I listed the assumptions fine but fumbled the classification part.
Start by clearly listing the main assumptions of linear regression (linearity, independence, homoscedasticity, normality of errors, no multicollinearity) and briefly explain each. Then, justify squared loss by connecting it to the Gauss-Markov theorem, maximum likelihood under Gaussian errors, and its convex optimization properties. Finally, mention practical implications and alternatives like robust regression when assumptions are violated.
Pro tip: Emphasize that squared loss is not just a convention—it's theoretically grounded in the Gauss-Markov theorem and MLE, but also acknowledge its sensitivity to outliers and when to consider alternatives like Huber loss. This shows you understand both theory and practical trade-offs, which is crucial at Amazon.
Enumerate the key assumptions: linearity, independence of errors, homoscedasticity, normality of errors, and no perfect multicollinearity. Briefly explain each in one sentence.
Discuss the Gauss-Markov theorem: under the assumptions, OLS (squared loss) gives the best linear unbiased estimator (BLUE). Also mention that minimizing squared loss is equivalent to maximum likelihood estimation when errors are Gaussian.
Note that squared loss leads to a convex optimization problem with a closed-form solution (normal equations), making it efficient and stable to compute.
Acknowledge that squared loss is sensitive to outliers and that assumptions may not hold in practice. Mention alternatives like robust regression (Huber loss) or quantile regression when appropriate.
Relate to real-world scenarios: when to use linear regression, how to check assumptions (residual plots, tests), and the impact on model performance and interpretability.
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I answered the log-likelihood part fine but blanked for a second on the two distinct places where a log appears.
Start by defining logistic regression as a linear model for binary classification that outputs probabilities via the sigmoid function. Then explain that the logarithm arises from the log-odds (logit) link function, which linearizes the relationship between features and the probability of the positive class. Finally, connect this to maximum likelihood estimation and the cross-entropy loss.
Pro tip: Emphasize that the log-odds interpretation makes coefficients interpretable as additive effects on the odds, which is crucial for explainability in production systems. Also, mention that logistic regression is a foundational building block for neural networks and is widely used at Amazon for its simplicity and scalability.
State that logistic regression is a supervised learning algorithm for binary classification that models the probability of the positive class using a linear combination of features passed through a sigmoid function.
Explain that the sigmoid squashes the linear output to [0,1], and its inverse is the log-odds (logit) function: log(p/(1-p)). This shows that the log-odds is linear in the features.
Describe how maximizing the likelihood of the data under the Bernoulli model leads to the cross-entropy loss, which naturally involves logarithms of the predicted probabilities.
Summarize that the logarithm appears because we model the log-odds as a linear function, and because the likelihood of binary outcomes involves log probabilities, making optimization tractable.
Mention that this formulation yields interpretable coefficients (additive effects on log-odds), convex loss for efficient optimization, and probabilistic outputs useful for decision-making.
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Start by defining a random forest as an ensemble of decision trees trained on bootstrapped samples with random feature subsets. Then explain the feature selection at each split: at each node, a random subset of features is considered, and the best split is chosen based on impurity reduction (e.g., Gini or entropy). Emphasize how this randomness reduces correlation among trees and improves generalization.
Pro tip: Mention that the number of features considered at each split (often denoted mtry or max_features) is a key hyperparameter; in practice, for classification, sqrt(p) is common, while for regression, p/3 is often used. Also note that feature selection at splits is not about selecting a global feature set but about local, randomized selection to decorrelate trees.
Explain that it's an ensemble learning method that builds multiple decision trees and aggregates their predictions (voting for classification, averaging for regression).
Each tree is trained on a bootstrap sample of the data, and at each node, a random subset of features is selected as candidates for splitting.
From the random feature subset, the algorithm chooses the feature and threshold that maximize impurity reduction (e.g., Gini impurity or information gain).
This random feature selection decorrelates trees, reducing variance and preventing overfitting compared to individual decision trees.
Discuss key hyperparameters like number of trees, max_features (features per split), and max_depth, and their impact on performance.
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Start by clearly explaining Adam's mechanism (momentum + adaptive learning rates) and vanilla SGD's simplicity. Then compare them across key dimensions like convergence speed, generalization, and hyperparameter sensitivity, and discuss real-world trade-offs in production ML systems. Finally, tie it back to practical scenarios, especially at scale like Amazon.
Pro tip: Emphasize that Adam often converges faster but can generalize worse than SGD with momentum, and mention that switching from Adam to SGD late in training can combine fast convergence with better final performance—a trick used in many winning Kaggle solutions and production systems.
Briefly explain vanilla SGD (updates parameters using a fixed learning rate times gradient) and Adam (combines momentum and RMSProp with bias correction). Highlight that Adam maintains per-parameter learning rates.
Contrast how SGD uses a single global learning rate, while Adam adapts learning rates based on first and second moment estimates. Mention that Adam includes bias correction for initial steps.
Explain that Adam typically converges faster and is less sensitive to learning rate, but SGD with momentum often achieves better generalization on many tasks, especially in computer vision.
Cover memory overhead (Adam stores two additional states per parameter), computational cost, and hyperparameter tuning. Note that Adam can fail to converge on some problems due to adaptive learning rates.
Tie to Amazon-scale scenarios: Adam is often preferred for quick prototyping and NLP tasks, while SGD might be used for large-scale recommendation systems where generalization and memory matter.
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Said the narrow one, which is right, but my first explanation leaned too hard on 'fewer parameters means fewer solutions' without really articulating the loss landscape argument.
First clarify that the question is about the loss landscape and optimization dynamics of two-layer networks. Then argue that narrow networks are more prone to poor local minima due to their limited capacity and highly non-convex loss surface, while wide networks have fewer bad local minima and are easier to optimize. Support your answer with theoretical insights and practical implications.
Pro tip: Mention that in practice, width often matters more than depth for avoiding poor local minima, and relate this to the success of overparameterized models in deep learning. Also, note that narrow networks may require more careful initialization and training tricks to escape suboptimal solutions.
Clarify what is meant by 'narrow' and 'wide' two-layer networks: narrow has few hidden units, wide has many. Emphasize that both are two-layer (one hidden layer) feedforward networks.
Discuss how the loss landscape of a narrow network is more rugged with many poor local minima, while a wide network has a smoother landscape with fewer bad local minima and more saddle points.
Explain that gradient-based methods can get trapped in poor local minima in narrow networks, but in wide networks, saddle points are the main obstacle and can be escaped more easily.
Cite results from optimization theory and deep learning: e.g., overparameterized networks have no bad local minima (under certain conditions), and wider networks have better conditioning.
Conclude that narrow networks are more likely to get stuck in poor local minima. Mention practical implications: wider networks are easier to train and often generalize better despite overparameterization.
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