A robot must collect a key before opening a door. Which state representation most directly fixes the missing information if position alone is not Markov?
A. Position only, with a smaller learning rate.
B. Position together with whether the key is held.
In REINFORCE, why can a state-dependent, action-independent baseline be subtracted from the return without changing the expected policy-gradient estimator?
A. The baseline must equal zero.
B. The expected score function at a fixed state is zero.
C. Every sampled gradient becomes zero.
D. The rewards are replaced by the baseline.
下面跟着题目逐步算
正确答案：B
固定状态 s 后，对动作分布求和：Σₐπ(a|s)∇logπ(a|s)=Σₐ∇π(a|s)=∇1=0。基线 b(s) 与动作无关，可以从求和中提出，因此被减掉的那一项期望为零。
A 不要求基线为零；C 说的是每个样本，实际上只有附加基线项的期望为零；D 不是替换环境奖励，而是调整梯度权重。
Two states have similar feature vectors in a linear value approximator. Updating the weights using one state may affect the estimate of the other state. This illustrates:
Explain prediction and control. Describe the two stages of policy iteration, and explain why a greedy policy improvement uses argmax rather than max. [5 points]
下面跟着题目逐步算
题目在问什么
先分清“评价已有做法”和“寻找更好做法”，再说明策略迭代怎样交替完成它们。
五个得分点（各 1 分）
① Prediction 在固定 π 下估计 vπ 或 qπ；它不负责主动换成另一策略。② Control 寻找最优策略或最优价值。
State both GLIE conditions. Is GLIE itself a learning update or an action-selection formula? Discuss fixed ε = 0.1 and explain why merely requiring εₜ → 0 is insufficient. [5 points]
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先说结论
GLIE 是一组关于行为策略序列的长期要求，不是 MC 的别名，也不是独立的 Q 更新公式。
评分与推理
① Infinite exploration（1 分）：在收敛定理所要求的状态动作范围内，各状态动作对被无限次访问。② Greedy in the limit（1 分）：行为策略在极限中把概率集中到当前价值估计的贪心动作上。
③ 分类（1 分）：MC、SARSA 是学习方式；ε-greedy 是选动作的方法；GLIE 是探索与极限行为的性质。
Give the standard DQN and Double DQN nonterminal targets. Explain which network selects and evaluates the next action in each case, and why Double DQN can reduce overestimation. [5 points]
Describe proportional prioritized experience replay. State a priority, a sampling probability and an importance-sampling weight. Explain the roles of the small positive priority offset and the correction exponent β. [5 points]
For an undiscounted episodic task, write a REINFORCE update with a state baseline. Then describe the actor and critic, and give a one-step TD-error signal commonly used by the actor. [5 points]
下面跟着题目逐步算
把两个角色分开
Actor 决定动作概率；critic 评价状态或动作有多好。REINFORCE 用完整回报作反馈，actor–critic 常用 critic 产生的自举信号。
There are nonterminal states A, B and terminal state T. γ=0.5. From A, the immediate reward is always 2; the next state is A or B with probability 0.5 each. From B, the reward is always -1; the next state is A or T with probability 0.5 each. T is absorbing with zero reward.
(a) Write the transition matrix in the order (A,B,T) and the expected reward vector. [2]
(b) For the observed episode A → B → T, compute G₀ and G₁. [2]
(c) Write and solve the Bellman equations for v(A),v(B). [2]
(d) From V₀=(0,0), calculate the synchronous vectors V₁ and V₂. [2]
γ=0.5. In S, action stay gives reward 1 and returns to S; move gives reward 0 and goes to U. In U, cash gives reward 4 and terminates; reset gives reward 0 and goes to S. Policy π selects each available action with probability 0.5.
(a) Write the Bellman expectation equations under π. [2]
(b) Solve for vπ(S) and vπ(U). [2]
(c) Compute qπ for all four state-action pairs, and give a greedy improved policy π′. [2]
(d) Evaluate π′. Is its action at S uniquely greedy with respect to vπ′? Explain. [2]
The grid has two rows and three columns. (1,1) and (2,3) are terminal. Four actions (up, down, left, right) are equiprobable. A move outside the grid leaves the state unchanged. Every move FROM a nonterminal state gives reward -1, including a move entering a terminal. γ=1; terminal values are fixed at zero.
(a) Write the synchronous random-policy update equation. [1]
(b) Starting from V₀=0, give the whole V₁ grid. [2]
(c) Give the whole V₂ grid. [2]
(d) Calculate V₃(1,2) and V₃(2,1). [2]
(e) Using V₂ for successor values, compute all four one-step action values at (1,2) and identify every greedy action. [2]
An episode is A --2→ B --(-1)→ A --4→ T. γ=0.5. For constant-step-size algorithms initialize V(A)=0, V(B)=2 and use α=0.5. Run each named algorithm independently from these initial values.
(a) Compute G₀,G₁,G₂. [2]
(b) Apply first-visit MC with the stated constant α once per visited state. [2]
(c) Using ONLY this episode and fresh zero counts, compute every-visit sample-mean MC estimates. [2]
(d) Apply online TD(0) in chronological order; show the value after each update. [3]
(e) Explain why the TD update at the second transition uses a value that has already changed. [1]
A nonterminal transition has current Q(S,A)=1, reward r=-1, γ=0.8 and learning rate α=0.25. Online next-state values for (a₁,a₂,a₃) are (5,2,0). The actual next action is a₂. The behaviour is ε-greedy with ε=0.3. Target-network next-state values are (1,4,3). Compute each algorithm independently.
(a) Give the behaviour probabilities of a₁,a₂,a₃. [2]
(b) Give the SARSA target and updated Q(S,A). [2]
(c) Give the Q-learning target and updated Q(S,A). [2]
(d) Give the standard DQN and Double DQN targets. Show which next action each selects. [3]
(e) If the transition instead terminates, give its target and tabular update. [1]
下面跟着题目逐步算
通用更新先写在纸上
只有目标 y 随算法改变。不要把 ε 当学习率，也不要先做 SARSA 再把结果送进 Q-learning。
γ=0.5. In X, safe gives reward 2 and goes to T; travel gives reward 0 and goes to Y. In Y, collect gives reward 4 and goes to T; cycle gives reward 1 and goes to X. Start policy iteration with π₀=(travel,cycle).
(a) Evaluate π₀ exactly. [2]
(b) Compute all four qπ₀ values and make a greedy improvement π₁. [3]
(c) Evaluate π₁, verify Bellman optimality, and list ALL deterministic optimal policies. [3]
(d) From V₀=0, give the synchronous value-iteration vectors V₁ and V₂. [2]
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