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2069Newton Needs a Nonzero Derivative at the Root 24Why is a simple root f'(r) != 0 the standard clean setting for fast Newton convergence?数学简单derivation未尝试免费2383Law of Total Variance in a Regime Simulator 13What two pieces make up Var(X) under the law of total variance when a simulator first samples a regime Z and then samples X conditional on Z?数学中等derivation未尝试免费2424Convexity of the Log-Cosh Loss 4Show that ell(r)=ln cosh(r) is convex in the residual r.机器学习中等derivation未尝试免费2433Pinball Loss Subgradient at the Kink 9For pinball loss rho tau(r)=tau r if r>=0 and (tau-1)r if r<0, what is the subgradient set at r=0?机器学习中等derivation未尝试面试订阅2434Why a Strictly Proper Loss Must Prefer Truthful Probabilities 10Why is it desirable for a probabilistic classification loss to be strictly proper rather than merely classification-accurate?机器学习困难derivation未尝试面试订阅2440Why Heavy-Tailed Noise Pushes You Away From Pure Squared Loss 15Why is pure squared loss often a poor default when the residual distribution has rare but huge outliers?机器学习中等derivation未尝试面试订阅2443Weighted Log-Loss Moves the Bayes Probability Toward the Costlier Class 16Why does class-weighted log-loss shift the optimal reported probability toward the class with the larger weight?机器学习中等derivation未尝试面试订阅2444Why Quantile Loss Is Useful in Risk Forecasting 17Why is pinball loss natural when the target is a VaR-like forecast rather than a mean forecast?机器学习中等derivation未尝试面试订阅2445Why Tail Forecasts Need Tail-Aligned Losses 25Why is it often a mistake to optimize plain squared loss when the operational task really cares about an extreme tail quantile?机器学习困难essay未尝试面试订阅2472Intercept From Sample Means and Slope 2Derive the OLS intercept in simple regression with an intercept once beta hat is known.机器学习简单derivation未尝试免费2477Why Centering Can Simplify OLS Algebra 7Why does centering features and targets often make OLS derivations cleaner when an intercept is present?机器学习中等essay未尝试免费2483Why Centering Leaves Slopes Unchanged 13Why does centering x and y leave the fitted slope unchanged in simple OLS with an intercept?机器学习中等derivation未尝试面试订阅2484Response Scaling 14If every target is multiplied by c, what happens to the OLS coefficient vector and intercept?机器学习困难derivation未尝试面试订阅2487Prediction Invariance Under Equivalent Parameterizations 16Why can two different coefficient vectors produce exactly the same OLS predictions when the design is rank-deficient?机器学习中等derivation未尝试面试订阅2493Projection Error Is Orthogonal to the Fitted Subspace 23Why is y - X beta hat orthogonal to every fitted vector Xv?机器学习中等derivation未尝试面试订阅2525One Newton Step for an Intercept-Only Logistic ModelAn intercept-only logistic model is fit to 7 positives and 3 negatives. Starting from b 0 = 0, what is one Newton step b 1 for minimizing the negative log-likelihood?机器学习困难数值题未尝试面试订阅2534One Gradient Step on a Tiny Logistic ProblemA one-feature logistic model without intercept uses beta = 0 initially, learning rate 0.2, data x = [-1, 0, 1], and labels y = [0, 0, 1]. What is beta after one gradient step on the negative log-likelihood?机器学习困难数值题未尝试面试订阅2535Decision Threshold Under Asymmetric Classification CostsA desk incurs cost 1 for a false positive and cost 5 for a false negative. Under a calibrated logistic probability p = P(Y=1|x), above what threshold should it predict class 1 to minimize expected cost?机器学习困难derivation未尝试面试订阅2540Intercept Shift for a Deployment Prior ChangeA logistic model was trained under class prior 0.5 and has intercept -0.4. At deployment the base rate falls to 0.2 while feature likelihood ratios are assumed unchanged. What adjusted intercept should be used?机器学习困难数值题未尝试面试订阅2541One Gradient Step on a Single Logistic Observation 22For one observation with x = 2, y = 1, current weight w = 0, and learning rate eta = 0.4, what is one gradient-descent update on the negative log-likelihood?机器学习简单数值题未尝试免费