain attention weights. Mention masking for causal attention if relevant.
Explain that the attention weights are used to compute a weighted sum of the values, producing the output for each token. Note that this output is then passed through a feed-forward network.
Briefly mention that multiple heads allow the model to attend to different representation subspaces. Discuss computational complexity O(n^2) and memory considerations, and how techniques like sparse attention address them.
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Explained the n-squared cost from computing all pairwise attention scores.
Start by explaining the quadratic complexity of self-attention: each token attends to every other token, leading to O(n^2) operations. Then compare with RNNs, which have O(n) sequential operations but struggle with long-range dependencies due to vanishing gradients and lack of parallelism. Conclude by discussing trade-offs in practice, such as memory bottlenecks and techniques like sparse attention.
Pro tip: Mention that while RNNs are theoretically linear, their sequential nature makes them slow on modern hardware, and self-attention's quadratic cost is often mitigated by parallelization and approximations. This shows awareness of practical deployment constraints.
Explain that self-attention computes pairwise interactions between all tokens, resulting in O(n^2) time and memory complexity for sequence length n.
State that RNNs process tokens sequentially, with O(n) time complexity per layer, but each step depends on the previous, limiting parallelism.
Discuss how RNNs suffer from vanishing gradients and difficulty capturing long-range dependencies, while self-attention directly models all pairwise interactions but becomes computationally expensive for long sequences.
Mention that self-attention's quadratic cost can be prohibitive for very long sequences, but its parallelism makes it efficient on GPUs; RNNs are linear but sequential, leading to slow training and inference.
Bring up approaches like sparse attention, linear attention, or chunked attention that reduce complexity, and note that RNN variants (e.g., LSTMs) still struggle with very long contexts.
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Ran through the use cases: classification and embeddings for encoder-only, autoregressive generation for decoder-only, seq2seq tasks like translation for the full encoder-decoder setup.
Start by defining the three transformer architectures based on their attention mechanisms and training objectives. Then, for each, explain the typical use cases and provide concrete examples of models and tasks. Finally, discuss trade-offs and selection criteria, emphasizing how the choice depends on the problem requirements.
Pro tip: Relate the architectures to real-world applications and mention specific models like BERT, GPT, and T5 to demonstrate practical knowledge. Also, highlight that encoder-only models are efficient for understanding tasks, while decoder-only models excel at generation, and encoder-decoder models are versatile for sequence-to-sequence tasks.
Briefly explain the structural differences: encoder-only uses bidirectional attention, decoder-only uses unidirectional (causal) attention, and encoder-decoder combines both with cross-attention.
Describe the typical pre-training objectives: masked language modeling for encoder-only, causal language modeling for decoder-only, and span corruption or sequence-to-sequence for encoder-decoder.
For each architecture, list common tasks: encoder-only for classification, NER, extractive QA; decoder-only for text generation, few-shot learning; encoder-decoder for translation, summarization, generative QA.
Compare computational efficiency, latency, and suitability for different data regimes. Mention that encoder-only is often faster for inference on understanding tasks, while decoder-only can be more flexible for generation.
Summarize how to choose: consider task type (understanding vs. generation), data availability (pre-training vs. fine-tuning), and deployment constraints (latency, memory).
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Covered sinusoidal encodings and why attention alone is permutation-invariant so you need to inject position somehow.
Start by explaining that positional encodings inject sequence order information into a permutation-invariant architecture. Then describe how residual connections and layer normalization address vanishing gradients and internal covariate shift to stabilize training. Finally, connect these concepts to practical benefits like faster convergence and deeper models.
Pro tip: Emphasize that positional encodings are added, not concatenated, to preserve dimensionality and enable parallel processing. For stability, mention that residual connections create shortcut paths for gradients, while layer normalization normalizes activations per sample, making training less sensitive to initialization and learning rates.
Explain that transformers lack inherent recurrence or convolution, so positional encodings provide information about the order of tokens in a sequence.
Mention common methods like sinusoidal functions or learned embeddings, and note they are added to input embeddings.
Discuss how residual connections (skip connections) allow gradients to flow directly through the network, mitigating vanishing gradients and enabling training of deep models.
Describe how layer normalization normalizes activations across features per sample, reducing internal covariate shift and stabilizing training.
Summarize how these components together improve convergence, allow higher learning rates, and reduce sensitivity to initialization.
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