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2501Equivalent L2 Radius From a Ridge Solution 6If the ridge optimum in R p is beta hat lambda, what radius t makes it also solve the constrained problem min RSS(beta) subject to ||beta|| 2 <= t?机器学习中等derivation未尝试免费2502Elastic-Net Coordinate Update in an Orthogonal Basis 7In an orthogonal coordinate with x T x = d and score z, derive the elastic-net coefficient with L1 weight lambda 1 and L2 weight lambda 2 when the coefficient is active and positive.机器学习简单derivation未尝试免费2503MAP Interpretation of Ridge 10Under a Gaussian-noise linear model, what Gaussian prior on beta makes ridge the MAP estimator?机器学习中等derivation未尝试面试订阅2505Ridge Never Flips Sign in the Orthogonal One-Feature Case 12In the orthogonal one-feature case with z = x T y, why does ridge preserve the sign of z for every lambda >= 0?机器学习困难derivation未尝试面试订阅2508Why Elastic Net Keeps the Lasso Threshold but Adds Ridge Shrinkage 14Why does elastic net still need |z| to clear an L1 threshold before a coordinate activates, but then shrink the active coefficient more than lasso does?机器学习中等derivation未尝试面试订阅2509Ridge Norm Shrinks Monotonically With Lambda 15Why should the ridge solution norm typically decrease as lambda increases?机器学习困难derivation未尝试面试订阅2510Zero Lambda Recovers OLS 16Why do ridge and lasso both reduce to OLS when their regularization parameter is set to zero?机器学习困难derivation未尝试面试订阅2513Why Correlated Features Frustrate Pure Lasso 17Why does pure lasso often behave erratically when several features are highly correlated and similarly predictive?机器学习中等essay未尝试面试订阅2514Equivalent L2 Radius in One Dimension 19In one dimension, if the ridge solution equals beta hat lambda, what radius t makes the constrained problem min RSS(beta) subject to |beta| <= t share the same optimizer?机器学习困难derivation未尝试面试订阅2515Why Small Lambda Means Weak Regularization 20Why does a very small lambda leave the regularized solution close to OLS?机器学习困难derivation未尝试面试订阅2518Ridge Shrinkage Ratio Numerically 23In an orthogonal coordinate with d = 6 and lambda = 2, what fraction of the OLS coefficient remains under ridge?机器学习中等derivation未尝试面试订阅2520Why L1 and L2 Pull Differently Near Zero 25Why does L1 regularization create a stronger qualitative push toward exact zero than L2 regularization near the origin?机器学习困难derivation未尝试面试订阅2521Intercept-Only Logistic MLE 1For an intercept-only logistic model with n 1 positives and n 0 negatives, what fitted probability p hat maximizes the log-likelihood?机器学习简单derivation未尝试免费2522Intercept From the Positive Rate 2In an intercept-only logistic model, if the fitted probability is p hat, what intercept b solves sigma(b)=p hat?机器学习简单derivation未尝试免费2523Gradient of Logistic Negative Log-Likelihood 3For one observation (x,y) with y in 0,1 and score z = w T x, what is the gradient of the negative log-likelihood with respect to w?机器学习中等derivation未尝试免费2524Why No Closed Form in Logistic Regression 5Why does logistic regression usually require iterative optimization rather than a normal-equation-style closed form?机器学习中等essay未尝试免费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2528Why Log-Loss Rewards Calibration 9Why does a well-calibrated probability forecaster typically fare better under log-loss than a forecaster that only gets rankings right?机器学习中等essay未尝试免费2532Hessian of Logistic Negative Log-Likelihood 4For one observation with score z = w T x, what is the Hessian of the negative log-likelihood with respect to w?机器学习简单derivation未尝试免费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