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2293LSM Implementation Judgment 13Why are tree benchmarks still useful even when the production engine is LSM Monte Carlo?数理金融困难essay未尝试面试订阅2294LSM Implementation Judgment 14Why does time-step placement matter in LSM beyond just making the simulation mesh 'finer'?数理金融困难essay未尝试面试订阅2295LSM Implementation Judgment 15Why is 'more paths' not a complete answer if the continuation specification itself is misshapen?数理金融困难essay未尝试面试订阅2550Optimal Leaf Label Under Asymmetric Trading CostsA classification leaf contains 6 positive cases and 14 negative cases. Predicting positive costs 1 per false positive, while predicting negative costs 4 per false negative. Which class should the leaf predict to minimize expected leaf loss?机器学习困难derivation未尝试面试订阅2555Best Valid Split Under a Minimum-Leaf ConstraintThree candidate splits on the same node have Gini gains 0.18, 0.16, and 0.11, with smaller-child sizes 3, 4, and 7 respectively. If the minimum allowed leaf size is 4, which split is actually chosen?机器学习困难derivation未尝试面试订阅2560Global Weight Rescaling Leaves Split Ranking Unchanged 5If every sample weight in a node is multiplied by the same constant c>0, how does each candidate split's weighted impurity decrease change?机器学习困难derivation未尝试面试订阅2564Validation Penalty Threshold for Keeping a SplitA stump has validation loss 30. Splitting it into two leaves lowers validation loss to 22 but adds an instability penalty lambda per extra leaf. For what largest lambda is the split still preferred?机器学习困难derivation未尝试面试订阅2570Surrogate Split Agreement Rate 8A primary split is missing for some rows, so a surrogate split is trained on the M rows where the primary feature is observed. If it sends A of those rows to the same side as the primary split, what is its agreement rate?机器学习困难derivation未尝试面试订阅2613L2-Regularized Region Update 7In one boosting region, choose a constant update gamma to minimize sum i in R (r i-gamma) 2 + lambda gamma 2. Let S = sum i in R r i and n = |R|. Derive gamma.机器学习困难derivation未尝试面试订阅2614Why the Initial Prediction Matters 18Why can the choice of the initial prediction F 0 matter for the early trajectory of boosting?机器学习中等essay未尝试面试订阅2617Two-Region Two-Round Boosting Calculation 25A boosting model starts from F 0=0 with learning rate eta=0.1. In round 1, region A gets update +2 and region B gets update -1. In round 2, region A gets update -0.5 and region B gets update +0.25. What are the final predictions for a point that always stays in region A and a point that always stays in region B?机器学习简单数值题未尝试免费2620A Bound on Total Function Movement 8Suppose every boosting round changes any one point's prediction by at most eta A in absolute value. What upper bound does this imply on the total prediction movement after M rounds?机器学习困难derivation未尝试面试订阅2743Scaled Uniform Values and Win ProbabilityThree bidders have i.i.d. private values from Uniform (20,80). In a first-price auction, find the symmetric equilibrium bid of a bidder with value 56. Also compute that bidder's probability of winning under equilibrium play.脑筋急转弯中等derivation未尝试面试订阅2745Reverse Auction With Two SuppliersA buyer runs a sealed-bid procurement auction with two risk-neutral suppliers. Each supplier's cost is i.i.d. Uniform (0,1), and the lowest bid wins and is paid its own bid. Derive the symmetric equilibrium bid function s(c), and compute the expected profit of a supplier with cost 0.2.脑筋急转弯中等derivation未尝试面试订阅2749English Auction Dropout LogicFour bidders have independent private values 40, 55, 70, and 90. In an English ascending auction with no reserve, who wins, what price is paid, and why is the outcome equivalent to a second-price auction in this private-value setting?脑筋急转弯简单derivation未尝试面试订阅2750Dutch Auction as a First-Price ClockIn a Dutch descending-price auction with four bidders whose i.i.d. values are Uniform (0,1), a bidder has value 0.8. Assuming risk neutrality and the private-value benchmark, at what clock price should that bidder stop the clock in symmetric equilibrium? Briefly explain why.脑筋急转弯简单derivation未尝试面试订阅2758First-Price Bidding With a ReserveTwo bidders have i.i.d. values Uniform (0,1), and the seller imposes a reserve price r\in(0,1). Derive the symmetric equilibrium bid function for values v\ge r in the first-price auction.脑筋急转弯困难derivation未尝试面试订阅2759Applying the First-Price Reserve FormulaIn the two-bidder first-price auction with reserve from the previous question, take r=0.5. What is the equilibrium bid of a bidder with value 0.8, and what is that bidder's expected utility?脑筋急转弯中等derivation未尝试面试订阅2760General Procurement Bid Function With Four SuppliersA procurement auction has four risk-neutral suppliers with i.i.d. costs Uniform (0,1). The lowest bid wins and is paid its own bid. Derive the symmetric equilibrium bid function, and compute the bid of a supplier with cost 0.3.脑筋急转弯中等derivation未尝试面试订阅2764Why Risk Aversion Breaks Revenue EquivalenceIn one or two paragraphs, explain why a first-price auction can yield higher expected revenue than a second-price auction when bidders are risk-averse, even though the two formats are revenue-equivalent under risk neutrality.脑筋急转弯中等derivation未尝试面试订阅