题目2433 · 机器学习
Pinball Loss Subgradient at the Kink 9
For 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?
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中文题目题目001 · 概率
Pinned Delegation Without the Rival Pair
A 3-person delegation contains a preassigned chairperson H and two additional members chosen uniformly from six others. Among those six, exactly one unordered pair is a forbidden rival pair. What is the probability the final delegation does not contain that forbidden pair?
打开 →题目011 · 概率
Pinned Surjection to Three Desks
Five labeled tasks are assigned to three labeled desks A, B, and C. Task 1 is forced to desk A. How many assignments use all three desks at least once?
打开 →题目5967 · 概率
Doubling Strategy and Optional Stopping Failure
A gambler starts with $0 net and makes fair $1-doubling bets (bet 1, then 2, then 4, ...) on a sequence of fair coin flips, stopping the first time they win a single flip (guaranteeing +$1 net). Let T be that stopping time. Compute E[net wealth at T], and explain whether E[net at
打开 →题目5977 · 概率
Optional Stopping on a Product Martingale
Let X_1, X_2, ... be i.i.d. fair +-1 steps and define the product P_n = prod_{i=1}^n (1 + (1/2) X_i), with P_0 = 1. Let N be any almost-surely finite stopping time. Treating P_n as a martingale, what is E[P_N]?
打开 →题目5963 · 概率
Wald's Identity from Optional Stopping
Let X_1,X_2,... be i.i.d. with mean 4, and let N be a stopping time (with respect to the X's) with E[N]=10. Using the martingale M_n = sum_{i<=n} X_i - 4n and optional stopping, find E[X_1+...+X_N].
打开 →题目5163 · 金融与交易
Why Bootstrapping Works
Why does curve bootstrapping solve short maturities first and then move outward one maturity at a time?
打开 →题目3686 · 随机过程
Why Bounded Stopping Times Make OST Safer
Why are bounded stopping times the cleanest setting for applying optional stopping in interview problems?
打开 →题目3564 · 随机过程
Why Optional Stopping Needs a Bounded Setup Here
Why are bounded intervals the natural place where optional-stopping arguments for Brownian hitting questions are most reliable?
打开 →题目3689 · 随机过程
Why OST Can Fail for Unbounded Stopping Rules
Why is it dangerous to apply optional stopping mechanically when the stopping rule is unbounded?
打开 →题目3214 · 概率
Centered Slippage Variance Under Random Stopping
Let $X_1,X_2,\dots$ be i.i.d. with mean $\mu$ and variance $4$. Let $N$ be independent of the increments and distributed as Geometric(\frac{1}{3}). Show that for the centered stopped sum $M_N=\sum_{i=1}^N (X_i-\mu)$, one has $E[M_N^2]$ equal to what value?
打开 →题目5931 · 概率
Discounted Offer Stopping
Each period an offer arrives, iid Uniform(0,1). If you accept an offer of value x at period t, you receive beta^{t} * x, where beta = 0.9 is a per-period discount factor (so waiting shrinks the value of any future acceptance). No recall, infinite horizon. Find the optimal station
打开 →题目5923 · 概率
Full-Information Uniform Stopping
You observe up to three independent draws from the Uniform(0,1) distribution, one at a time, and after each you may stop and collect the value just seen or discard it and continue (no recall of discarded values). If you reach the third draw you must take it. Knowing the distribut
打开 →题目5987 · 概率
When the Stopping Rule Looks at the Last Draw
Draw i.i.d. values $X_1,X_2,\dots$ uniform on $\{1,2,3\}$ (so $E[X_i]=2$). Define $N$ as follows: keep drawing and stop the first time you draw a $3$; let $N$ be the number of draws. Let $S_N=\sum_{i=1}^N X_i$. A candidate computes $E[N]E[X_1]=3\cdot 2=6$ and claims $E[S_N]=6$. C
打开 →题目4314 · 机器学习
Before Trusting Early Stopping
Your validation metric is noisy day to day. Before treating the first local peak as the stopping point, what should you calibrate?
打开 →题目2643 · 机器学习
Clipping Plus Weight Decay on a Vector 25
A parameter vector is w_t=(3,4). Its gradient is g=(6,8), whose norm is 10. Apply global-norm clipping with threshold 5, then a decoupled weight-decay step with learning rate eta=0.1 and lambda=0.1. What is the new parameter vector?
打开 →题目2622 · 机器学习
Global-Norm Clipping Formula 2
A gradient vector g has norm ||g|| greater than clip threshold c. Derive the clipped gradient under standard global-norm clipping.
打开 →题目3540 · 随机过程
Reflection Versus Optional Stopping
In one paragraph, contrast what reflection principle and optional stopping are each best at in Brownian barrier problems.
打开 →题目082 · 概率
The Sleeping Beauty Problem
Sleeping Beauty participates in the following experiment. On Sunday she is put to sleep. A fair coin is flipped. If it lands Heads, she is woken on Monday only. If it lands Tails, she is woken on Monday and again on Tuesday (with her memory of the Monday awakening erased before T
打开 →题目2068 · 数学
What a Good Stopping Rule Uses 23
What two diagnostics do desks commonly monitor together before stopping a Newton solve?
打开 →题目2641 · 机器学习
Why Clipping Helps Exploding but Not Vanishing Gradients 23
Why is gradient clipping a natural remedy for exploding gradients but not for vanishing gradients?
打开 →题目2619 · 机器学习
Why Flat Late-Round Validation Gains Still Suggest Stopping 22
If the validation gain per boosting round becomes tiny and erratic late in training, why is that often a strong argument for stopping?
打开 →题目2645 · 机器学习
Why Global-Norm Clipping Preserves Direction 14
Why does global-norm clipping change the magnitude of a gradient vector but not its direction whenever clipping is active?
打开 →题目2662 · 机器学习
Why Overlapping Validation Windows Complicate Score Aggregation
Why should a practitioner be careful when averaging performance over overlapping validation windows?
打开 →题目5903 · 概率
Capping the Single-Bet Drawdown
You bet a fraction $f$ of wealth on an even-money coin with win probability $p=0.65$, but a risk rule forbids any single losing bet from cutting your wealth by more than $20\%$. What fraction should you bet, and for which win probabilities $p$ does this drawdown rule actually con
打开 →题目2458 · 机器学习
Choosing Early Stopping by the Test Curve
A team trains one model, plots test loss by boosting round, and reports the round with the best test value. Why is the final test score no longer a valid final check?
打开 →题目3497 · 随机过程
Covariance of Two Overlapping Increments
What is Cov(W_4 - W_1, W_6 - W_3)?
打开 →题目4378 · 机器学习
Early Stopping Leakage
You can afford only 30 evaluations, and experience suggests that only a few hyperparameters matter strongly while the rest are weak. Would grid search or random search usually deserve the first try?
打开 →题目4385 · 机器学习
Early-Stopping Patience
One CV fold is much smaller than the others and dominates the variance of the average score. What tuning-related design concern should you address first?
打开 →题目1793 · 统计
Elastic Net Grouping
Two features are almost duplicates but both are economically meaningful. Why does Elastic Net often behave better than pure Lasso here?
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