I knew the assumptions cold but fumbled the justification for squared loss until I remembered to frame it as MLE under Gaussian noise.
Start by clearly listing the assumptions of linear regression (linearity, independence, homoscedasticity, normality of errors, no multicollinearity) and explain why each matters for valid inference. Then, justify squared loss by connecting it to maximum likelihood estimation under Gaussian errors and its mathematical convenience, while briefly noting alternatives like absolute loss for robustness.
Pro tip: Mention that squared loss is sensitive to outliers, so in practice at Amazon you might consider robust alternatives or regularization, showing awareness of trade-offs beyond textbook assumptions.
Enumerate the key assumptions: linearity, independence of errors, homoscedasticity, normality of errors, and no perfect multicollinearity. Briefly explain each.
Discuss how violations affect coefficient estimates, standard errors, and hypothesis tests, leading to biased or inefficient results.
Connect squared loss to maximum likelihood estimation under Gaussian errors and its convexity, which ensures a unique global minimum.
Acknowledge that squared loss is sensitive to outliers and may not be ideal for all data; mention alternatives like absolute loss or Huber loss.
Tie the discussion to real-world scenarios, such as when assumptions are violated and how to address them (e.g., transformations, robust regression).
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Went straight to the Bernoulli likelihood angle.
Start by defining logistic regression as a linear model for binary classification that outputs probabilities via the sigmoid function. Then explain that log loss (cross-entropy) is the proper loss function because it directly measures the quality of predicted probabilities and penalizes confident misclassifications, unlike raw probability which is not a proper scoring rule. Emphasize that log loss is derived from maximum likelihood estimation under a Bernoulli assumption, making it statistically principled and convex for efficient optimization.
Pro tip: Mention that log loss is a proper scoring rule, which encourages calibrated probabilities—crucial in real-world applications like Amazon's where decision-making depends on reliable confidence estimates. Also, note that using raw probability as a loss (e.g., mean squared error on probabilities) can lead to vanishing gradients and poor calibration.
Explain that logistic regression models the probability of a binary outcome using a linear combination of inputs passed through the sigmoid function. It's a discriminative model that directly estimates P(y=1|x).
State that we need a loss function to measure how well the model's predicted probabilities match the true labels. Log loss, or binary cross-entropy, is the standard choice.
Discuss that raw probability (e.g., using accuracy or MSE on probabilities) is not a proper scoring rule and doesn't penalize overconfident wrong predictions enough. Log loss heavily penalizes confident misclassifications and is derived from maximum likelihood estimation.
Mention that log loss is convex, ensuring a unique global minimum, and aligns with the probabilistic foundation of logistic regression. It also provides well-calibrated probabilities.
Emphasize that in applications like Amazon's, calibrated probabilities are essential for decision-making, and log loss encourages that. Also, note that log loss is the natural loss for Bernoulli distribution.
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The sqrt(p) thing is one of those details I always have to remind myself is actually the point, not just a hyperparameter quirk.
Start by defining Random Forest as an ensemble of decision trees trained on bootstrap samples with random feature subsets, then explain the feature selection at each split: a random subset of features is considered, and the best split is chosen based on impurity reduction (Gini or entropy). Emphasize that this randomness reduces correlation among trees and improves generalization, and connect it to trade-offs like bias-variance and computational efficiency.
Pro tip: Mention that the number of features considered at each split (mtry) is a key hyperparameter that controls the trade-off between tree diversity and individual tree strength, and that in practice it's often set to sqrt(p) for classification and p/3 for regression.
Explain that it's an ensemble learning method that builds multiple decision trees and aggregates their predictions (majority vote for classification, average for regression).
Mention bootstrap sampling (bagging) and that each tree is grown on a different bootstrap sample, with random feature selection at each split.
At each node, a random subset of m features (out of p total) is selected, and the best split among those is chosen based on impurity reduction (Gini impurity or entropy).
Highlight that this randomness decorrelates trees, reducing variance and preventing overfitting, while maintaining low bias.
Mention hyperparameters like number of trees and mtry, and how they affect performance, computational cost, and interpretability.
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I had drilled the four update lines enough that I could write them out without hesitating.
Start by clearly defining Adam's update rule, emphasizing its adaptive per-parameter learning rates and bias correction. Then compare it to SGD across key dimensions like convergence speed, generalization, and hyperparameter sensitivity, citing empirical evidence and practical considerations. Conclude with guidance on when to choose each optimizer based on problem characteristics.
Pro tip: Mention that while Adam often converges faster, SGD with momentum can achieve better generalization in some vision tasks, and recent optimizers like AdamW fix Adam's weight decay issue—showing you're up-to-date with research.
Explain that Adam computes adaptive learning rates for each parameter using estimates of first and second moments of gradients, with bias correction.
Briefly describe SGD and its common variant with momentum, noting that it uses a single learning rate for all parameters.
Contrast Adam's fast convergence, robustness to hyperparameters, and suitability for sparse gradients against SGD's better generalization, lower memory, and simplicity.
Highlight scenarios where each shines: Adam for NLP/transformers, SGD for computer vision; mention AdamW and learning rate schedules.
Summarize that the choice depends on the task, and suggest starting with Adam for rapid prototyping, then trying SGD for final performance.
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This one is designed to catch people who assume more parameters means more complexity means harder optimization.
Clarify that the question is about optimization difficulty in non-convex loss landscapes, then compare the narrow and wide networks in terms of parameter count, redundancy, and loss surface geometry. Argue that the narrow network is more likely to get stuck in local minima due to its limited capacity and more rugged loss surface, while the wide network has smoother, more connected minima.
Pro tip: Acknowledge that in practice, with modern optimizers and overparameterization, local minima are often not the main obstacle; instead, saddle points and plateaus dominate. This shows depth beyond textbook knowledge.
Explain that local minima are suboptimal points where gradients vanish but are not global minima, and that they can trap gradient-based optimization.
Discuss how narrow networks have fewer parameters and less redundancy, leading to a more rugged loss surface with many isolated local minima, while wide networks have smoother, more connected surfaces.
Mention that wide networks are overparameterized, which tends to eliminate bad local minima and create paths to good solutions, as supported by recent theoretical and empirical work.
Note that in practice, local minima are less problematic than saddle points, and that other factors like initialization, learning rate, and batch size also influence optimization.
State clearly that the narrow network is more likely to get stuck in local minima, but emphasize the nuances and practical considerations.
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