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2003Execution Cost With Stronger Capacity Wall 8The blow-up term is steeper, but the same convexity logic should still go through. Show that f(q) = 2 q 2 + 4/(1-q) is strictly convex on q<1.数学中等derivation未尝试免费2009Log Barrier Plus Ridge Penalty 14The desk penalizes approaching a utilization cap and also adds a quadratic regularizer. Show that r(x) = -ln(1-2x) + 1x 2 is convex on x < 0.5.数学困难derivation未尝试免费2014Why the Smooth-Max Proxy Is Convex 19Explain in one sentence why log(exp(a 1 T x)+...+exp(a k T x)) is convex.数学困难derivation未尝试免费2015Inventory Cost Near a Hard Capacity Limit 20The PM wants a formal convexity check before using the function in an optimizer. Show that f(q) = 5 q 2 + 2/(1-q) is strictly convex on q<1.数学困难derivation未尝试面试订阅2025Reciprocal Funding Buffer Is Convex 5Let phi(L)=1/(1+L) on L>-1. State the Jensen inequality relation between E[phi(L)] and phi(E[L]).数学困难derivation未尝试面试订阅2027Log Barrier Jensen Direction 7A utilization score explodes as it nears 1, so the convexity direction matters for stress design. Let u(x)=-ln(1-x) on x<1. If U is random and almost surely below 1, compare E[u(U)] and u(E[U]).数学中等derivation未尝试免费2034Conditional Jensen Lower Bound 14If phi is convex, what inequality holds between E[phi(X)|F] and phi(E[X|F]) almost surely?数学困难derivation未尝试面试订阅2035Funding Buffer Gap With Unequal Scenario Weights 15The high-leverage state is rarer, but still materially affects the convex average. A funding-buffer model uses phi(L)=1/(1+L). Suppose L takes values 1 and 4 with probabilities 1/4 and 3/4. Compute E[phi(L)] and phi(E[L]).数学困难数值题未尝试面试订阅2040Three-Scenario Square-Root Impact Gap 20Suppose V takes values 0, 3, and 8 with equal probability. Compute E[sqrt(1+V)] and sqrt(1+E[V]).数学困难数值题未尝试面试订阅2043Log Carry Gap From Two Scenarios 23The desk wants to see the exact concave Jensen gap, not just the inequality direction. A desk scores carry through psi(x)=ln(1+x). Suppose X takes values 0 and 3 with probabilities 1/2 and 1/2. Compute E[psi(X)] and psi(E[X]).数学中等数值题未尝试面试订阅2396Variance of an Equal-Weight Correlated EnsembleFive base models each have prediction variance 4, and every pair of model predictions has correlation 0.25. If you average the five predictions equally, what is the ensemble variance?机器学习简单derivation未尝试免费2398Bias Budget Implied by a Variance ReductionA regularization change reduces a model's variance term from 0.30 to 0.11 while leaving irreducible noise unchanged. How much extra bias squared could you add before the total MSE stops improving?机器学习中等derivation未尝试免费2399Optimal Weight on a Noisy Unbiased ModelModel A is unbiased with variance 9. Model B has variance 1.44 and fixed bias 0.6. If you blend them as P w = wA + (1-w)B and treat their errors as independent, what weight w minimizes MSE?机器学习困难derivation未尝试面试订阅2400How Many Independent Fits to Hit a Variance TargetEach independently trained model has variance 2.4 and negligible bias. How many equally weighted independent fits must you average to bring the variance term below 0.3?机器学习中等derivation未尝试免费2402Second Crossover With a Lower-Bias Flexible ModelA flexible model has excess error 0.02 + 24/n, while a simpler model has excess error 0.14 + 6/n. At what sample size do they tie?机器学习中等derivation未尝试面试订阅2403Variance of a Correlated Five-Model CommitteeFive models each have variance 1.6 and pairwise correlation 0.4. What is the variance of their equal-weight average?机器学习中等derivation未尝试免费2404Data Multiplier Needed to Push Variance Below a Noise Floor FractionA model's variance term is currently 0.30, and irreducible noise is 0.05. If variance scales exactly like 1/n, by what factor must the dataset grow so the variance term falls to 0.05?机器学习中等derivation未尝试面试订阅2405Recover the Irreducible NoiseA model has test MSE 0.92, bias 2 0.15, and variance 0.27. What irreducible noise term is implied?机器学习简单数值题未尝试面试订阅2408Variance of a Three-Model Independent AverageThree independently trained models each have variance 1.8 and negligible bias. What is the variance of their equal-weight average?机器学习简单数值题未尝试面试订阅2409Why More Data Usually Helps a Variance-Dominated Model FirstWhy does collecting more data usually help a high-variance model more than a high-bias model?机器学习困难essay未尝试面试订阅