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Why does convexity matter for gradient descent?

Unlike linear regression, logistic regression trained with gradient descent:

In a setting where treatment is not randomly assigned, the naive estimator (mean outcome of the treated minus mean outcome of the untreated) can be decomposed as the Average Treatment effect on the Treated (ATT) plus:

Which linkage criterion merges the pair of clusters whose combination causes the smallest increase in within-cluster variance?

Required sample size per group in an A/B test scales as 1/delta^2, where delta is the minimum detectable effect. If you halve the minimum detectable effect you want to reliably detect (holding power and alpha fixed), the required sample size per group:

Why does backpropagation make it computationally tractable to train networks with many layers?

Which of the following IS actually guaranteed by the universal approximation theorem?

DBSCAN with a single eps merges two nearby tight clusters into one blob whenever eps is set large enough to also capture a separate, diffuse cluster elsewhere in the data. Which algorithm is specifically designed to fix this failure mode?

DBSCAN is run with min_samples = 5. A point lies within eps of 6 other points (7 total, including itself). This point should be classified as:

Evaluate the limit using algebraic or trigonometric methods when applicable. If the limit is infinite, write  or .  If the limit does not exist and is not infinite, write DNE. Use the math editor (“Insert Math Equation” as needed on the toolbar) to enter your final answer. Show all work on your paper. Find   using algebraic and trigonometric methods.